Going Past the Chemist's

Penney's Game puzzle

2011-09-01T12:58:11Z

Penney's Game is a non-transitive probability game involving coin tosses. You invite your friend to name a sequence of coin tosses (at least three long), and you can always name a sequence that is more likely to come up first.

See this link which suggests a card-game variant for a good explanation and a diagram.

Now, what happens if you have five players. Player 1 names HTT as his sequence. Player 2 names HHT in order to beat Player 1. Player 3 names THH to beat Player 1. So Player 4 names TTH to beat Player 3. Player 5 now names HTT to beat Player 4.

If Player 2 usually beats P1, P3 usually beats P2, and P4 usually beats P3, then how does P5 - equivalently P1 usually beat P4? P1 < P2 < P3 < P4 < P1 ??

Bit baffling at first, but I think it's fairly easy to figure out what I've partly neglected to point out.

All this comes from a discussion on the JREF forums of this vos Savant column - which prompts an entirely separate discussion of random strings and probability...

On academic publishing

2011-09-01T11:59:13Z

There's been substantial discussion lately over academic publishing. Most recently, George Monbiot put out this on Comment Is Free, to be followed by this response on the Scholarly Kitchen (a blog from the Society of Scholarly Publishers). I can't remember where I saw many other discussions of this - if you know any feel free to comment (edit: Ahah - the ever excellent Peter Coles had this blog).

Let me say first I find both a little aggressive in tone, but there's important points raised by Monbiot, and some sensible responses by Kent Anderson at the Scholