## The Value of Replication

Daryl Bem is a respected psychology researcher who decided to try his hand at parapsychology. Last year he published a series of studies in which he claimed evidence for precognition — for test subjects being influenced in their choices by future events. The studies were published in a peer-reviewed psychology journal, the Journal of Personality and Social Psychology. This created somewhat of a controversy, and was deemed by some to be a failure of peer-review.

While the study designs were clever (he simply reversed the direction of some standard psychology experiments, putting the influencing factor after the effect it was supposed to have), and the studies looked fine on paper, the research raised many red flags — particularly in Bem’s conclusions.

The episode has created the opportunity to debate some important aspects of the scientific literature. Eric-Jan Wagenmakers and others questioned the p-value approach to statistical analysis, arguing that it tends to over-call a positive result. They argue for a Bayesian analysis, and in their re-analysis of the Bem data they found the evidence for psi to be “weak to non-existent.” This is essentially the same approach to the data that we support as science-based medicine, and the Bem study is a good example of why. If the standard techniques are finding evidence for the impossible, then it is more likely that the techniques are flawed rather than the entire body of physical science is wrong.

Now another debate has been spawned by the same Bem research — that involving the role and value of exact replication. There have already been several attempts to replicate Bem’s research, with negative results: Galak and Nelson, Hadlaczky, and Circee, for example. Others, such as psychologist Richard Wiseman, have also replicated Bem’s research with negative results, but are running into trouble getting their studies published — and this is the crux of the new debate.

According to Wiseman, (as reported by The Psychologist, and discussed by Ben Goldacre) the Journal of Personality and Social Psychology turned down Wiseman’s submission on the grounds that they don’t publish replications, only “theory-advancing research.” In other words — strict replications are not of sufficient scientific value and interest to warrant space in their journal. Meanwhile other journals are reluctant to publish the replication because they feel the study should go in the journal that published the original research, which makes sense.

This episode illustrates potential problems with the scientific literature. We often advocate at SBM that individual studies can never be that reliable — rather, we need to look at the pattern of research in the entire literature. That means, however, understanding how the scientific literature operates and how that may create spurious artifactual patterns.

For example, I recently wrote about the so-called “decline effect” — a tendency for effect sizes to shrink or “decline” as research on a phenomenon progresses. In fact, this was first observed in the psi research, as the effect is very dramatic there — so far, all psi effects have declined to non-existence. The decline effect is likely a result of artifacts in the literature. Journals are more inclined to publish dramatic positive studies (“theory-advancing research”), and are less interested in boring replications, or in initially negative research. A journal is unlikely to put out a press release that says, “We had this idea, and it turned out to be wrong, so never-mind.” Also, as research techniques and questions are honed, research results are likely to become closer to actual effect sizes, which means the effect of researcher bias will be diminished.

If the literature itself is biased toward positive studies, and dramatic studies, then this would further tend to exaggerate apparent phenomena — whether it is the effectiveness of a new drug or the existence of anomalous cognition. If journals are reluctant to publish replications, that might “hide the decline” (to borrow an inflammatory phrase) — meaning that perhaps there is even more of a decline effect if we consider unpublished negative replications. In medicine this would be critical to know — are we basing some treatments on a spurious signal in the noise of research.

There have already been proposals to create a registry of studies, before they are even conducted (specifically for human research), so that the totality of evidence will be transparent and known — not just the headline-grabbing positive studies, or the ones that meet the desires of the researchers or those funding the research. This proposal is primarily to deal with the issue of publication bias — the tendency not to publish negative studies.

Wiseman now makes the same call for a registry of trials before they even begin to avoid the bias of not publishing replications. In fact, he has taken it upon himself to create a registry of attempted replications of Bem’s research.

While this may be a specific fix for replications for Bem’s psi research — the bigger issues remain. Goldacre argues that there are systemic problems with how information filters down to professionals and the public. Reporting is highly biased toward dramatic positive studies, while retractions, corrections, and failed replications are quiet voices lost in the wilderness of information.

Most readers will already understand the critical value of replication to the process of science. Individual studies are plagued by flaws and biases. Most preliminary studies turn out to be wrong in the long run. We can really only arrive at a confident conclusion when a research paradigm produces reliable results in different labs with different researchers. Replication allows for biases and systematic errors to average out. Only if a phenomenon is real should it reliably replicate.

Further — the excuse by journals that they don’t have the space now seems quaint and obsolete, in the age of digital publishing. The scientific publishing industry needs a bit of an overhaul, to fully adapt to the possibilities of the digital age and to use this as an opportunity to fix some endemic problems. For example, journals can publish just abstracts of certain papers with the full articles available only online. Journals can use the extra space made available by online publishing (whether online only or partially in print) to make dedicated room for negative studies and for exact replications (replications that also expand the research are easier to publish). Databases and reviews of such studies can also make it as easy to find and access negative studies and replications as it is the more dramatic studies that tend to grab headlines.

**Conclusion**

The scientific endeavor is now a victim of its own success, in that research is producing a tsunami of information. The modern challenge is to sort through this information in a systematic way so that we can find the real patterns in the evidence and reach reliable conclusions on specific questions. The present system has not fully adapted to this volume of information, and there remain obsolete practices that produce spurious apparent patterns in the research. These fake patterns of evidence tend to be biased toward the false positive — falsely concluding that there is an effect when there really isn’t — or at least in exaggerating effects.

These artifactual problems with the literature as a whole combine with the statistical flaws in relying on the p-value, which tends to over-call positive results as well. This problem can be fixed by moving to a more Bayesian approach (considering prior probability).

All of this is happening at a time when prior probability (scientific plausibility) is being given less attention than it should, in that highly implausible notions are being seriously entertained in the peer-reviewed literature. Bem’s psi research is an excellent example, but we deal with many other examples frequently at SBM, such as homeopathy and acupuncture. Current statistical methods and publication biases are not equipped to deal with the results of research into highly implausible claims. The result is an excess of false-positive studies in the literature — a residue that is then used to justify still more research into highly implausible ideas. These ideas can never quite reach the critical mass of evidence to be generally accepted as real, but they do generate enough noise to confuse the public and regulators, and to create an endless treadmill of still more research.

The bright spot is that highly implausible research has helped to highlight some of these flaws in the literature. Now all we have to do is fix them.

**Posted in:**
Neuroscience/Mental Health

Sorry if this sounds harsh, it is meant to be harsh. What this episode shows is that the journal JPSP is not a serious scientific journal. It is fluff, it is pseudoscience and entertainment, not a journal worth publishing in, and not a journal worth reading, not a journal that has scientific or intellectual integrity.

“Professor Eliot Smith, the editor of JPSP (Attitudes and Social Cognition section) told us that the journal has a long-standing policy of not publishing simple replications. ‘This policy is not new and is not unique to this journal,’ he said. ‘The policy applies whether the replication is successful or unsuccessful; indeed, I have rejected a paper reporting a successful replication of Bem’s work [as well as the negative replication by Ritchie et al].’ Smith added that it would be impractical to suspend the journal’s long-standing policy precisely because of the media attention that Bem’s work had attracted. ‘We would be flooded with such manuscripts and would not have page space for anything else,’ he said.”Scientific journals have an obligation to the scientific community that sends papers to them to publish to be honest and fair brokers of science. Arbitrarily rejecting studies that directly bear on extremely controversial prior work they have published, simply because it is a “replication”, is an abdication of their responsibility to be a fair broker of science and an honest record of the scientific literature. It conveniently lets them publish crap with poor peer review and then never allow the crap work to be responded to.

If the editor consider it impractical to publish any work that is a replication because they would then have no space for anything else, then they are receiving too many manuscripts. If the editor needs to apply a mindless triage of

“no replications”, then the editor is in over his head and is overwhelmed. The journal should either revise the policy and replace the overwhelmed editor, or real scientists should stop considering the journal a suitable place to publish.A not bad other paper about it was

http://www.ncbi.nlm.nih.gov/pubmed/21573926 ,

“A Bayes factor meta-analysis of Bem’s ESP claim”

Psychon Bull Rev. 2011 May 15.

And they say things about p-values accurately, using phrases like “Although such a ratio constitutes evidence for the alternative, it is not as substantial as might be inferred by such a small p-value.” and “as might naïvely be inferred from the p-value”. That is, the problem is what is inferred from the p-value by some people more than the p-value itself. For example if I compare expression of a gene between 10 vs 10 samples and get p=.001, I have some interest, but if it was 300 vs 300, I’m not, cause the difference is almost nothing.

And I can see that paper, but my university doesn’t subscribe me to see JPSP, maybe for reasons given in comment #1.

Example of publication bias driving me nutters lately: the “significant but trivial” problem. Abstracts often emphasize

statistical significancewhile conveniently failing to mention an underwhelmingeffect size. (I described this in a short article, with a little help fromxckdstick people.)A good recent example is a paper that reports a “significant” correlation between foot and shin stress fractures and vertical loading rates in running (velocity of impact, how “jarring” the step is). Their result is statistically significant, but the actual results aren’t summarized! Gee, I wonder why? Just add a dash of

post hoc ergo propter hocfor a hopelessly misleading abstract, dutifully and uncritically reported, even by experts who should know better: “Loading rates cause shin splints!”A close relative of the “significant but trivial” problem is the “statistically significant but not clinically significant” problem. Vitamin B supplements lower blood homocysteine levels by a statistically significant amount, but they don’t decrease the incidence of heart attacks. We must ask if a statistically significant finding actually represents a clinical benefit for patient outcome, if it is POEMS – patient-oriented evidence that matters.

Yep: even a big, real effect is not necessarily meaningful to patients. Or not meaningful for long. There are lots of surgical examples. Big ol’ effect sizes in some back pain surgeries, but cruddy long term outcomes.

Just because you can measure a thing doesn’t make it matter. Tooth fairy science!

There were a lot of “statistically significant” analytical tests for mercury in autism which had zero significance from a clinical standpoint.

When the ranges completely overlap and the distribution is skewed by a few high values, the mean or median doesn’t mean much. Especially when the difference is known to be tiny compared to normal intake.

“If the standard techniques are finding evidence for the impossible, then it is more likely that the techniques are flawed rather than the entire body of physical science is wrong.”

–I don’t have a problem with Bem pulling this off. This does not mean that science does not work, or using p values does not work.

I get much more bent out of shape when I see naive authors report a “p value” of “.0000.”

Bem had an unexpected finding. Same as many an original study. And just the same, if some phenomenon of the universe has been illuminated, replication should, logically, be possible.

That is, if we, as scientists, stay commited to our faith-based belief that the universe is an orderly place, where things happen in a consistent manner suitable to replication studies.

The Bem findings have already been repeated, and those reps have failed to reproduce the result.

Should Bem never have tried?

There are dearly-held hypotheses out there right now, that some would declare are impossible to deny that, with the correct study design and opportunity, will be shown to be wrong.

We should encourage researchers to investigate the “impossible” when warranted to some degree, and to the degree that other relevant questions are not neglected.

Deying the time-space continuum: relativity indicates how this can be done. And has been done. Two objects can get into notably different time frames.

And things have been made “invisible” by strategically taking advantage of what is known of quantum physics and the multiverse.

It is dangerous to declare that some topic cannot be investigated by an empirical study because someone, somewhere, beliaves the underlying hypothesis to be “impossible.”

Bayesian analysis is often very fitting. but if a line of resaerch has been done the wrong way, those “priors” do not fit the new circumstance. Myself, that is my difficulty with Bayesian analysis.

I am not so hot on the agnostic null-hypothesis-testing model either. But a false positive now and then does not make null hypothesis testing irrelevant.

So, let Bem conduct all of the studies he wants. Just so long as he is honest and transparent. We have a built-in “check” for that – it is called “replication.”

So relax.

Reproducing results is difficult under the best of circumstances. http://bioinformatics.mdanderson.org/Supplements/ReproRsch-All/Modified/ENAR/banksNotes.pdf has a nice review of some of the problems that arise. The author points out that few applied papers are perfectly reproducible, and that if enough detail were provided in a submitted paper to lend itself to reproducibility testing, the paper would be too long and tedious to publish. The author points out:

“Only a handful of studies can be deeply checked, and these are usually the ones that make the most spectacular claims. Additional barriers are that such review is slow, the work is not valued unless it discredits important publications, the process of obtaining the original data and code can be onerous, and there is not much joy in calling foul on the work of respected scientists and colleagues. “

“Example of publication bias driving me nutters lately: the “significant but trivial” problem. Abstracts often emphasize statistical significance while conveniently failing to mention an underwhelming effect size. (I described this in a short article, with a little help from xckd stick people.)”

Yes, that is particularly frustrating. I sometimes feel there is an unholy alliance, though, between scientists reporting their findings and science journalists. When reading articles in the NY Times, I regularly see the announcement that the finding was statistically significant, with no mention of effect size. I’m not sure how this happens (do science journalists not know that they should report this?) but very often I look at the paper itself and am underwhelmed.

MedsVsTherapy, stating that Bem “had an unexpected finding” seems to presuppose the validity of his conclusions. Perhaps if he’d done his job with a less glaring enthusiasm for what he hoped to find, his findings wouldn’t have been quite so “unexpected.”

I think wiseman should not be pitching his study as a replication, but rather an improvement over data collection methods, since he used the internet to recruit labs.

Some journals are worse than others at seeking attention-grabbing damn-the-truth results. It’s one reason why most people in astrophysics don’t take

Natureas seriously as astrophysics-only journals.Parapsychology! So we are back to the Rhine studies at Duke’s. Five ESP symbols randomly mixed in a fixed set of 20 items containing four occurrences of each symbol. What is the likelihood of person A (the psychic) transferring their thoughts to person B located within the environs of person A? The answer – highly!

We are high-order devious creatures with deviousness (not to be confused with deviation – standard or otherwise) built into our genetics so coughs, sneezes, silence, blinking and a myriad of verbal and non-verbal communication can take place to attain a desired outcome. The misdirection (for that is what it is) is the keenness of the evaluator’s to embroil them selves in statistics – so getting in their own road by supporting their own flawed conclusions.

But then there is probability analysis (you can show your p-values the door). Here in Britain we have a fixed-set lottery comprising 49 consecutively numbered balls; from which 6 balls are ‘randomly’ drawn. What is the likelihood of two consecutive numbered balls occurring in the 6 drawn from the set? The answer is – highly!

Over here we have a greed-driven show called ‘Deal or no deal’. In it, a contestant randomly selects and retains a box from a fixed set of 16 boxes. Each box contains a monetary ‘prize’ ranging from a penny (dime) to £250,000 (around $300,000 as the crow flies).

There are ‘rounds’ whereby the contestant then randomly ‘eliminates’ boxes one by one – which are then opened to reveal the prize contained within. After 3 ‘selections’ are eliminated, the ‘Banker’ offers a sum of money for the contestant’s box and the MC asks one question ‘deal or no deal’?

The contestant can either take the money offered – or – seeing bigger prizes in boxes not yet revealed; in the hope that their box contains a ‘Biggy’ – a word here which means a worthwhile sum of money; they continue ‘rounds’ to the end game where only TWO boxes remain – theirs and the other remaining box still ‘in play’.

When the end game is reached and the money offered for the box is declined in favour of the much larger sum ‘in play’, invariably the contestant is given the option to ‘swap’ their box with the remaining box. And so: – a three part question. A) Given that there are TWO boxes remaining, what are the odds of the better prize being in the contestant’s box? And B) – is it in more favourable for the contestant to SWAP their box for the one remaining – and if so – why?

These are NOT trick questions – but – ‘are the answers likely to surprise many of you?’ The answer to that question is – highly!

Ready?

Then please read on.

To A: ‘Given that there are two boxes remaining, what are the odds of the better prize being in the contestant’s box?’ The answer is NOT 50-50! Intuitive maybe – but reality is different, and unless you are applying Fisher or Taguchi – stats sucks!

The initial selection was from SIXTEEN boxes – i.e. a FIFTEEN to ONE chance of THAT box NOT containing the ‘Biggy’.

So given that, is it in more favourable for the contestant to swap their box for the one remaining? The answer is YES!

This is a little more tricky to comprehend but can be proved with 3 playing cards – a King (representing the Biggy) and 2 low-value ‘spot’ cards (e,g, a 2-spot and a 3-spot), and a little judicious ‘cheating’ to illustrate principle.

PREPARATION

Turn the King face-down on the table – a term here which means hide the value and reveal the pattern on the back of the card. Now surreptitiously ‘mark’ the card by placing a little pencil dot in one corner so that when all of the cards are face-down, YOU know which one is the KING.

PRESENTATION

All three cards ‘in play’ represent the ‘boxes’. Show the 3 cards – King and two ‘spots’. Turn the three cards face down and thoroughly mix them up. Lay them out on the table and then invite the contestant to select ANY face-down card (but not reveal it). They have a three-in-one chance of selecting ‘the Biggy’ (the King).

Now: (this bit is important) turn over a SPOT card. At this juncture, there may be two in play – there is a high probability that there will only be one because the contestant has selected the other – but – because you know which is the King – irrespective of who has it – you can easily select a spot card (the value is not important) for your first ‘reveal’.

What the ‘contestant’ sees is a ‘spot’ card, so what they DEDUCE (correctly) is that the King is still in play.

Now offer the ‘swap’.

The chances of their SELECTED card NOT being the Biggy is TWO chances from three – or 66% AGAINST. This means, that the chances of the Biggy still being in play is ALSO 66% – this time – in favour of the Banker. Or to put it another way – SWAPPING gives you a TWO to ONE advantage of capturing the Biggy.

Go on – surprise yourself – try it!

Oh – nearly forgot: what I have just illustrated is Bayer’s theorem!

Have fun.

“if standard techniques are finding evidence for the impossible . . .”

Well, it means either standard techniques are flawed, or that the impossible is plausible, no? Not that I support the general idea of psychics, but as skeptics, it is pretty hard to sell it both ways, right? Either the techniques of science work or they don’t. I don’t buy the results, but this is hard for me to sell.

Also, in terms of effect size, let’s remember the results on aspirin and heart attacks – a darned small effect size, if i remember correctly, but not one that we should ignore, so sometimes small effect sizes can be meaningful. We can’t just ignore findings with small effect sizes out of hand.

I’m afraid the Wiseman paper is actually a Stuart Ritchie / Richard Wiseman / Chris French study; it’d be nice to give full credit to everyone involved.

I’m pretty sure Ritchie is the lead author of the study, as well.

OK, I would like to throw out a provocative notion, so I accept that the replies may be equally strong – and I welcome those.

I work in nutritional products research. Most of my compounds or products come to me with a considerable history of use, which mainly means that I have far more baseline data to draw upon than a new chemical entity (drug). If I establish efficacy and general mechanism in preclinical studies, I will do a pilot clinical study to establish effect size. Should this succeed, I will do an adequately powered, well-controlled RCT to test for efficacy. If the clinical demonstrates efficacy, I will approve my company using the results to support a label claim.

This is not the drug model, but drugs do not have history of use. In Steven’s example, a study unsupported by the weight of evidence should not be taken en face without replication. I think that is a good point. My work generally has a weight of preclinical and pilot study evidence behind it, but the turning point study is typically one study. Am I violating the admonition of replication? Is it relevant under these circumstances?

My deeper thoughts on nutritional product studies are on my blog (not shameless self promotion, more letting people know where my more in-detail thoughts can be found) – http://www.nutrition-industry.blogspot.com.

@Tell it like it is,

Given the rules you describe for Deal or No Deal, there is no benefit to switching if you manage to get down to 2 boxes.

In the playing card example, switching is indeed better, but the two situations are not comparable. With the cards, the dealer selects which card to show, and always picks a spot card. The dealer knows which card is the King, and uses that information in deciding which card to reveal. Thus, the dealer’s choice is not random, and that affects the odds of switching.

In the game show, the contestant picks which boxes to reveal, without knowing where the best prize is located. Thus, his choice is effectively random. That makes all the difference.

Try your playing card example again, except follow the Deal or No Deal rules – where you do all the selections with no knowledge of where the King is located. You’ll find the advantage to switching has disappeared.

qetzal is correct. In DOND, all choices are purely random and there is no forcing. Hence it is indeed the case that each of the N remaining values has a 1/N chance of being in each individual box.

The classic example of forcing making it beneficial to switch is Let’s Make a Deal (aka the Monty Hall problem). Three doors, one with a good prize. You pick one at random. The host opens one of the other two to reveal a bad prize, and you can choose to keep your original door or switch to the other remaining door. In this case you’ve forced the host’s choice if you happened to pick a bad prize, and therefore you have a 2/3 shot of getting the good prize by switching.

But the crucial element of forcing is not applicable to DOND. This happens to be why I despise the show; there are about two meaningful binary decisions per hour (credible choices to take or reject the deal) and everything else is meaningless fluff with no impact whatsoever on the final result.

Ed Whitney: thanks so much for that link. I follow that subject carefully, but had not seen that document.

In DOND – in the American version, the host does not know what dollar value is in which case. So the choices are random, and offers are statistical, and no new information if offered.

In some of the European versions, the host does know which cases contain which dollar amounts (or at least where the biggie is), and the offers the host makes gives the player information.

This is not exactly the Monty Hall problem, but it is somewhere in between.

I did not say the Bem or anyone else should not research psi – he can research whatever he wants.

However – I am free to criticize his methods. He is on record as advocating methods that are hugely biased toward generating a false positive. So it’s no surprise that he generated some false positives here.

He also published results which are highly unlikely, meaning highly likely to be spurious or artifactual – without doing sufficient follow up to see if they should be taken seriously. In other words – he was insufficiently skeptical of his own claims.

I can also criticize the journal for published exploratory results like Bems, without adequate review, and then to fail to publish negative replications.

But also keep in mind – the article is about what we should do systemically within the science publishing industry to better represent the evidence. The Bem episode is just an example of how the current system failed, or at least could have worked much better.

Steve,

As you’ve said before, it’s not the answers that matter as much as it is the process.

Doing psi research? Ok, fine. To even say it produces an answer that’s probably wrong, how do we actually know that much? Process.

Here we see both the breakdown of process twice – in the initial publication and when replication is discouraged.

The first is a problem we should be happy to face, as we get to discuss statistical methodology, experimental methodology, etc.

The second is a problem that is chilling to face, to consider that all the things just discussed are effectively irrelevant because any replication experiment that seeks to better control these factors is allegedly doomed to never be published.

Dr. Novella (rightfully) criticizes JPSP for their unwillingness to print attempted replications of a study they published. I have to admit, however, that I’m having a hard time determining good criteria for deciding when replications should or should not be published. Even in the digital age there are practical limitations to what a journal can review and publish.

It would probably be useful if for each article a journal published it maintained an electronic database (obviating the need for using actual journal space) tracking all attempted replications.

I’ve recently abandoned clinical medicine and have been looking for a new project to start. Since I don’t think it’s likely that journals will start to maintain a database akin to the one above any thoughts from the community about the possible merits of an eJournal dedicated to study replication?

Just wondering what y’all think.

@ quetzal, Scott and Steven Novella: First – a big thank you for reading and consolidating my blog on probabilities, the foolhardiness associated with solely relying on statistics, and the need to replicate tests. As discussed in the forum, what is coming out is that statistics needs a complete revamp if it is to be of any value and not cast doubt or aspertions.

Certainly the approach used by Fisher and Taguchi pave a very good way forward for clarity, robustness, and swiftness. Both men use replication – and while Taguchi uses Fisher as his primary driver, he goes one step further than Fisher in that he also introduces ‘noise’ factors’ into the replications to gain a deeper insight into what is going on and WHY.

The clever bit is that the Fisher/Taguchi test arrays use a LOG 2 factorial subset to prove ALL permutations. Simply put, if there are 16 factors being evaluated and each factor has two levels (dose A, dose B – dose – no dose …) then we would require 2 to the power of 16 permutations (65,536) repeated twice to validate a 16 factor test. A lot of time-consuming work.

Using the Taguchi arrays and associated maths, the test would require just SIXTEEN replicated tests. And the result indicates the ‘best’ permutation – even if THAT permutation was NOT evaluated. Replicating the indicated test would permit swift validation. If you want to try it out then invest in a copy of ‘Minitab’ (around $60).

If you want to see the potency then watch the very moving movie ‘Lorenzo’s oil’ starring Susan Sarandon and Nick Nolte.

Now down to business. You are all correct in that on the DOND Show, we are informed that the MC has no prior knowledge of where any of the prizes are – so cannot directly affect the outcome using ‘prior knowledge’ in the way I advocated.

The reason I advocated the ‘prior knowledge’ of where the King lay was to illustrate the PRINCIPLE of what is actually taking place when a pre-selection is made from a set containing more than two items.

In the 16 box set used in DOND the chance of picking the biggy is 15 to one AGAINST. That is to say, you only have a 16 to one chance of picking the biggy so there are fifteen ‘non-chances’ stacked against you. As each outside box drops out, even if the prize isn’t a ‘worthwhile sum of money’, the chance of the box still in play contains the bigger value is greatly increased – so it is in the contestant’s best interest to swap.

Should you wish to ‘play out’ a larger version a few times then include ALL of the cards in an entire suit – i.e. from 1 through to 13 – where the Knave has a value of 11, the Queen a value of 12, and the King a value of 13; play it in a just manner (i.e well shuffled set and. no prior knowledge of who has what); lay out the cards in a row; offer the ‘draw’; disregard all cards in play except for one you like the look of; reveal the contestants card and the one still in play, and observe the ‘end game’. You will discover that there IS benefit to switching when two cards remain – this is what Beyer’s theorem proves.

With your reference to ‘Monty Hall’ Scott and Steven, I presume this was the name of an MC who played the game as I advocated – a term here meaning he had prior knowledge as to where the biggie was – and he used access to boxes or rooms via ‘doors’ to reveal what was won or lost. Same meat – different gravy. If it was 16 rooms then you would need a bigger studio. We had a similar one over here (which I detested). The contestants were steel workers, miners, and car workers who worked in atrocious conditions, were short-lived, and needed a break – and for the most part, all they got was ‘look what you could have won’ as the contestant’s faces became crest-fallen staring at a prize of a caravan as the doors were opened. SICK!

And for completeness. To use Beyer to maximise your OWN advantage, select a horse in horse race with more than two horses running and back (a term which here means put a monetary bet on) that horse to LOSE!

Although the odds offered will be the REVERSE of the odds offered for it to WIN, you have effectively backed ALL of the other horses to WIN.

Although you will not win as much as if you had backed that same horse to WIN and it ‘came in’ (unless it was an ‘odds-on’ ‘favourite’ – in which case, if the horse lost you would stand to make more), you will make a tidy profit – the return on your investment will be greater than leaving your money in a bank for a year.

The risk of you LOSING your bet is exactly as you stated Scott – the reciprocal of the number of horses in the race. It’s your call! Good luck guys.

@ SloFox While on paper your proposal appears sound, I believe that a journal to study replication would be unwieldy, difficult to maintain, contentious, and not offer much in the way of return for all of the investment time and effort.

Replications MUST be carried out by the person(s) confirming their findings for any study to have real any merit and be taken seriously, otherwise, if the test cannot be replicated and the same results derived, how can we possibly endorse it?

Listen to ‘Thin ice’ from Pink Floyd’s album ‘The Wall’. Come to that – set aside an hour and listen to the entire album. Not recommended when driving on the freeway.

Regards

@ TILIS — Thanks for the honest response. I’m sure such an effort would be difficult but I’m not sure that it wouldn’t be worth the effort. I’d essentially be creating a data repository for studies that couldn’t be published elsewhere because of their ‘lack of novelty.’

I think it’s less likely that all studies can be replicated in whole or in part by the original authors (despite the fact that it should nevertheless be encouraged). This is particularly difficult with large randomized trials since it’s often difficult enough to obtain enough funding and recruit enough subjects in the first place.

There are a lot of studies, however, that at least partially replicate previous research efforts and not all are published. For example, a group of researchers may publish a study comparing Drug A to placebo. Another group may then conduct a trial comparing Drug B to Drug A (and possibly also to placebo). If Drug B is equivalent to Drug A the study may be difficult to publish in many journals but not in mine since that’s exactly my target study. More importantly, it may be that in the second study both Drugs A and B were less or more effective than in the original study. I think this would be important data to make public.

Lastly, whenever new methodology is used (usually in more basic science research) other research groups will often try to adopt the new methodology into their ongoing research. I’ve had to do this myself. Attempts to do so are not always successful but often chalked up to technical problems. Maybe, however, the issue is with the data from the group pioneering the new methodology (e.g. Bem’s psi work). This is another target group for me.

Ultimately the incentive for the authors is to have something to show for the effort they put into the negative results.

Maybe this isn’t the best idea but I’m going to continue to push it a while longer. More critiques are encouraged.

NB–I try to dedicate at least one day a month for listening to Floyd alone. I’ve been listening to the Wall for about 30 years now and enjoy it best in a comfy chair or sofa with no distractions.

Many cheers.

@ SloFox Thank you for taking the time and patience to reply.

Whilst it may be so that the cost of carrying out some studies can be prohibitive and therefore inhibit carrying out replication of same, and what you propose is very laudable, I would urge you to consider your proposition in ALL lights by performing a thorough ‘Risk analysis’ – not just on the tangibles – but also on the intangibles. Begin with a SWOT analysis – a term here which means evaluate the Strengths; Weaknesses; Opportunities; and Threats.

Carrying out a thorough Threat Analysis will prevent you from setting yourself up for a fall – an expression which here means be personally and publicly ridiculed and possibly sued for publishing a white paper (which by default means that you accept what is being presented) that later turns out to be a load of hogwash – or worse – harmful.

Your RA should focus on due care and attention and therefore to reduce the threat of exposure you MUST consider:

- preventative measures

- control measures

- Due diligence (to yourself)

- Litigation

And you should also take ‘precautions’ a word which here means sought advice and acted upon it.

One last thing, prevention is better than cure – but – you must also have ‘recovery systems’ in place for when things go wrong – which is highly likely.

Momma loves her baby

And daddy loves you too.

And the sea may look warm to you babe

And the sky may look blue

But ooooh Baby

Ooooh baby blue

Oooooh babe.

If you should go skating

On the thin ice of modern life

Dragging behind you the silent reproach

Of a million tear-stained eyes

Don’t be surprised when a crack in the ice

Appears under your feet.

You slip out of your depth and out of your mind

With your fear flowing out behind you

As you claw the thin ice.

(Roger Waters)

And on ego: Gather ye rosebuds while ye may; Old Time is a-flying: And this same flower that smiles to-day; To-morrow will be dying. (Robert Herrick)

I wish you the very best in your quest for enlightenment

@Tell it like it is

Thanks for the advice. Right now, this is just an idea. I hope I can turn it into a concept in the upcoming months and a brain child by the end of the year. I’ll certainly keep your guidance in mind if I ever produce a business plan.

@SloFox You are very welcome. If you are compiling a business plan then you may also wish to include an Impact Analysis – a term here which means define the internal and external factors that will impact to your business and how you plan to deal with them.

Strategy should drive the entire pitch to outshine your competitors. There are numerous strategies. What you require is a ‘Win Strategy’ (AKA ‘Capture Strategy’) on which you base your entire argument to increase your likelihood of raising capital and capturing business by attending to all details so as to maximise the odds in your favour.

My approach to carrying out and effective IA is to use the acronym MORTAR AND PESTLE as your driver.

The MORTAR, a term which here means Management Of Revenue, Time and Resources, evaluates the INTERNAL ‘impacts’ associated with running your business. Think of it like this: Does the mortar hold the bricks together – or keep the bricks apart?

The PESTLE is a utensil that evaluates the EXTERNAL factors that, like a pestle, will pound (AKA ‘impact’) your business.

What are these external influences, and what are their ‘impacts’?

These are contained in the term PESTLE – a term which here means consider the following influences and impacts on your business:

Political

Environmental (how green are you?)

Social (include competitors, contributors, and customers)

Technological (talking to Smart-phones, etc.)

Legislation – here is OK – but what about elsewhere? What laws affect my operations? Is appropriate and effective due diligence in place?

Economics (is it viable? is it doable? is funding available? what is my break-even point? and when do I achieve it? what are the monetary impacts of failure? how do I recover? etc.)

You are obviously committed to your quest and I trust this helps you to oil the wheels.

@TILIS:

No, you won’t. As long as all selections are random, there won’t be any benefit to switching.

Try it for yourself. (I have.)

@TILIS You’re completely ignoring all of the times that the contestants reveal the biggy during the game before they get to the two final boxes.

To run the experiment accurately (even with just three cards) you’d need to *randomly* get rid of all but one of the non-selected cards. Tally the number of times that the biggy is thrown out, the number of times that the biggy is the final non-selected card and the number of times that the biggy is the selected card and I’m sure you’ll find that the biggy ends up being the selected card just as often as it ends up being the remaining non-selected card.

In other words you’re right that the vast majority of times you won’t select the biggy but the vast majority of times the biggy will also be thrown out.

@ quetzal, thejmii et al: Thank you all for your interest and your replies.

First – an apology – for a misspelling. I have cited ‘Beyer’s theorem’ – I actually meant BAYES’ theorem – my typo – my bad.

Before I provide an answer to your obvious impasse, please permit me an ‘aside’ that I trust will bring some awareness and comprehension prior to your enlightenment.

In 1742 the Reverend Thomas Bayes was elected a Fellow of the esteemed scientific body of its day – the Royal Society. In 1763, Bayes had a paper posthumously accepted for publication by the Royal Society entitled ‘An essay towards solving a problem in the doctrine of chances’.

Just like filing a patent, given what had happened to Karl (Carl?) Scheele; who discovered chlorine 36 years before Humphry Davy and was never credited – and who then went on to discover oxygen in 1772 – 2 years prior to Joseph Priestley – who gets the credit; to bring rigour and thoroughness to the validation process, there is an inevitable publication delay whilst checks are made to see if the work is original ‘beyond all reasonable doubt’, and has not been plagiarised – a term which here means ‘bootlegged from someone else before THEIR work is granted publication’ (which in Bayes case it hadn’t – the work was (is) original).

Interestingly, based on the premise that there are ‘universal principles’ out there and if these principles could be identified and codified, they could be taught to people to make the process of invention and discovery more rapid and predictable, in 1946 Genrich Altshuller and his colleagues in the former USSR developed a technique called TRIZ.

TRIZ involves searching through ALL of the pending patents of the entire WORLD (which have to be made public prior to the granting of a patent to draw out other contenders) and using the knowledge gained to greatly accelerate development.

By applying TRIZ the Russians were the first nation to put a person in space (Yuri Gagarin) and the first nation to build and commission the (now) International Space Station – so it is no ‘accident’ that the Russian Space Shuttle looks remarkably like the American version – and pretty much works the same way – so much for filing a patent!

So what? I hear you say. So what indeed – for thanks to the physicist R.T. Cox, the sleeping text of Bayes is now the REVELUTION in the sciences: forming the backbone and driving force of scientists in EVERY discipline who are realising that science itself is the application of Bayes’ Theorem – and not the seriously flawed statistical models in current vogue – as originally proposed by Gauss.

Bayes’ Theorem shows that any system of inference that fits certain requirements can be mapped onto ‘probability’. It describes what makes an entity ‘evidence’ when one performs an experiment; but it goes much further: it also describes the ‘strength’ of the evidence, and so Bayes’ Theorem not only tells us ‘what’ to revise, but by ‘how much’.

Statistical models fade away into insignificance when the Bayesian method is applied. This is because, unlike statistics, the Bayesian method accurately defines the evidence as always being the result of the ‘differential’ between two ‘conditional’ probabilities – in other words – the ‘odds’ are based upon ‘likelihood’ measured against priory data – a term which here means ‘what we already know’.

The reason statistics fails time after time after time – with the inherent delays, prohibitive costs, and serious HARM that this causes, is because the entire system is based upon SUBJECTIVE ASSUMPTION. As a consequence, those trained in classical statistics begin with a delusion and continue to delude themselves into believing that ‘factor X’ indicates that a condition holds true purely based upon how well the ‘X evidence’ appears to ‘validate’ the sought condition. Such a determination is dubious!

To illustrate: A common statistical misconception is the ‘Null Hypothesis’ test – the comparing of ‘means’ (AKA averages) of various ‘data sets’ to see if they are the same.

This approach is seriously flawed because you may conclude that if the means are the same then there is no difference – when really – there is; or – you may conclude that there IS a difference – when really (you’ve guessed it) – there isn’t.

The impact is that you will treat solution options as identical – even though they aren’t; or, you will reach FALSE conclusions. In both cases you will implement FLAWED solutions.

It goes without saying that some form of ‘fog’ index is needed to hide this joke from the layman. This is provided in the ‘confidence level’. This is akin to a dealer in used cars offering you a ‘deal’ at one price (the high confidence value) – and then – when you refuse the deal – with a wink and a cough, dropping the price by as much as 70% (the lower confidence value). Would YOU ‘trust’ this salesperson to sell you a second-hand automobile? What is the ‘real’ price? How ‘confident’ are you in the dealer’s statement?

Unlike statistics, Bayes ‘strong evidence’ is NOT based on an assumption that there a very high probability that A leads to X (associative), but that the evidence is based upon a very ‘low’ probability that NOT-A could have led to X (disassociatative). In other words, ‘A’ cannot be both ‘A’ and ‘NOT-A’ so if you attempt to present both ‘A’ and ‘NOT-A’ as ‘bearing out’ your theory then the Bayesian rules state that you have a paradox (see null hypothesis above). To increase the probability of a theory being likely requires you to carry out tests that can potentially ‘decrease’ its probability. Falsification is much stronger than confirmation.

Confused? Then let’s work a simple example that is relevant and fitting for this forum.

Let’s begin by saying that due to most of us having a poorly grounded set of ‘mental’ rules, everything about Bayesian reasoning is very counterintuitive.

Now at this point I could say “Stop reading this and go and bury your head in a copy of ‘Language Truth and Logic’ by A W Ayres and get to grips with ‘logical positivism’.”

But changing your mindset would be classed as immoral and unreasonable because you would view the world through different eyes and would see that such things as ‘Buy one – get one free’ really means ‘buy TWO’, and that there is no such thing as ‘half price’ because a price is merely an ‘offer to buy’ and has no corrolation whatsoever to what the previous price was (see ‘car salesperson’ allegory).

So rather than say the grammar might be impecable; the rhetoric might be forceful; but if the LOGIC is wrong then ITS WRONG – PERIOD – which would appear conceited; at this juncture, all I will say is that Bayes’ theorem is valid in ALL common interpretations of probability and is more easy to comprehend when expressed in terms of ‘odds’ – hence my earlier suggestion to back a horse to LOSE – its counter intutive – but it pays DIVIDENDS.

KEY IDEA

The key idea is that the PROBABILITY of an event A given an event B (for example, the probability that one has breast cancer given that one has tested positive in a mammogram) depends not only on the relationship between events A and B (that is, the accuracy of mammogram screening) but ALSO on the ‘marginal probability’ of occurrence of each event (a person actually having cancer).

Having read that, you might think that you have just joined Alice in Wonderland. Well – you HAVE! Just as Alice understood that ‘Jam yesterday – jam tomorrow – but never jam today’ informs us that we have a TENDENCY to look back at the sweet things, and look forward to the niceties coming up – but the day-to-day REALITY is very different, we must turn to the very valuable lesson Alice learned from the ‘The mad Hatter’ – I quote: ‘Why you might as well say “I eat what I like is the same thing as I like what I eat” … or you might as well say “I breathe when I sleep is the same as I sleep when I breathe”.

The probability that a woman with a positive mammography has breast cancer is not at all the same thing as ‘the probability that a woman with breast cancer has a positive mammography’; they are as unlike as Alice and the mad Hatter.

Finding the SOUGHT answer, ‘the probability that a woman with a positive mammography has breast cancer’, uses all THREE pieces of information – the PRIOR probability that a woman has breast cancer, the PROBABILITY that a woman with breast cancer gets a positive mammography, and the PROBABILITY that a woman who does NOT have breast cancer ALSO gets a positive mammography.

And now – laydees and gentlefolk – give it up and show your love for – ‘BAYERS THEREOM’

Ta dah!

The PRIORY

100 out of 10,000 women at age forty who participate in routine screening have breast cancer.

80 out of every 100 women WITH breast cancer will get a positive mammography.

Out of the 9,900 women who do NOT have breast cancer, 950 will ALSO get a positive mammography.

If 10,000 women in this age group undergo a routine screening, what proportion of women with positive mammographies will actually have breast cancer?

Are you sitting comfortably? Then let us begin.

10,000 women participated in routine screening for cancer so the TOTAL POPULATION is 10,000.

From the same 10,000 women:

80 out of every 100 women WITH breast cancer will get a positive mammography.

9,900 will NOT have breast cancer and of those 9,900 women, 950 will ALSO get positive mammographies.

This makes the total number of women with positive mammographies 950 + 80 = 1,030.

Of those 1,030 women with positive mammographies, 80 will have cancer. Expressed as a PROPORTION, this is 80/1,030 = 0.07767 = 7.8% – or approximately one in thirteen.

This is the answer a doctor should give to a POSITIVE-MAMMOGRAPHY patient if she asks what are the CHANCES she has breast cancer. If thirteen patients ask this question, it is likely that only ONE person out of those 13 will have cancer. In other words – the odds are TWELVE to ONE that the patient testing positive does NOT have cancer!

What did YOU determine it to be?

The vast majority of doctors in these studies (75% – or THREE out of FOUR*) believed that if around 80% of women with breast cancer have positive mammographies, then the probability of a women with a positive mammography having breast cancer must be around 80% – WAY off beam. * See Casscells, Schoenberger, and Grayboys 1978; Eddy 1982; Gigerenzer and Hoffrage 1995.

Determining the CORRECT answer always requires all THREE pieces of information – the percentage of women WITH breast cancer, the percentage of women WITHOUT breast cancer who receive ALSO false positives (i.e. the not-cancer), and the percentage of women WITH breast cancer who receive positives that are sadly true.

Back to DOND. You are a ‘doubting Thomas’. Try this:

1 Remove the ACE from your suit (you will see why in a mo).

2 Thoroughly shuffle the suit of cards.

3 With the cards face-down, deal off the cards in six PAIRS, where the first card dealt represents the contestant’s choice, and the second the remaining ‘end-game’ ‘box’.

4 Now – reveal each pair.

5 For every occurrence, examine the number of times the contestant ‘wins’ and the number of times the contestant ‘loses’.

6 Tally up the ‘wins’ and ‘loses’.

7 Subtract the smaller number from the higher value (they cancel out) and log the OUTCOME to THAT round – the contestant wins – or – the contestant loses.

8 Repeat the sequence from step 2 ONE HUNDRED times.

You now have a data sample consisting of 100 rounds – some of which the contestant won, the rest in which the contestant lost.

Tally up the ‘wins’ and ‘loses’.

Subtract the smaller number value from the higher value (they cancel out) and observe what remains (wins or losses).

The number that remains will predict the LIKLIHOOD of winning or losing as a PERCENTAGE. For example if 3 ‘losses’ remain then you have a 3% probability of LOSING – and therefore – a 97% chance of NOT-losing – i.e. WINNING.

Divide this value into 100 and you have the ODDS of winning (or losing).

Irrespective of what prizes remain, should you SWAP THE BOX?

@TILIS If you ended up with three more losses than wins (would it even be possible to end up with an odd number?) how in the world would that make winning more likely than losing (especially to the degree you suggest)?

After 600 games (100 round of 6 games) if the difference between the wins & losses is 3 that means there were 298.5 wins and 301.5 losses. If we were to assign a probability of winning and losing based on purely the results then this would mean the probablity of losing would be 50.25% and the probability of winning would be 49.75%. Of course we can work out the probability based upon all possible occurences and not just from actual results and the probability would actually be 1/2 for winning or losing.

In DOND the probability of selecting the biggy is 1/22 (it seems there’s 22 boxes). When the biggy is selected the probability of it being one of the last two boxes in play is 1. Thefore the total probability of ‘the biggy being selected and being one of the last two boxes remaining’ is 1/22*1=1/22.

The probability of not selecting the biggy is 21/22. When the biggy is not selected the probability of it being one of the last two boxes in play is 1/21. Therefore the total probability of ‘the biggy not being selected and being on of the last two boxes remaining’ is 21/22*1/21=1/22.

The probability for both scenarios is the same. If you find yourself in a position where the biggy is still in play you have no more reason to assume that it is because you lucked out at the beginning than you do to assume that you lucked out in your subsequent selections.

I doubt you’ll agree with any of that and if you don’t then could you answer the following question:

For every twenty-two times that the biggy is one of the final boxes how many times do you expect that the biggy will be the box the player did NOT initially select (e.g. out of twenty-two occurences how many times would it have been beneficial to swap)?

If you stil disagree with what I have said I’d really like to know your answer to this question.

@TILIS,

I’m familiar with the statistics of diagnostic testing. I agree it can be counterintuitive, and I’m aware of the data suggesting many physicians don’t adequately understand it. None of that helps resolve our disagreement.

Your suggested card game is much more complicated than necessary. Let’s just stick with a scaled down version of DOND.

1. Take an Ace (= biggie) and two deuces. Mix them up well and deal them face down so you have no idea which one is the ace.

2. Pick one card to be “yours.”

3. Pick either of the remaining cards and expose it. If it’s the Ace, of course you’ve lost.

4. Assuming you didn’t expose the ace above, reveal the remaining two cards. If the card you originally picked is the ace, score it as a win for “STAY” (i.e. staying with your original choice). If the other card is the ace, score it as a win for “SWITCH.”

5. Repeat steps 1-4 above enough times, and you will find that STAY and SWITCH win equally often. However, after only 100 trials, an apparent odds ration of up to 1.44:1 (i.e. 59 SWITCH : 41 STAY or vice versa) wouldn’t be statistically signficant. If instead you run 500 trials, any outcome over 1.19:1 (i.e. 272 SWITCH : 228 STAY) becomes significant.

Now let’s do the Monty Hall version:

1. Have someone else mix up the cards and arrange them face down so that he knows which is the ace, but you do not.

2. You pick one card to be yours.

3. He picks one of the remaining cards

that he knows to be a dueceand reveals it. Obviously he can always do this.4. Reveal the remaining two cards and score as STAY or SWITCH like before.

5. Repeat many times, and now you will find that SWITCH wins twice as often (on average) as STAY. In this case, 100 trials will likely be enough to statistically reject the null hypothesis of 1:1 odds.

As I said before,

I have done thisand confirmed those two results empirically. (Although I did it using Excel and random number generation, not with physical cards.)Thank you one and all for taking the time and trouble to follow my thread, read my humble contributions, and pass your comments. For clarification, I am not attempting to answer ‘what is the likelihood of the ‘biggy’ being in the end game’ because you can compute that for yourselves with relative ease. Throughout I have posed the question ‘Is it beneficial to SWAP the boxes when only two remain?’

To address the question posed by thejmii (thanks) “If you ended up with three more losses than wins (would it even be possible to end up with an odd number?) how in the world would that make winning more likely than losing (especially to the degree you suggest)?”

You are not ending up with 3 more losses than wins – or three MORE of anything. The three remaining are from a TOTAL data set of 100 items and represent PERCENTILES – but it is encouraging to see that you have noticed that using PARITY as the differentiator to eliminate redundant data on a data set containing an EVEN number of data items, you will end up with either ZERO ‘differential’ (i.e. the odds are 50:50 – no better than tossing a coin) OR a ‘remainder’ that is ‘even’. The converse is true for a data set containing an ODD number of data items.

Remaining with the ‘three loss’ scenario; doing the maths: for EVERY game we have a 3% ‘differential’ – call it an ‘edge’ if you wish. This translates to 51.5 PERCENT LIKLIHOOD of a ‘win’ and a 48.5% chance of achieving a ‘not-win’ (i.e. ‘lose’). This gives 51.5 – 48.5 = 3% chance in FAVOUR of WINNING.

There may be a DESIRE to pursue EVERY permutation of any two from 13 in the 13-card-set (N!/(N-n)! X 2 = 312) – where big ‘N’= the total number of items in the set (13) and little ‘n’ is the number of items selected (2), and ‘!’ is the mathematical symbol for ‘factorial’.

In story-form this would read as: I drew the King – you drew the Ace; I drew the King – you drew the deuce; I drew the King – you drew the trey; all the way through to – I drew the Ace – You drew the Knave; I drew the Ace you drew the Queen … – HOWEVER – what I promote, illustrated, and recommended was a ‘six-pair’ exercise per round,

This was to demonstrate the notion of FRACTIONAL FACTORIALS. Basically getting rid of REDUNDANCY.

Now I realise that some of you may now be screaming “Why has TILIS DOUBLED the ‘any two from 13’ result – that can’t be right?”

If this is your retort then you are again about to join Alice in Wonderland. Those who refute Bayes, and stick solely with classical statistics are siding with ‘The King of Hearts’. He is the guy Alice met at the ‘Trial’ of the Knave of Hearts (who allegedly stole some tarts). No matter how much evidence the King is shown to the contrary (the tarts were not stolen in the first place – and were openly displayed in the courtroom as ‘exhibit A’) – the King is NOT for budging – a word here which means he is so full of his own self-importance and harebrained beliefs that he is RIGHT in ALL things and so takes pleasure in, and is quick to shoot down ANY party who DARES to have the audacity to challenge him. No doubt YOU know someone like that? A politician perhaps?

Off the record, the King and Queen are Sophists and their purpose is to teach Alice (Liddell) how to spot absurdities in people’s thinking. Sophists are a devious bunch of people who play a mindless game called ‘one-upmanship’ by using fallacious and salacious reasoning in an attempt to outwit their opponents. As they lack the intelligence to discuss the merits of a philosophical argument, their techniques include questioning the motive of the person making the argument; attempting to deflate the opposition by exaggerating the propositions stated; misrepresenting the other person’s words; and attacking a straw man instead. The most captivating heroine in all of literature Alice – who does not need a ‘fairy Godmother’ – deflates the King and Queen and then defeats them all with the classic line – “You are nothing but a pack of cards!”

Back to the plot. Yes there are just 156 permutations of any two from thirteen – BUT – for every pair drawn there are TWO conclusions because the card drawn FIRST affects the OUTCOME in TWO ways – not ONE!

Let’s assume that I am the contestant and, going first, I draw a King (Hearts perhaps?) and an arbitrary value is then drawn from the data set by you.

If I draw the KING then I ‘WIN’ – PERIOD – making ALL other ‘your card’ options REDUNDANT because they are – and can therefore be disregarded (12 winning chances – yaaay).

If YOU draw the KING then I ‘LOSE’ – PERIOD – making ALL other ‘my card’ options REDUNDANT (booooo!).

Expanding, we have:

If I draw the QUEEN (value 12) then the permutations where I WIN = ALL ‘your card’ options that are BELOW mine – Ace through Knave (11 winning chances – yaaay)

Similarly, if I draw the EIGHT then the permutations where I WIN is ‘you drew the ACE’ through to ‘you drew the SEVEN’ (7 chances) – making all other options REDUNDANT. Hang on – the lower the VALUE on my card – the MORE LIKELY it will be a ‘not-WIN’ card and YOU will draw the ‘WIN’. Oh dear!

If I draw the ACE I lose – pure and simple – I am in the ‘Penny club’.

What is determined is that there are 312 ‘possibilities’ – and only 78 ways of winning. Therefore the CHANCE of winning = 78/312 = 0.25 = 25% = ONE winning chance in FOUR – or – THREE winning chances in FOUR – IF I SWAP!

To your question thejmii (thank you): “For every twenty-two times that the biggy is one of the final boxes how many times do you expect that the biggy will be the box the player did NOT initially select (e.g. out of twenty-two occurrences how many times would it have been beneficial to swap)?”

DOND has TWENTY TWO boxes in its arsenal so to determine the odds of having a winning box at the end, we must compute:

The total number of permutations (N!/(N-n)! X 2) = 924

The total number of ‘winning’ combinations = 231

With only 231 ways of winning from 924 possibilities, the CHANCE of winning = 231/924 = 0.2738 = 25% = one CHANCE of winning in FOUR – or – THREE CHANCES of winning in FOUR – IF I SWAP!

If you doubt this then watch 100 different episodes of DOND, cancel out the ‘WINS’ and ‘not-WINS’ (loses) and observe the result for yourself. On the way, apply Bayes’ Theorem to determine at what point YOU would DEAL! To remind you – there will be THREE ‘pops’ at the ‘targets’ remaining.

At any one time, you can determine what the total is in the ‘pot’, you can see how many items remain, you know the highest and lowest values in the pot, and you can determine the likelihood of any value being zapped – so if you have REALLY grasped Bayes you should be able to accurately calculate the ‘Bankers’ ‘offer’.

If you feel so inclined, why not purchase the board game (complete with talking telephone) – it pulls all members of the family together and is much more entertaining than ‘Call of Duty’ where you become extremely insular and very aggressive defending a hole that – quite frankly – isn’t worth the bother to protect, and consumes a HUGE amount of your waking time on the planet simply watching flashing imagery and listening to loud clatter as you develop extreme dexterity in operating knobs on your controller.

On Fractional Factorials

The notion of ‘fractional factorials’ – a term which here means ‘only evaluating what is relevant’, is KEY to SWIFTLY obtaining a ‘break-through’ – a term which here means throwing out the redundant data and getting the win.

With reference to your comment qetzal: “after only 100 trials, an apparent odds ration of up to 1.44:1 (i.e. 59 SWITCH : 41 STAY or vice versa) wouldn’t be statistically signficant. If instead you run 500 trials, any outcome over 1.19:1 (i.e. 272 SWITCH : 228 STAY) becomes significant.” May I say that I am not talking about statistics in the same way as you are so I am not seeking ‘statistical significance’ or any other ‘rule’. I am talking about the application of Bayes principles to determine the LIKELIHOOD that there is a correlation between factor X and factor B – and by HOW MUCH. This a total paradigm-shift from statistical practices in vogue at present.

You will observe that current mammogram tests DO use large data samples, but by crunching to just the data of interest, having proven Bayes in the ‘real’ world, as counter-intuitive as it may seem, the same result would be revealed with a much smaller data-set.

My experience with the Excel ‘random number’ generator is that it is a sinusoidal function that does not generate sufficient ‘randomness’, so has limited application.

Try this:

- Declare an array of six items numbered 1 through 6, and assign some arbitrary information to the six data fields (example items in the refrigerator)

Then:

- Create (say) half a dozen cells that generate a random integer between 1 and 6.

Next:

Use this integer in the ‘lookup’ function to randomly select a data item from your ‘data base’.

Finally, press F9 to refresh the program and watch the generator in action.

You saw (and I trust understood) in my previous blog how a huge data sample of 10,000 women was crunched down to just 1,030 with positive mammography’s to arrive at the proportion that will (sadly) have cancer. What this, and the examples above demonstrate, is that the amount of REDUNDANT data in any data set is VAST. The larger the data set – the more junk.

It is no ‘accident’ that pigs are bred in Yorkshire, lambs in Wales, and potatoes in Lincolnshire; and that Kent is the ‘garden of England’. Following massive food shortages after the First World War, it was Fisher who determined and proved the notion of ‘fractional factorials’ and his approach enabled the nation to grow food in the most economical way possible, and thus, prevent famine.

This notion of ‘fractional factorials’ has been further expanded by Genichi Taguchi – himself swiftly giving his country solutions to economic success following the dropping of two nuclear bombs on Japan to finally put an end to World War 2 – a war – like World War One – that was gaining nothing and putting humankind in jeopardy.

Application of these concepts have brought terrific success to Toyota, Sony, Seiko, Toshiba, Zanusi, and many other Japanese ‘household name’ companies – who use Bayes to make huge technological leaps forward and steal a march on the West by swiftly providing superior products at lower cost. Apple inc now use this approach and their products have all but destroyed the Nokia Corporation and severely dented Sony-Erickson.

Huge amounts of time, money, and effort is spent by Western corporations embroiled in ‘Six Sigma’ initiatives that return peanuts! Oh look – our $200,000 year-long Six-sigma initiative saved Ford $1.2 million. Ford’s annual turnover is in excess of $700 BILLION – not exactly a good return on investment methinks.

People wallowing in statistics spend the majority of their short time on our amazing planet floundering in massive amounts of data THAT IS NOT FAULTY and therefore what they play with is irrelevant and it tells them next to NOTHING.

It is the OUTLIERS that point to the problems – not the data that is ‘within range’ that, like the ‘win’ ‘not-win’ data I described how to eliminate earlier, can be cancelled out – and yet – it is the OUTLIERS – a word here which means ‘outside of defined boundaries’, which statististitions make EVERY attempt to IGNORE – why is this? Answers on a postcard to: Hugo Mad, Suite 666, Sucker Corporation, Crazy Street, Dupe Town. Or call toll-free on STATSSUCKS.

Well that’s me done here.

Thanks for the fun good people.

See you on the thin ice!

THE THIN ICE

Momma loves her baby

And daddy loves you too.

And the sea may look warm to you babe

And the sky may look blue

But ooooh Baby

Ooooh baby blue

Oooooh babe…

If you should go skating

On the thin ice of modern life

Dragging behind you the silent reproach

Of a million tear-stained eyes

Don’t be surprised when a crack in the ice

Appears under your feet.

You slip out of your depth and out of your mind

With your fear flowing out behind you

As you claw the thin ice.

(Roger Waters – The Wall)

Have FUN and live your dream! You are a long time DEAD.

TILIS the next time.

“The number that remains will predict the LIKLIHOOD of winning or losing as a PERCENTAGE. For example if 3 ‘losses’ remain then you have a 3% probability of LOSING – and therefore – a 97% chance of NOT-losing – i.e. WINNING.”

“Remaining with the ‘three loss’ scenario; doing the maths: for EVERY game we have a 3% ‘differential’ – call it an ‘edge’ if you wish. This translates to 51.5 PERCENT LIKLIHOOD of a ‘win’ and a 48.5% chance of achieving a ‘not-win’ (i.e. ‘lose’). This gives 51.5 – 48.5 = 3% chance in FAVOUR of WINNING.”

—

“For clarification, I am not attempting to answer ‘what is the likelihood of the ‘biggy’ being in the end game’ because you can compute that for yourselves with relative ease. Throughout I have posed the question ‘Is it beneficial to SWAP the boxes when only two remain?’”

Thankyou for the clarification, based on your initial card experiment where you removed all but the biggy and one spot card I had it in my head that you were trying to suggest that when you are down to two boxes and one of them definitely is the biggy then it would be better to swap – you clarified this in your next post with a different set up of the card experiment and I guess I wasn’t paying attention.

Having said that if we look at the broader question as to whether it is beneficial to swap when you are down to the final two cases I still disagree with you: whatever two cases remain in play have an equal chance of being in either case. Rather than keeping this in the ether lets go to the data.

Your predication is “the CHANCE of winning = 231/924 = 0.2738 = 25% = one CHANCE of winning in FOUR – or – THREE CHANCES of winning in FOUR – IF I SWAP!”, my prediction is approximately 50% either way.

Now to the data: ilovedealornodeal.co.uk has information on all of the shows from season 2 onwards. Looking at the first 100 episodes we find that in the final two boxes the amount in the player’s box is higher 51 times and the amount in the other box is higher 49 times (http://goo.gl/mE02J).

There is no benefit to swapping the boxes when only two remain.

Completely false. You’ve done the math wrong. N!/(N-n)!n! are *combinations*, where ordering does not matter. N!/(N-n)! is the correct number of *permutations*, where ordering does matter.

What you’ve done is calculate the number of permutations, assume it’s the number of combinations, and then multiply by the ratio of the number of permutations to combinations to get permutations. But you already HAD permutations.

There are 156 possible outcomes to the game. In 78 of them you win. 50% odds. That really is all there is to it. Claiming that there are 312 is simply wrong. Seriously, if you think there are 312 then write them all down. You’ll run out at 156.

Seriously, go take an intro-level probability course. You need it.

Oh, forgot to note…

In

N!/(N-n!)n!

the n! is in the denominator, not the numerator. You’ve put it in the numerator, getting more combinations than permutations when it’s the other way around.

@TILIS:

It’s not perfectly random, but it’s adequate for these purposes. (Note that I’m using a fully updated Excel 2003, not one of the earlier versions with less robust RAND functions.) When I use it to model 1000 trials of the three-card version of DOND, I get 343 wins for STAY versus 318 for SWITCH. (In the remaining 339 cases, the first exposed card was the biggie.) That gives us a ratio of 1.08:1 in favor of STAY, which is statistically indistinguishable from the predicted 1:1 (p=0.93 by chi squared test).

In contrast, when I model 1000 trials of the Monty Hall version, I get 334 wins for STAY versus 666 wins for SWITCH. (The first exposed card in Monty Hall is never the biggie, of course.) That’s 1.99:1 in favor of SWITCH, indistinguishable from the predicted 2:1 (p=0.96), but clearly not compatible with 1:1 odds (p less than 0.001)

I can refresh the model with F9 and get different exact numbers, but they stay in the same ranges – about 1:1 for DOND versus 2:1 for Monty Hall.

So, are you implying that Excel’s RAND function is so biased that it’s somehow wrongly making STAY seem equivalent to SWITCH for DOND, even though it gets the correct 2:1 odds for Monty Hall? If I thought you were open to persuasion, I’d repeat the modeling using a better source of random numbers. However, you seem unwilling to actually look at the evidence or consider that you might be wrong, so I don’t see any point. If I’m wrong on that, then I urge you to do your own modelling, with whatever source of random numbers you think is acceptable. If you model things correctly, I’ll bet you a “biggie” that you’ll find there’s no benefit to switching.

@Scott et al Whilst I stand corrected on the fact that ALL permutations – including ‘ordering’ are included in the formula N!/(N-n)!, and therefore acknowledge I have made a blooper, I do not RESPECT you. I do NOT recall in any of my posts where I requested insults.

May I suggest that there is a high probability that you are a sophist who is solely out to ‘prove a point’ – even if it means retorting to demoralisation. To reiterate what I stated in a previous blog, sophists are a devious bunch of people who play a mindless game called ‘one-upmanship’ as they lack the intelligence (and integrity) to discuss the merits of a philosophical argument in a rational and civilised manner. The majority end up lonely!

@ jimii Thanks for pointing out my failure to fully expand the statement relating to “3% probability of LOSING – and therefore – a 97% chance of NOT-losing – i.e. WINNING” vs “This translates to 51.5 PERCENT LIKLIHOOD of a ‘win’ and a 48.5% chance of achieving a ‘not-win’ (i.e. ‘lose’). This gives 51.5 – 48.5 = 3% chance in FAVOUR of WINNING.” where you allude that I appear to have contradicted myself.

At the time of writing, I was well ‘in the zone’, and as with any ‘live’ system, trying to ‘proof-read as you go’ is not easy. Let me clarify. Imagine if you will a line on an ‘X’ (or ‘Y’) axis. Let us call this a ‘probability’ line. At one extreme we have ‘Low probability’ (indefinite), at the other end – ‘High probability’ (Almost certain).

Let us now place our marker bang in the middle of this line – the 50:50 position.

Now let us re-examine the ‘3% differential’ scenario. This translates to 51.5 percent likelihood of a ‘win’ and a 48.5% chance of achieving a ‘not-win’ (i.e. ‘lose’).

What we have done is moved our marker closer to the ‘high probability’ end of the graph.

This does two things a) it increases the likelihood that a win will ensue; and b) it decreases the likelihood that a loss will ensue.

The ‘shift’ under discussion gives 51.5 – 48.5 = 3% differential and therefore if we subtract our 3% from 100% we have: 100 – 3 = a 97% chance IN FAVOUR of WINNING – the more often we play, the more times we will win. Is that better put?

If we consider a piece of buttered toast and we hold it at arms length by one corner and let it fall to the ground, what are the chances it will land ‘butter-side down’?

Because the butter adds more weight to one side of the bread than the other, there will be a tendency to encourage the ‘weighted’ bread to fall ‘butter-side down’ – not always – but – assume it is a 3% differential – this ‘differential’ gives a 97% chance IN FAVOUR of WINNING. Try it.

To see this ‘differential’ in action, observe the effect the ‘zero’ (or TWO ‘zero’s in some states) has on a roulette table. In EVERY game the ‘zero’ changes the odds to ‘underpay’ ALL winners, and when the ball lands on it, another win ensues – a percentile of ALL stake money is removed! This is ‘dammed if you do – dammed if you don’t’.

On whether it is beneficial to swap when you are down to the final two cases, in the light of my error and the evidence you very kindly have taken the trouble to evaluate and present you may well be correct. By the way, thank you for the link – my niece can’t get enough of it.

At the top of this blog I highlighted a moral issue when replying to people to whom you have no direct affiliation with. Returning to the mammogram scenario, I pose the question “If we (humankind) are not yet in a position to reduce the likelihood of cancer, is there any benefit to improving the mammogram test?”

Before I give you my viewpoint, please permit me an ‘aside’ that I trust you will find worthy of note.

Imagine if you will that a certain substance abuse test is 99% sensitive and 99% specific.

That is to say, the test will correctly identify a substance abuser as testing positive 99% of the time, and will correctly identify a not-substance abuser as testing negative 99% of the time.

The PRIORY

5 out of 1,000 people at a university use banned substances

All persons are tested for substance abuse

The testing technique is 99% sensitive

The testing technique is 99% specific

Given a positive substance abuse test, what is the likelihood of misclassifying not-substance abusers as substance abusers?

We begin with how many substance abusers will test positive.

Since the test is 99% sensitive, this is 5 X 0.99 = 4.99 (essentially ALL of them).

Out of the remaining 995 people, how many who do NOT abuse substances will test ‘false-positive’?

Since the test is 99% specific we determine those that will NOT test ‘false-positive’ as 995 X 0.99 = 985, which means there will be 995 – 985 = 10 not-substance abusers who WILL test positive.

Totalling up we have 5 substance abusers and 10 not-substance abusers = 15 ‘positives’.

Given the data from the substance abuse test, what is the likelihood of misclassifying a not-substance abuser as a substance abuser? 10/15 = a whopping 66% or 2 out of 3.

It is likely that only ONE person out of THREE will be indulging in the self-harm under evaluation.

Be honest to yourself. What did YOU determine it to be?

What this illustrates is that on paper the specificity and sensitivity of the test would seem to indicate a high detection rate. What the application of Bayer reveals is that the exact OPPOSITE is true.

This, and the mammogram example also establish that the RARER the condition for which we are testing, the greater the percentage of positive tests that will be false positives.

So back to the moral dilemma. You may recall approximately one in thirteen women with positive mammography’s will have cancer.

If we (humankind) are not yet in a position to reduce the likelihood of cancer, is there any benefit to improving the mammogram test?

My take is YES because although the morality of improving screening reduces the initial HOPE (only one in 13 who test positive actually have it – so maybe I am clear?) – it does NOT reduce CANCER – so it is best to know ‘as soon as’ and NOT be given false hope.

In medieval times there were four ruling classes. These classes are depicted in the Tarot and modern playing cards as follows:

1. The Church (Crown) (The Chalice or Holy Grail is Cups in the Tarot and the suit of Hearts (everlasting and unconditional love) in a pack of cards)

2. The Military to protect the church (represented by Swords in the Tarot and the suit of Clubs in a pack of cards)

3. The Merchants (represented by Pentacles (coins – depicting prosperity) in the Tarot and the suit of Diamonds (worldly goods) in a pack of cards)

4. The Farmers (represented by the Staff (workforce) in the Tarot and the suit of Spades (tilling the earth) in a pack of cards)

This was known as the feudal system. As time passed, the Church eventually gave way to the Monarchy and the feudal system became capitalism.

Given the unwarranted rebuffs from Scott et al, you will have determined my reason for reference to ‘The Wall’. I trust it is not immoral to take you …..

Outside the wall

All alone, or in two’s,

The ones who really love you

Walk up and down outside the wall.

Some hand in hand

And some gathered together in bands.

The bleeding hearts and artists

Make their stand.

And when they’ve given you their all

Some stagger and fall, after all it’s not easy

Banging your heart against some mad bugger’s wall.

Isn’t this where….

(Waters – ‘The Wall’)

PS: qetzal: Where do I send the King (or Ace – your choice)? Would you prefer: Cups, Hearts, Swords, Clubs, Pentacles, Diamonds, Staffs, or Spades?

So explaining how you’re wrong, suggesting how you can obtain accurate information, and doing that in a perfectly civil tone is somehow unfounded, insulting, and sophistry?

Do you at least admit that you were in fact completely wrong and it makes no difference whether you switch or not?

@TILIS:

No need. I’m glad to see that I was wrong and that you will, in fact, acknowledge possible error here. I will happily take that as my ‘winnings’ and I apologize for impugning your character.

@Scott Hi again Scott

Whilst I accept “Completely false. You’ve done the math wrong” as a constructive and welcome criticism, I frown upon your remark “Seriously, go take an intro-level probability course. You need it.”

Such a remark is unwarranted, discourteous, and offensive; and does not add anything to the DISCUSSION.

Let me focus you back ‘on-topic’

Eric-Jan Wagenmakers and others questioned the p-value approach to statistical analysis, arguing that it tends to over-call a positive result and suggested BAYES as the right way forward – PERIOD!

The article prompted us to discuss the merits of REPLICATION; evaluate BAYES as an alternative strategy to STATISTICS; to present a little intellectual by-play along the way; to stimulate debate; and in so doing, provide enlightenment and FELLOWSHIP. THAT is what I have done!

Having just read your response, which has absolutely nothing to with any aspect of the topic in hand, you seem to have an eccentric desire to continue to drive home the point that I misinterpreted a formula, had the error pointed out, empirically confirmed it for myself by working through a smaller data-set in longhand to see exactly what was happening (factorials are like Topsy – and I doubted my spreadsheet), satisfied myself why I was incorrect in doubling the result, and publicly admitted the error – what has this got to do with the TOPIC?

To answer your curt “Do you at least admit that you were in fact completely wrong and it makes NO difference whether you switch or not?” I say this Scott – as our eChums have established USING BAYES PROBABILITY MODELS – it DOES make a difference whether you switch or not IN CERTAIN CIRCUMSTANCES!

ATTESTATION

At the penultimate part of the ‘end game’ we see the values in all THREE boxes – so – like ‘Monty Hall’ we have some ‘prior knowledge’.

Now, given ONLY the scenarios where we see a ‘biggy’ and two insignificant values remaining AND we see ONE of the insignificant values tumble – leaving a biggy and an insignificant value ‘in play’ (i.e. we are only selecting the games that replicate the SAME scenario as ‘Monty Hall’), should we swap the box?

By the way Scott – this is all meant to be FUN – LIGHTEN UP MATE!

@ Qetzal You have humbled me sir. No apology necessary because no crime has been committed – merely misunderstandings that is all (we are two GREAT nations divided by a common tongue). When Alice meets the Duchess she is advised ‘take care of the ‘sense’ and the sounds will take care of themselves.’

I have enjoyed the dialogue immensely.

On ‘buttered toast’ – felines are one of the few creatures that exploit the ‘displaced weight’ to advantage. Should they fall they almost INSTANTLY place all four appendages straight out in front of their stomachs (as when they walk). This small weight differential causes them to land on their FEET 99.99% of the time.

Olympic divers are trained to thrust their arms UPWARDS in relation to their bodies to ensure they enter the water head-first. Its only a small differential – but it has a HUGE impact on winning.

Now – how do flies land on ceilings? Do they fly up and then ‘flip’?

How do they get off? Do they free-fall and then ‘flip’?

Is it true that a butterfly is actually ANOTHER CREATURE that uses the caterpillar as a ‘host’ to feed off while it grows?

And finally – when we parse a mathematical equation we are taught the ‘rules of precedence’ and we remember them using the acronym BODMAS – Brackets; Of; Divide; Multiply; Add; and Subtract.

When parsing a Bayes expression, things become like the White Queen in Wonderland who lives backwards and remembers forwards, for the ‘order of implication’ changes depending on which way you PARSE the expression.

If we consider the relationship between Red box l Biggie, if you parse the expression from left to right as one would read English, then the ‘l’ symbol means the probability that the box is painted red, GIVEN that the box contains a ‘Biggie’.

Moving your eyes from right to left as one would when reading Hebrew or Arabic, then the ‘l’ symbol means the probability that a box containing a ‘Biggie’ IS painted red.

It’s all good stuff.

Our existence, like Alice’s is marked with ceaseless change, by an ever-growing radical altering of the self. As we plummet from one part of OUR Wonderland into another, we know we will never be the same.

I believe it was G K Chesterton who wrote: ‘We have always kept this sharp distinction between the science of mental relations, in which there really are laws, and the science of physical facts, in which there are no laws – only weird repetitions.

There’s a nice knock-down argument for you.

“I shouldn’t know you if we DID meet” said Humpty Dumpty “you are so exactly like other people.”

Have fun.

TILIS

@TILIS

No. Perhaps you’re thinking of this paper, which claims that the caterpillar-to-butterfly metamorphosis is evidence of some kind of weird merger of an ancient worm with an insect. It’s a nonsense paper that should never have been published in the peer reveiwed scientific literature. (See here for a discussion of the flawed process that allowed it to be published.)

Also, your ideas on displaced weight are, as best I’m aware, incorrect. Cats land on their feet by twisting their bodies in mid-air. See this Wikipedia article for a discussion and references. Similarly with divers. It’s not about weight displacement per se; it’s about using body movements to control angular momentum.

Think about it this way. A cat’s feet and legs represent only a small fraction of it’s mass. The majority of the mass is in the head and body. So if it was just a matter of mass distribution, cats should still land body-first in the vast majority of falls.

On cats and caterpillars – now there is a knock-down argument for you.

OK everyone, lets get down to business.

Who agrees with Eric-Jan Wagenmakers who suggests that STATISTICS has little merit and BAYES is the right way forward?

Before you vote, please indulge me as I put before you the case as I see it.

Let’s begin by saying that although everything about Bayesian reasoning is very counterintuitive, Bayes is now the REVOLUTION in the sciences: forming the backbone and driving force of scientists in EVERY discipline who are realising that science itself is the application of ‘Bayes Theorem’ – and not the seriously flawed statistical models in current vogue.

KEY IDEA

The key idea is that the PROBABILITY of an event A given an event B depends not only on the relationship between events A and B but ALSO on the ‘marginal probability’ of occurrence of each event to determine the ‘strength’ of the evidence, and so ‘Bayes Theorem’ not only tells us the LIKELIHOOD that there is a correlation between factor A and factor B and by HOW MUCH – it also tells us ‘what’ to revise, and by ‘how much’. This a total paradigm-shift from statistical practices in vogue at present.

Because statisticians start off with the flawed premise of ASSUMING most variables in life are INDEPENDENT, I explained that the reason statistics fails time after time after time – with the inherent delays, prohibitive costs, and serious HARM that this causes, is because the entire system is based upon SUBJECTIVE ASSUMPTION.

NONE of the correspondents disagreed. How could they – they KNOW that when you make a cup of coffee the TASTE is affected by the coffee bean, the minerals in the water, the temperature of the water, the water to coffee-granule ratio, the water to milk ratio, the amount of sugar/chocolate added, etc.

I showed how the concept of the ‘Null Hypothesis’ and other subjective assessments such as ‘Pair-wise comparison’, and the ‘Chi-square’ test, is seriously flawed and the impact is that you will treat solution options as identical – even though they aren’t; or, you will reach FALSE conclusions – or worse – you will miss a trick. In all cases you will implement FLAWED solutions – which is the point Eric-Jan Wagenmakers makes!

I then pointed out, and one of the correspondents also picked this up, how ‘fog indexes’ and ‘statistical validation rules’ (such as the ethereal ‘p-value’ rule) are embroiled in the system to make it ‘work’ (?).

As I demonstrated using real-world examples, because ‘Bayes Theorem’ shows that any system of inference that fits certain requirements can be mapped onto ‘probability’ and describes what makes an entity ‘evidence’, and what is the STRENGTH of that evidence, BAYES does not need, or require, or use such ‘frigs’.

I explained that unlike statistics, Bayes ‘strong evidence’ is NOT based on an assumption that there a very high probability that A leads to X (associative), but that the evidence is based upon a very ‘low’ probability that NOT-A could have led to X (disassociatative). Falsification is much stronger than confirmation.

BAYES requires just THREE pieces of information – the PRIORY and the two ‘conditionals’. An example: Fire requires Fuel (the priory), heat, and oxygen. Because they are all INTERELATED, if you MODIFY any one then the BEHAVIOUR of the fire changes. REMOVE ANY one and the fire goes OUT.

If you think of this like three sides that form a triangle, the triangle doesn’t change shape if you increase or decrease the size of the triangle: ergo – the probabilities don’t change if you change the SAMPLE-SIZE. All you MUST ‘ENSURE’ is that there is at least ONE occurrence of the THREE FACTORS in your data sample – otherwise – just like the fire …

On sample-size – if the priory says that there is just ONE occurrence of (say) cancer in 1,000 in a particular demograph, then you will require 1,000 data samples from THAT demograph. If the occurrence is one in 10,000 – well – form a TEAM to gather the ten thousand data items required.

One of the BAYES drivers is the notion of only evaluating what is ‘relevant’ because this is KEY to SWIFTLY obtaining a ‘break-through’ – a term which here means throwing out the redundant data and getting the win.

I demonstrated the notion of FRACTIONAL FACTORIALS. Basically how BAYES is used to get rid of REDUNDANCY – both in terms of ‘crunching’ to remove redundant data within the data-set, and ‘capturing’ to reduce the need for large data samples.

I explained how the Russian’s applied BAYES to OBJECTIVELY benefit mankind by quickly searching through ALL of the pending patents of the entire WORLD and extracting the PERTINANT knowledge to greatly accelerate development – including being the first nation to put a person in space and the first nation to build and commission the (now) International Space Station.

I then went onto explain that it is no ‘accident’ that the Russian Space Shuttle looks remarkably like the American version – and pretty much works the same way – and so does their ‘Concordski’ – which is still flying over China at TWICE the speed of SOUND whilst ours was de-commissioned following ONE incident (which wasn’t caused by any fault in the aircraft) – how many Boeings have fallen out of the sky?

I then went on to explain how application of BAYERS brought terrific success to many Japanese ‘household name’ companies – who provide superior products at lower cost, and how Apple inc, using this approach, have all but destroyed the Nokia Corporation and severely dented Sony-Erickson.

I then illustrated how huge amounts of time and effort, and VAST amounts of money that could be put to better use, is wasted by Western corporations embroiled in ‘Six Sigma’ initiatives that return peanuts because the statisticians spend the majority of their time floundering in IRRELEVANT data tells them next to NOTHING.

All he way through I showed that it is the OUTLIERS that point to the problems – the very thing that statististitions make EVERY attempt to IGNORE. I was even brave enough to say STATISTICS SUCKS.

Throughout ALL of this discussion, not ONE correspondent has come back showing PROOF that STATISTICS is not only quicker than BAYES – it is MORE ACCURATE.

Statistical models fade away into insignificance when the Bayesian method is applied. This is because, unlike statistics, BAYES WORKS! This is NOT fallacy – this is REALITY – and the odds are heavily stacked in favour of using BAYES, based upon ‘what we already know’.

And so good people – the poll.

All of those who accept that BAYES is wickedly potent – a term here that means SUPERBLY EFFECTIVE, and are willing to take the time and trouble to give it a go alongside your statistical evaluations please post “I HAVE REMOVED A BRICK FROM MY WALL!”

All of those who believe otherwise, please post the following:

Another Brick in the Wall Part 2

We don’t need no education

We dont need no thought control

No dark sarcasm in the classroom

Teachers leave them kids alone

Hey! Teachers! Leave them kids alone!

All in all it’s just another brick in the wall.

All in all you’re just another brick in the wall.

“Wrong, Do it again!”

“If you don’t eat yer meat, you can’t have any pudding. How can you

have any pudding if you don’t eat yer meat?”

“You! Yes, you behind the bikesheds, stand still laddy!”

I don’t need no arms around me

And I don’t need no drugs to calm me.

I have seen the writing on the wall.

Don’t think I need anything at all.

No! Don’t think I’ll need anything at all.

All in all it was all just bricks in the wall.

All in all you were all just bricks in the wall.

(Waters – ‘The Wall’)

See you on the thin ice.

TILIS

And finally …

Einstein worked in the PATENT office and saw Plancks papers and much besides.

It was his WIFE who crunched the numbers – using ……

@TILII

I am not sure this is the right audience to parse Bayesian statistics – although I have been a proponent of their use in clinical trials because it fits the “learning while doing” paradigm so effective in many other areas of public health.

Bayesian and “conventional” statistics both rely on assumptions to make the math work. If assumptions of correlation, normality, independence, etc. are violated, the statistics become less reliable but not unreliable.

One must weigh how improbable the outcome appears to be, how the biology fits the result, and how much the researcher had to torture the statistics to get a significant results. On balance, you decide if the result has crossed over from reliable to unreliable. The findings are weighed by the statistics, but biology isn’t math.

@JPZ Thank you for your support. I agree wholeheartedly with what you say – in all aspects – including casting pearls before swine – and add: wouldn’t it be wonderful if biology could be cracked solely through the application of maths?

Because Bayes Theorem measures correlation, (but sadly not causation – the ‘why?’), even in the strange wonderland world of biology, if the data is available, applying Bayes Theorem is a potent tool to quickly establish the most likely outcome to an event, and in my own experience it beats statistical analysis hands-down and can be swiftly applied to any situation; and what is wonderful is the same algebraic expression is used to make inferences of every kind. It doesn’t get better than that.

On cracking biological turmoil, watch the very moving film ‘Lorenzo’s Oil’.

It would seem that in our curiosity to see Alice through the looking glass and what she found there, we have accompanied her. I say this because it seems we are likely to be holding the White King’s pencil; for in the story the White King complains “My dear! I really MUST get a thinner pencil. I can’t manage this one a bit; it writes all manner of things I don’t intend.”

While these Bloggs are fun and we may be fortunate enough to find an eCompanion, we all know that, for the most part, what we write will not change anything – except – perhaps – our mood and the mood and MIND of someone else (isn’t that why we do it?)

Having lost both of my parents, I am an orphan. I am not here to change the world, and I am not here to leave any lasting legacy – I can’t change those facts. Al I can do is be thankful for the existence I have been granted, have fun, and perhaps be fortunate enough to remain healthy, give what I can, fall in love, pro-create, and pass on my genes.

Warmest regards

TILIS

PA: For those of you who have not read Lewis Carroll’s masterwork, and are curious to know what Alice found, she found a book that is the entire crux of the many many lessons in logic and human behaviour contained within the story.

I cannot say if the ‘steam’ is worth a thousand pounds (£1,000) a puff – but maybe the words are worth a thousand pounds a puff after all? Who knows? (see chapter 3 ‘Looking glass insects’)

@ JPZ Grief is a weird thing – it’s like poison – you can’t know what it is like until you experience it, and when you do – it’s too late – they’ve gone FOREVER. But grief is also a sly old fox, ever playing ‘hide and seek’, and has a nasty habit of popping in when you least expect it. Last night I had another visit from my mum. This is what she had to say.

“Use your literary gift and write a muse on the opening lines of ‘Return of the Native’ by Thomas Hardy” So I HAVE – and I trust it genuinely contains some merit.

As I approached the lab observing that a bleak rainy Saturday afternoon in a bitter cold June was approaching the time of the cheerless twilight, and the vast tract of rain-drenched unenclosed sodden wild known as Egdon Heath embrowned itself moment by moment as the vile wind howled, and the vicious arctic rain lashed down in torrents against the backdrop of a dark, thunderous, lightening streaked sky, drenching everything around it until it was completely soaked through, contemplating the summer we are all enjoying as I rubbed the steamed-up window of my car with my coat sleeve and stared across the bleak barren landscape, and having just removed my Scottish Tweed jacket to better facilitate putting my lab coat on in the confines of the cabin of my car (automobile), as a misdirection, in readiness to slip past Matron who is sat in her usual place fast asleep on a chair by the main entrance to the lab, as she snorts in time to the drool dribbling onto the sleeve of her cardigan, as she waits in vain to catch latecomers; and having read, marked, and inwardly digested your comments confirming the difficulty in locating wanted data amongst vast tracks of collected data, exacerbated by the fact that the statisticians cry of ‘more data’ is reminiscent of watching Alice through the looking glass observing the Tweedles fighting over a resource that is of no use to either of them, nohow and contrary wise; and appreciating the fact that there is more gold ore to be obtained from extracting the gold contained in a ton of mobile phones (cell-phones) than what there is extracting gold from a ton of rock in a gold mine, and realising that the cyanide used as a catalyst to leach out the gold probably causes more harm than the harm caused by processing the mobile phones, which made me begin to postulate the paradox as to whether or not two wrongs make a right?, I was blessed with a thought that decided to chose to visit ME, and enter my mind, and I was filled with rapturous joy and elation as it suddenly pushed its way forward to the forefront of my conciousness and both suggest to me and encourage me to conjecture the notion that biology is more like Newton’s second law of motion, which states that for every action there is an equal and opposite reaction, and therefore, I put the proposition to you, and anyone else that may have more than a passing interest in such things, that, based upon these empirical observations and the privilege of having a thought take the time and trouble to seek ME out and enter MY head, that, rather than data-mining, maybe we should be applying Newtonian algorithms to our quest for answers and solutions, in light of the fact that, as confirmed by the findings of Augusto Odone, shown so beautifully and clearly in the superb film ‘Lorenzo’s Oil’, the biological landscape immediately modifies itself the moment we attempt to change something ON that biological landscape.

As I share the thought that is currently residing with me, I welcome any thoughts that may have visited you.

Regards

TILIS

If you are reading this then what may on the face of it look like I am talking to myself has now been rewarded by someone who is unthreading this thread in an attempt to gain wisdom.

It was Aristotle, the founder of logic, who gave us the most basic requirements of REASONING: “The same thing cannot at the same time both belong and NOT belong to the same object and in the same respect. If animals can speak then they can speak.” If Alice cannot get into the garden because she is too big to get through the door then she is too big to get through the door (she is in fact an OUTLIER).

“I know what you are thinking about,” Said Tweedledum, “but it aint so, nohow.”

“Contrariwise,” continued Tweedledee, “if it was so, it might be; and if it were so, it would be; but as it isn’t – it aint.”

This is REASONING and it is the application of good reason that BUILDS strong arguments – they don’t just assemble themselves – not even in Wonderland. If we FAIL to do this and simply accept everything on ‘blind faith’ then we would not make any HEADWAY.

As I have demonstrated throughout this thread, probabilities are not what MOST people BELIEVE them to be, and I provided worked examples, together with strong evidence to the contrary at the outset and throughout, using DOND as the foundation and asking just ONE question ‘SWAP or NOT SWAP?’ I shall return to that further down.

My mum knew that I wished to conclusively demonstrate in a fair and entertaining way that the amount of REDUNDANT data in any data set is VAST, and so I believe that she visited me to give me her advice so that I could create a valid example to show that the larger the data set – the more junk it contains. The REDUNDANT data may be interesting – fascinating even – but none the less – it is junk.

Referring to the sizeable sentence in my last post:

Sentences = 1

Total words = 484

Characters without spaces = 2,299

Characters including spaces = 2,782

Commas = 27

Periods = 1

PERTINENT DATA: “Biology is more like Newton’s second law of motion. Maybe we should be applying Newtonian algorithms to our quest for solutions, in light of the fact that the biological landscape immediately modifies itself the moment we attempt to change something ON that biological landscape.”

Sentences = 1

Total words = 44

Characters without spaces = 236

Characters including spaces = 279

Commas = 2

Periods = 2

As some observe in this discussion, and despite evidence all around us to the CONTRARY, statisticians start off with the FLAWED premise of ASSUMING most variables in life are INDEPENDENT. As a result, the statistical methods statisticians use look for ASSOCIATIONS – and when they find one – oh boy – we get everything from scaremongering to bullsh!t! Vitamins cause autism (yeah right), lack of iron in the mother causes cleft-lip to occur in the child (how fascinating), and on and on and on ad nausea.

How many pregnant women were STARVED of vitamins, and of those, what proportion gave birth to autistic children?

How many pregnant women were given the CORRECT portion of vitamins, and of those, what proportion gave birth to autistic children? (i.e. false positives)

How many pregnant women were OVERDOSED with vitamins, and of those, what proportion gave birth to autistic children?

Was the AGE of the mother or some underlying factor in the DEMOGRAPHY considered (e.g. fathers worked at a chemical/nuclear plant)?

Likewise – on cleft-lip:

How many pregnant women were deliberately STARVED of iron, and of those, what proportion gave birth to children with cleft-lips?

How many pregnant women were given the CORRECT portion of iron, and of those, what proportion gave birth to children with cleft-lips? (i.e. false positives)

How many pregnant women were OVER-DOSED with iron, and of those, what proportion gave birth to children with cleft-lips?

Was ANY Mendelian modelling carried on the GENETICS of the GRANDPARENTS? And again – was some underlying factor in the DEMOGRAPHY considered?

If I was an obstetrician would I emulate Dr. Slop – the male midwife in Laurence Sterne’s masterwork ’Tristram Shandy’? As a supporter of preformationism (the mother carries the eggs of HER mother – and so the child born is the child of the PARENT of the mother – hence why the HUGE bond between a child and its GRAND parent exists – and why genetic malfunctions ‘skip’ a generation), I raise the question of whether once a foetus becomes a homunculus (fully formed human) it should be allowed to be baptised whilst in the womb. Why should a baby be born – OR (God forbid) killed or terminated – before it can be christened? What is all of THAT about? Ah hah – what I am now writing is both digressive and progressive – both at the same time! Who is up for opening up a new thread and broadening THIS argument?

BACK TO BAYES

The REAL beauty of Bayes is, as Aristotle suggests, that his method makes every effort to DISPROVE the data presented. As we saw when the sophists attempted to deploy brinkmanship and INSISTED UNREMITTINGLY to assert the premise ‘in DOND “it makes NO difference whether you switch or not”’, I pointed out that this was a FALACY, and the resulting ‘egg on face’ verified that FALSIFICATION is much STRONGER than confirmation.

Let me put this fallacy to bed once and for all so that we may develop the argument ORIGINALLY presented at the top of the discussion.

Out of the 1,540 ‘three-box’ penultimate-game scenarios possible, there are 3 chances out of 22 = 210 cases where the biggy would be one of the 3 in the penultimate game.

Of those 210, there is a 2 in 3 chance that the biggy will NOT be knocked out on the final ‘pop’. Ergo, as all participating correspondents have demonstrated, when this scenario occurs, there are 139 cases where SWAPPING brings advantage – giving a probability that this would occur around 9% of the time – or around one game in eleven. There’s glory for you!

And for the sophists, I ask you to speak out loud the answer to the question Humpty Dumpty asks Alice – viz:

How old did you say you were?

WRONG – you never said a word like it!

HELLO MONTY HALL

A scratch-card is offered with a 1 in 3 chance of winning. Write down how many scratch cards you should purchase to guarantee a win?

If you wrote down THREE please give the pencil back to the White King because you are DELUDED into believing you will win once out of every three tries. In fact the odds are heavily stacked AGAINST you. In reality, you have TWO chances of LOSING each and EVERY time you play – ask Monty Hall.

THE GAMBLERS FALLACY

Gamblers, and those of you using the White King’s pencil believe that the ‘odds’ of an event taking place is related to the number of ATTEMPTS. For example, they believe that because the probability a coin landing on ‘heads’ is always 1 in 2 (i.e. it either will or it wont) if you toss a coin 60 times it should give 30 results out of 60 events. This is a FALLACY! Here is why.

Because the SOUGHT OUTCOME (in this example the sought outcome is ‘heads’) does not occur on EVERY occasion (e.g. every toss of a coin, throw of a dice or spin of a roulette wheel), as the total number of attempts (e.g. coin tosses) gets LARGER, the LIKLIHOOD that the occurrence of the sought outcome (say ‘heads’) will happen actually REDUCES. For simple proof, for those that gamble money on the outcome of an event, out of the total number of ‘plays’ of your favourite game – be it bingo or roulette, how many times do you come home with your pockets laden with gold?

The probability (likelihood) of a coin landing on ‘heads’ is 1 chance in 2 – each and EVERY time you attempt it – NO MATTER HOW MANY TIMES YOU ATTEMPT IT.

Going deeper, presented with the suit of Hearts, what is the probability (P) of drawing a KING from this well shuffled set of THIRTEEN cards?

Event E = ‘A King is drawn’ = 1 (the suit of Hearts only has one King)

The total number of outcomes = 13 (The number of cards in the suit)

P(E) = 1/13 = 1 chance in 13 = 0.0769 = (0.0769 X 100) = around 8% probability (likelihood) of drawing a KING – each and every time you attempt it!

Now let us INCREASE the SIZE of the data-set and present all FIFTY TWO cards for the same evaluation.

Presented with ALL FOUR SUITS, what is the probability (P) of drawing a KING from this well shuffled set of FIFTY TWO cards?

Event E = ‘A King is drawn’ = 4 (there are FOUR ‘Kings’ in the pack)

The total number of outcomes = 52 (The number of cards in the pack)

P(E) = 4/52 = 1 chance in 13 = 0.0769 = (0.0769 X 100) = around 8% probability (likelihood) of drawing a King from a pack (deck) of cards – each and every time you attempt it. EXACTLY the same as the SMALLER data-set.

What you should now appreciate is that:

- The lower the number of sought outcomes, in proportion to the total number of outcomes (events), the lower the PROBABILITY of the event taking place.

- The outcome of PREVIOUS events CAN affect the outcome of any FUTURE event. If my train is late most days then it is HIGHLY LIKELY that my train will be delayed TODAY.

- A CHANCE event is NOT influenced by the events which have gone before. If a true die has not shown 6 for 20 throws, the probability of throwing a 6 is ‘1 in 6’ on the 21st throw, and all SUBSEQUENT throws. As the White Queen says, “It’s a poor sort of memory that only works backwards.”

Once you grasp these three very simple concepts you will be master of one of the most fundamental tools to determine everything where there is UNCERTAINTY – from what are the odds that my train will be delayed to what are my chances of winning the lottery – and every other relationship you care to consider.

One wonders if this simple concept upsets some basic human instinct, because it is not difficult to find professional gamblers who will agree with all the above remarks – only to continue to bet in an IRRATIONAL manner based upon ‘the law of averages’.

ON GAMBLING SYSTEMS

Casinos have SECRETS up their sleeves which will – in the long run – BREAK ALL players!

Roulette (French for ‘Small Wheel’) is the glamour game that made Monte Carlo the world’s most famous casino.

A system based upon the ‘Law of Averages’ assumption called the ‘D’Alembert’ is very popular with Roulette players all over the world. It operates as follows: An ‘evens’ bet is placed for the ball to land on (lets’ say) RED. If the player LOSES their bet they add a chip to their stake on the next spin. If the bet is WON they reduce their stake on the next spin by one chip.

THIS DOES NOT WORK!

Let us examine the ‘D’Alembert’ system to see why. On paper it looks good, however, like all other ‘systems’, sooner or later the player is brought down to earth by the fact that the player eventually goes broke.

Here is why! If a player on a ‘losing streak’ continually increases his bets, several things happen:

- Constantly increasing the size of the bet will force the player to risk a large amount of money to try to recoup losses.

- By increasing each stake the player will swiftly reach the limit of their own bankroll.

- If they have the means to reach it, the player will eventually be stopped by the Casino Limit – so they can NEVER regain what was previously lost – EVER!

- When a player tries to break the bank at a casino, they are fighting the Casino EDGE as well as an opponent with MUCH larger resources.

- Once they reach the limits of their bankroll it is GAME OVER they are forced to quit!

Do NOT be fooled good people – this situation applies to ALL ‘Gambling Systems’!

Millions of Roulette players all over the world refuse to believe this!

Many haunt the casinos day after day, endlessly recording STATISTICS – seeking how often certain numbers, colours or combinations have appeared or not appeared. What I trust I have demonstrated and that you have grasped for yourself, and what people do not seem to readily grasp is:

1 – The lower the number of sought outcomes, in proportion to the total number of outcomes (events), the lower the PROBABILITY of the event taking place.

2 – A CHANCE event is not influenced by the events which have gone before.

3 – Statistics are based upon ‘averages’ and are therefore almost meaningless.

4 – Gambling ‘systems’ are pointless.

The reality is – players are pitting their wits and nerves against LEGAL cheating!

Prior to decimalisation of the British currency, the Pound (£) was divided into 240 parts called ‘pence’. There were several coins which were ‘consolidated’ values. One of these was the ‘shilling’ which was equivalent to 12 X 1/240 DECIMALISED pence – or 5 NEW pence. Another was the ‘sixpence’ – equivalent to 6 X 1/240 decimalised pence – or 2.5 NEW pence.

With regards to the ‘Gamblers Fallacy’; in ‘David Copperfield’ Charles Dickens, through Mr Wilkins Micawber makes the point: “Annual income twenty pounds, annual expenditure nineteen pounds nineteen and six (£19.975), result HAPPINESS. Annual income twenty pounds, annual expenditure twenty pounds ought and six (£20.025), result MISERY.”

As I trust these simple illustrations have demonstrated, the ‘law of averages’ used EXTENSIVELY in STATISTICS is a FALLACY!

Anyone like to see Bayes applied to roulette?

TILIS

@TILIS

No, it’s not a fallacy, it’s simply a fact.

Agreed. So the chance that the biggie is in still in play with only 3 boxes left is 210/1540. Not surprisingly, this reduces to 3/22.

Yes, so the chance that the biggie is still in play with only 2 boxes left is (2/3)*(210/1540) or 140/1520. Note that this reduces to 2/22.

WRONG. That would imply that of those 140 final possibilities cases, you initially picked the wrong box 139 times versus picking the right box only once. But that’s not correct. It’s an even split at that point – 70 cases where you picked the wrong box (and you’d need to switch to win), versus 70 cases where you picked the right box (and switching would cause you to lose.

IOW, if you switch, you win in 70/1540 = 1/22 situations. But if you don’t switch, you also win in 70/1540 = 1/22 situations. Switching makes no difference.

Think of it this way. Suppose I take 22 playing cards: one king and the rest spot cards (2-10). I mix them throughly and lay them face down in a line in front of you. If you simply pick one card and turn it over, there is a 1/22 chance that it will be the king. Agreed?

Now instead, imagine you identify one card as yours, but don’t turn it over yet. Instead, separate your card and two others from the remaining 19 (still not turning any over). This represents getting to the 3-box stage above, except we haven’t exposed any cards yet.

Now set aside one of the two cards that was grouped with yours. This represents going from 3 boxes to 2 boxes.

Now, if you turn over just your card, the chance that it’s the king is still 1/22, right? Simply moving the cards around on the table won’t change the odds. Suppose instead you turn over just the other card that’s grouped with yours. It’s chance of being the king is also 1/22, right?

What if you could turn over all the cards simultaneously. The chance that the king is in any particular place is still 1/22, right? And that applies to both your card, and the one that’s still grouped with it.

Finally, what if you turn over all the cards in the initial group of 19, and then the one card that was rejected from the group of 3, leaving only your card and the one other face down. If your analysis above were correct, suddenly the other card is much more likely to be the king (139/140) than is your card (1/140). But that would imply that the simple act of exposing the cards in groups affects the odds, which is clearly wrong.

Once again, the key here is that all choices are made randomly by you, and you have no knowledge about which card is where (or what’s in each box, for DOND). I might know where the king (or the biggie) is, but that doesn’t matter (as long as I don’t give you hints), because you make all the choices. As long as your choices are random, it can’t make any difference which cards (or boxes) are exposed in which order.

This contrasts to the Monty Hall situation, where Monty knows where the biggie is, Monty picks which box to open first, and Monty always chooses to open a losing box. Monty’s choice is NOT random, so that DOES change the odds for the remaining boxes.

Are you sitting comfortably?

Then I will begin.

To answer you, I open by saying human creatures have shared needs and interests, and concepts, language, and SHARED thought brings us TOGETHER, whereas – HOSTILITY is POINTLESS and drives us APART.

Thoughts visit us, and deliberation emerges in the first place through interpersonal communication in a shared material world, and continues to develop as we engage each other in FRIENDLY and CONSIDERED dialogue, and that LANGUAGE depends upon COMMUNICATION – and not vice versa.

This is the triangulating situation – two creatures communicating about a common world. As explainers, we require TWO mutually irreducible vocabularies – body and mind.

Now at this juncture we could discuss the relations between language and the world, speaker intention and linguistic meaning, language and mind, mind and body, mind and world, mind and other minds, and ask: “Can a ‘current-belief view’ make room for human thought without attempting to destroy an ALTERNATIVE view by reducing it to something trivial?”

But would doing this bring comprehension or stimulus? For although humans are acquainted directly with the world, not indirectly via some intermediary such as sense-data, representations, or language itself, and we accept that ‘epistemic theories of truth’ – a term here which means ‘MY proof shows different’ – are all FAILURES; we may wish to turn to Tarski’s ‘Truth theory’ to trace out the empirical connections between the concept of ‘truth’ and ‘observable behaviour’.

As the philosopher David Hume postulated in his discussion of Newton’s Third Law when his attention was drawn to the behaviour of one billiard ball striking another, “You EXPECT the billiard ball to move! May I not conceive that a hundred different events might as well follow from that cause? May not both of these balls remain at absolute rest? May not the first ball return in a straight line, or leap from the second in ANY line or direction? All of these SUPPOSITIONS are consistent and CONCEIVABLE. Why then should we give the preference to ONE – which is no more consistent or conceivable than the rest? All of our reasoning a priori (prior to the event happening) will never be able to show us ANY foundation for this preference. Causes and effects are discoverable – not by reason – but by EXPERIENCE.”

As you continuously protest, and Hume clearly teaches, there is an ENDLESS number of ways in which an interpreter can assign objects to the interpretent’s expressions, however, if we are to find a language intelligible, we must find it capable of COMMUNICATING, and we must appeal to the entities we think exist, which are just the entities that belong to the ontology of our LANGUAGE – including courtesy, good sense, and sound judgement.

MEANING is a function of what a speaker intends, and this intention includes what the speaker intends their HEARER to understand. Inevitably, there will be errors – but, as Socrates expounds, and Bayes ENSURES, these errors should provoke examination of the ‘falsity’ to arrive at the ‘idea’ or NOTION of truth.

The triangulation of language, thought, and INTERPRETATION does not rest with ‘shooting down’ apparent errors in expression; it must EXAMINE the ROOT PRECEPT and not be sidelined by trivia. In this case we are asking the question ‘Is Bayes better than statistics?

Continuously deriding something prevents derivation of a proper discussion that provokes thought that enables us to draw SOUND conclusions to the ideas being presented by CORRECT interpretation of what the WORDS mean.

As history continues to show us, the MEANING of words will vary from person to person, culture to culture, and century to century. It is for this reason I have periodically painstakingly taken time to give DEFINITIONS of SENSE where I believed what I was attempting to CONVEY could be misconstrued by a different culture to my own.

As a simple illustration, I offer what the White Queen says to Alice when Alice VENTURES to ask “Are five nights warmer than ONE night?” “Just so” cried the Red Queen “five times as warm, and five times as cold – by the same RULE – just as I am five times as RICH as you are – and five times as CLEVER!” (Chapter 9 ‘Queen Alice’ ATTLG)

There are several things to take note of here.

The first premise is centered on the notion of something being ‘warmer’ or ‘colder’ – warmer or colder than WHAT exactly? It is the ABSUDITY that a temperature scale should start at some ARBITARY point – as defined by the Dutch instrument maker Daniel Gabriel Fahrenheit in 1717, and the Swedish astronomer Anders Celsius in 1742.

Each devised their own arbitrary ‘start point’ and derived totally unrelated GRADATION of SCALE. In the world of Celsius the ‘freezing’ point of water is Zero degrees and the boiling point of water is 100 degrees. In Fahrenheit’s world, the ‘freezing’ point of water is 32 degrees and the boiling point of water is 212 degrees. Going further, when Celsius ORIGINALLY devised his ‘scale’, he made the freezing point of water = 100 degrees, and the boiling point of water = 0 degrees. It was Lord Kelvin who gave us the ABSOLUTE ZERO scale (measured in KELVINS (‘K’) (more further down).

Back to Alice. The second point to take note of is the play on the words ‘rich’ and ‘clever’. The now quaint expression ‘That’s rich’ is the equivalent of Bart Simpson’s “Doh!” – and the word ‘rich’ is the OPPOSITE of CLEVER. So the Red Queen was alluding that both she and Alice were as INTELLIGENT as each other.

Ho now! Is the task of philosophical discussion to clarify, reconcile, unearth, and criticise beliefs and ASSUMPTIONS with which one begins one’s investigation? Or is it to lead to a discovery of TRUTHS that were NOT is sight at the START?

So far, with the exception of ONE correspondent – not ONE single person – including your good self – has ATTEMPTED to lead to a discovery of truths that were NOT is sight at the START – choosing instead to show how rich you all are!

Please permit me this important ‘aside’.

AN IMPORTANT ASIDE (See – I TOLD you it was important)

This part relates to the INITIAL proposition that stimulated the ENTIRE discussion thus far – that of PARAPSYCHOLOGY. AND IT IS RELEVENT TO THIS DISCUSSION!

The ancient scriptures of the Sumerian Cuneiform Scripts from the middle Uruk period around the 30th Century BEFORE CHRIST (BC) and the sacred Hindi Vendantic Scriptures ‘The Upanishads’ describe many MYSTIC arts – including the ability to ‘see’ through SOLID matter and transform water from a liquid to a solid and a solid to a gas – and back again – merely by ‘uttering’ appropriate FREQUENCIES.

Heisenberg hinted at ‘mind over matter’ many decades ago in his ‘uncertainty principle’, and it is only now being recognised as the missing link between modern science and ancient mysticism. Much of the theory of Heisenberg and Schrödinger is described in the ‘The Upanishads’.

The Upanishads also fully describe ‘super-string theory’ and a TEN-dimensional cosmological model. This document was hidden during the ‘dark age’ time between the collapse of classical civilisation and the beginning of the Italian Renaissance (new birth).

The whereabouts of these documents and an ‘index’ to the contents was discovered in a mediaeval Aramaic dating back many centuries, and, after two centuries of attempt, finally deciphered in the thirteenth century.

These recovered documents form the precept behind modern “entanglement theory”.

‘Entanglement’ is the core primeval belief that everything is INTERCONNECTED to everything else in some mystical way (an echo being a phenomenon to demonstrate this); and that mankind has a spiritual quest to seek atonement with the universe.

A short digression back to Alice through the looking glass is called for here.

Alice finds a book and not understanding many of the Medieval English expressions contained within the text, she turns to Humpty Dumpty and asks him to aid her to read ‘The Jabberwocky’. Oh – nearly forgot to define the term. A ‘wocky’ is a particularly nasty kind of ‘sprite’ that enters the mind and resides there – and it incessantly ‘jabbers’ (talks rubbish) and causes a possessor who hears the ‘voice’ (or ‘voices’ if two get in and start arguing – heaven forbid) to experience fear and throw fits of rage. Many enter older people who are off their guard (medical fact).

Alice reads “Twas brillig” (soup making time) “and the slithy toves” (crafty, sly, feral children); “did gyre” (from gyrate – to spin round and round with arms outstretched – which Victorian society viewed as a form of madness) “and gimble” (screw themselves into the floor – an implication that such children would end up paupers and trash-mothers like the Duchess) “in the wabe.” Wabe is pronounced ‘way be’ and refers to the shadiness of toves is akin to the wabe (shadow) cast by a hill – casting long sinister shadows across the countryside that eventually and inevitably ensnare the unwary. Carroll’s physical reference to the wabe is in reference to a sundial – not the shadow cast by the gnomon – which indicates the hour of the day – but the sundial as an edifice casting it’s own ‘sinister’ wabe across a garden.

As Alice reads, she encounters the term ‘slithy toves’, and she asks Humpty Dumpty to explain what ‘slithy’ means. He replies: “It means ‘lithe and slimy’. You see it’s a ‘portmanteau’ – TWO meanings packed into one word.”

The word ‘atonement’ is a VERY special word that ALSO has TWO meanings contained within it – depending upon how you PRONOUNCE it. When pronounced as ‘a-TONE-ment’ it refers to divine punishment that is inflicted upon one because of our misdeeds – the ‘God never pays debts back in money’ allegory. When pronounced as ‘at-ONE-ment’ it refers to mankind’s spiritual quest to seek at-one-ment with the universe.

Both Newton and Kelvin were deeply gripped with these early ‘noetic sciences’ and it was Kelvin who believed (and later proved) that all matter ‘vibrates’ and these vibrations reveal themselves as electromagnetic waves in the VISIBLE spectrum – each to its own colour according to its TEMPERATURE. This can be observed by watching a horseshoe in a farrier’s furnace begin to glow and change its colour as the temperature changes – black – its cold – red – its hot – yellow – its seriously hot.

Put simply, the EXPANSION of a body is due to excitement of the molecules as they are heated. The hotter a body gets, the quicker the molecules ‘want’ (desire?) to move apart from each other. The increased frequency creates more ‘bounce’ energy which causes the molecules to bounce further apart, thus causing the object to increase in physical size AND change colour. It’s weird – but true.

These ‘vibrations’ are what causes energy transfer to take place and the sensations we call ‘hot’ or ‘cold’ are merely faster or slower frequencies impacting our skin. Fast ones BURN!

Kelvin affirmed that all matter, with ONE EXCEPTION, expands when it is heated, and contracts when it is cooled. The exception to the ‘rule’ is WATER, which expands when it is heated, and ALSO expands as it solidifies – when it is cooled. At around 4 degrees Celsius water is a solid, a liquid, AND a gas. This weird state is called the ‘triple point’.

Kelvin devised his “absolute” temperature scale (the ‘K’ scale) and formulated the Laws of Thermodynamics, which loosely state that work is heat and heat is work, and heat cannot of itself pass from one body to a HOTTER body.

Kelvin had access to the ancient Mycenaean Greek texts written around the late fifteenth century BEFORE Christ (BC) where all of this is described, and which Kelvin unravelled and made sense of. We know that microwave vibrations cause water to boil. Presently, just as we know that X-ray AND ultrasonic sounds allow us to ‘see’ through solid matter, we must conclude that there is a frequency that turns water to ice (I HAVE seen it done when I visited Egypt).

The methods are from a secret Sect referred to as being ancient – even in the early Anglo Saxon chronicles. Upholding routes in noetic science, its members may have included: Nicolaus Copernicus, Leonardo Da Vinci, Sir Isaac Newton, William Halley, Voltaire, Goethe, Antoine-Laurent Lavoisier, Nicola Tesla, and William Thompson (Lord Kelvin – he of the LAWS of thermodynamics).

There are many questions – far more than answers.

COMING FULL CIRCLE

Turning briefly to roulette, at this juncture in my dissertation, you may be wondering what you have just read got to do with a ball hurtling around the outside perimeter of a roulette wheel as the wheel spins in the opposite direction to the travel of the ball? The answer – LOTS!

I sated earlier that casino’s have many SECRETS. I am about to share some with you.

The European Roulette wheel has 37 pockets.

THIRTY SIX pockets have a unique number from a consecutive set of numbers – One through 36 distributed in a SPECIAL SEQUENCE based upon a SECRET formula which was calculated in CHINA over four thousand years ago.

A GREEN pocket has the value ZERO, the remaining pockets alternate RED – BLACK.

WHERE the GREEN pocket is placed in relation to all of the other values is ALSO a well kept secret. It is where it is for a REASON! Know this: the ZERO and its LOCATION play a HUGE part in boosting the profits of the casino!

Newton’s THIRD Law of MOTION (momentum), Lagrange’s ‘conservation of momentum’ equations (kinetic energy), the D’Alembert ‘inertia principle’ (oh yes he is in there), and Emmy Noether’s FIRST Theorem (principle of least action) are ALL involved! And ALL are fully described in ancient texts! That is all I am prepared to say!

THE ZERO EFFECT

What if the ball lands on the ZERO? Answer – the Casino removes a proportion of the stake wagered, or put another way – EVERY player loses!

For reasons stated above, the probability of this happening is a GUARANTEED ONE chance out of every 37 spins – giving the Casino a 1/37 gain (around 2.7%) on ALL money staked.

What if the ball lands on a number between ONE and 36 (36:1 payout)?

Answer: Because of the ZERO, the Casino is paying you UNDER the TRUE odds of 37 to one – or put another way – it is SHORT-CHANGING every winner by one unit – around 2.71% of their stake money for EVERY wager won (I alluded to this in the ‘3% ‘edge’ gives 97% win’ discussion).

In addition, the Casino wins all of YOUR ‘losing’ wagers PLUS all OTHER single-number wagers. The probability of losing is (1 – 1/37) = 97.29%. You are NOT permitted to bet that a number will NOT come in (I wonder why?)

What if the ball lands on a RED pocket (Evens payout)? Answer: All RED-Pocket wagers receive a chip for every chip staked – the Casino wins all BLACK-Pocket wagers – PLUS – because of the ZERO – the Casino is again short-changing the winners by a MASSIVE 5.3%.

Here is how it is calculated?

The player believing he is betting on an EVEN-CHANCE wager is betting that one of 18 events will win (i.e. black-red odd-even, low-high, etc.). There are however, 18 events – PLUS the ZERO that will LOSE for him. The Casino is short-changing the winner one NINETEENTH of their win – or – put another way, the gambler is paying the Casino 5.26% of their winnings for every wager placed. This situation applies to ALL ‘even-payout’ bets!

The American Roulette wheel has 38 pockets laid out in a BLATENT ‘secret sequence’. TWO green pockets have the value ZERO.

Do the sums for YOUR wheel and be prepared to be VERY disappointed!

BACK TO THE MAIN THEME

Should you take the time to examine Socrates elenchus you will see that it includes the dialectic of conversational ‘give and take’. Truths emerge only in the context of frank discussion and communication. Communication is the root to a SHARED UNDERSTANDING – and is NOT achieved through the use of sophist(icated) BRINKMANSHIP – which serves NO purpose whatsoever!

Causal relations are EXTENSIONAL relations which hold between SINGULAR EVENTS – no matter HOW they are described. If ‘A’ causes ‘B’ under the attribute of extension, and ‘B’ is identical with ‘C’ under the attribute of THOUGHT, then how can we DENY that ‘A’ ALSO causes ‘C’? This is the crux of ‘Bayes theorem’.

Throughout history it has been shown that every TRUE singular causal statement relating to two events is backed by a LAW that covers those events WHEN PROPERLY DESCRIBED. Examples include Newton’s Laws, Kepler’s Law, Ohms Law, Boyles Law, Charles’ Law, Kelvin’s Laws, Boole’s Laws, and BAYES Laws.

My INTENT was NOT to create a FOG or construct a CHASM, or tarnish TRUTH by having a continual tedious discourse on DOND – which – quite frankly, has not broadened the discussion one jot; my INTENT was to elucidate and develop the NOTION postulated by Eric-Jan Wagenmakers and others – that STATISTICS have NOT served us well, and that NOT applying Bayes is missing a trick. THAT and ONLY THAT is the ROOT argument I have taken the time and considerable effort to postulate, elucidate, and DEMONSTRATE.

As you are unable, or unwilling to discuss and compare the one with the other in a reasonable manner then let us leave things where they are and as they are – you can lead a horse to water but you cannot make it drink.

And finally, MY last word on DOND.

I have TWO questions – viz:

In all of the games where the contestant LOST, what percentage of contestants would have WON if they HAD swapped their box?

In all of the games where the ‘biggy’ was present, AND the contestant LOST and HADN’T swapped their box, what percentage of THESE games would examination of the PENULTIMATE scenario to obtain the PRIOR KNOWLEDGE, have suggested that the BEST STRATEGY would be to SWAP?

In other words – irrespective of what odds of winning you or I believe or don’t believe exist; does the 3 box ‘Monty Hall’ situation of seeing ONE biggy and TWO irrelevant values PRIOR to the end game; and then seeing an irrelevant box tumble (i.e. ‘Monty’ shows a LOW value) – leaving the biggy and an irrelevant value ‘in play’, bear out the BEST strategy to INCREASE the LIKELIHOOD of winning from around the 50% to around SIXTY SIX PERCENT – is to SWAP in the ‘end game’?

THAT was the sticking point that has held us back throughout the entire discussion on evaluating ‘Bayes Theorem’ as an alternative strategy to STATISTICS.

Causes and effects are discoverable – not by reason – but by EXPERIENCE.

IN SUMMATION

I trust that you have enjoyed what you have read here. I trust too that I have clearly communicated concepts and provided insight, and you have come away with a better understanding of the strange world which we experience.

Right now, we face a critical branch-point in our history. What we do RIGHT NOW will propagate down through the centuries and powerfully affect the destiny of our descendants. It is well within our power to destroy our civilisation and perhaps our species as well.

“If we continue to surrender to superstition or greed or stupidity – we can plunge our world into darkness much deeper than the ‘dark age’ time between the collapse of classical civilisation and the Italian Renaissance. But we are also capable of using our compassion, and our intelligence; our technology and our wealth to make an abundant and meaningful life for EVERY inhabitant on this planet, to enhance enormously our understanding of the Universe; and to carry us to the stars!” (Carl Sagan)

Dr Carl Sagan was a super intelligent man who communicated difficult and complex ideas easily to the masses through his TV series ‘Cosmos’.

Cosmos is regarded as the greatest science documentary in history and a seminal moment of TV. As he travels through time and space in his dandelion spaceship he reflects on humanity’s successes and mistakes when it came to enlightenment.

It has been fully digitally remastered and released in the DVD format through brightvionentertainment.net. and, no matter what your faith or beliefs, if you possess an open mind then this is the DVD set for you.

Bon chance in all of your endeavours

TILIS OUT

TILIS writes:

If you’re asking specifically about contestants who lost after getting down to just two boxes, then obviously all of them would have won if they’d swapped. However, there would be just as many contestants who WON after getting down to two boxes, because they did pick the biggie at the beginning. All of them would have LOST if they’d swapped.

Zero. As I’ve tried to explain in as many ways as I can, THERE IS NO BENEFIT TO SWAPPING. Not as long as the contestant has no knowledge of what’s in any box, and the contestant is the one deciding which box to keep and which to reject.

If a contestant makes it to the 2-box stage with the biggie still in play, there is a 50% chance that his box holds the biggie, and a 50% chance that the other box holds the biggie. His chance of winning is EXACTLY THE SAME whether he swaps or not.

I’m afraid your trust is misplaced. You have NOT clearly communicated. Instead, you post huge walls of text that are almost entirely unresponsive to the actual discussion. The few paragraphs that are on topic betray your unwillingness and/or inability to understand the points I’ve been making.

I thought you were willing seriously consider what I and others have tried to show you. I now see I was quite wrong.

Bon chance to you as well. I doubt I will bother replying to you again.

@ qetzal Thank you for taking the time and effort required to peruse my muses’ sir – particularly my last one because it contains concepts that are, for many, extremely difficult to grasp.

Although we both agree to differ, and although the argument between the application of Bayes theorem as an alternative to statistics, which have NOT served us well, was not drawn to its logical conclusion (or challenged), you have been a worthy opponent and I hope to have the privilege and pleasure of another roustabout in the juxtaposition of life as we KNOW it – not life as we BELIEVE we know it.

Newton’s laws of motion – covering everything from centripetal force to gravity, and everything else in-between, as described in the ‘Principia’ are NOT based upon some arbitrary set of calculations. As Hume informs – they are DERIVED from OBSERVATION. In other words, what Newton has done in giving the world the calculus, is define a set of ‘sums’ (summations) that closely fit REALITY.

Newton’s masterwork, which took two years to write, started life as an enquiry from Edmond Halley (who has a Comet named after him).

Working for the British Government has many perks. In matters relating to the Principia I have taken the time on your behalf to track down the reference from the archives of the British Library. It is recorded in an ancient account from Newton’s confidante Abraham DeMoivre. I have reproduced the two pertinent stanzas, and I trust you find them of interest. They are as follows:

In 1684 Dr Halley came to visit at Cambridge (and) after they had spent some time together the Dr asked him (Newton) what he thought the curve would be that described the Planets supposing the force of attraction towards the Sun to be the reciprocal to the square of their distance from it. Sir Isaac replied immediately that it would be an ellipse.

The Doctor, struck with joy and amazement, asked him how he knew it. “Why” saith he, “I have calculated it”, whereupon Dr Halley asked him for his calculation without further delay. Sir Isaac looked amongst his papers, but could not find it and offered to recalculate it.

Newton did as he promised – and MUCH more besides, for what flowed from the mans mind was the ‘Philosophiae Naturalis Principia Mathematica’ – the mathematical principles of NATURAL philosophy.

Whether you wish to determine the temperature of your coffee in five minutes time using Newtons’ Law of cooling, or determine the particles of pollution per cubic centimetre of air in a city, in ALL life sciences it is to Newton’s calculus we turn – and his ‘improper’ integrals based upon OBSERVATION return some VERY surprising results that defy INTUITION.

Bayes Theorem is simply a unique sub-set of this great man’s masterwork.

Should you be in London, then why not visit Westminster Abbey and ask to see Newton’s grave. The monument will inspire you!

ON ROULETTE

Roulette contains a number of elements that make it the ideal casino game.

- It is easy to learn.

- The slow pace and simple rules make it a relaxing game to play and enjoy.

- It’s a game where winnings build up quickly – much faster than many other games of ‘chance’.

- It offers the gambler with a modest budget the opportunity to gamble for a long while with a small ‘investment’ (i.e. the amount of money you are prepared to forfeit if you are on a ‘losing streak’).

- It is a game where a persons’ favourite ‘lucky number’ is a very real, very romantic factor.

- It is the casinos number one spectator sport because when a high better gets hot, SPIRITUAL TENSION fills the air – and everyone watching is as excited as the hot better. This is to do with momentary penetration of the 6th wall in the ‘Seven Walls’ in which we reside.

All of these factors are what makes the game of Roulette so fascinating and so popular – and also, as I trust I communicated, make it the perfect game to study probability.

As an aside, although they may not realise it, thespians are attempting to penetrate the FOURTH of these walls. Many will speak of ‘the fourth wall’ but few do not have any concept of what the first three are – let alone the other 3 above – and hence, until enlightened, they PRESUME the first three is a reference to the three walls on the stage.

And finally

SIX TIPS TO ENJOY ROULETTE

1 Never purchase a ‘Gambling System’. They do not work and merchants that request payment for such things are attempting to defraud you – so beware!

2 Given a CHOICE between the European Roulette table and the American Roulette table, only ever play the EUROPEAN Roulette Table.

The European Roulette Table has One ‘Zero’ – giving the casino a 2.63% advantage over the player. There are TWO Zeros on the American Table. The double-zero Roulette Wheel realises 1 win out of every NINETEEN spins – considerably increasing the casino’s overall advantage to around 8% – 5.5% on ‘even’ bets, plus an additional 2.6% on single numbers.

3 Focus your resolve on generating profit, NOT beating the casino – you never will! Only ever bet on the outside wagers: Red/Black, Odd/Even, High/Low, Columns, Rows, or Dozens. Although the rewards are MARGINAL you will enjoy your experience – and walk away a winner!

4 Never EVER play on-line! The systems use ‘random number generators’ to determine where the BALL has allegedly ‘landed’. As the probability formula PROVES when we demonstrated with coins, dice, and cards, the larger the number of events (i.e. bank of ‘random’ numbers), the less chance you have of winning.

In addition, although the virtual ‘roulette wheel’ is visible on the screen, and to all intents and purposes appears ‘normal’ – the majority of on-line casinos usually include THREE or more ‘zeroes’ in the ‘random number’ bank. These are NOT visible on the ‘virtual roulette wheel’ – so the systems are heavily stacked AGAINST you! As an exercise, calculate the effect THREE zeroes has on your chances of winning.

5 Establish on the SIZE of your stake for a given investment. To calculate the size of your stake, divide your investment by the number of times you wish to enjoy the thrill of the play. For example, if you have an investment of £80 and you wish to enjoy 40 games of Roulette then each stake should not exceed 80/40 = £2 PER STAKE – around 2.5% of your investment per play. This will give you FORTY chances of winning.

Do not be tempted to increase each stake beyond the calculated amount – if you do so, you will swiftly reach the limits of your investment and take away the enjoyment!

6 Always pocket the money you win and only play until you exhaust your investment. Example: You begin with an investment of £50. After 1 hour of play you possess a total of £120 equating to a profit of £70. Set aside your winnings and put them into your bank account!

7 Never bet more than you can afford to lose! If you lose your investment within a short period, do not be tempted to make further wagers – WALK AWAY!

These tips are your Bible for ensuring a lucrative profit from the game of Roulette.

Bon chance my friend

TILIS

PS One Newton is the weight of an APPLE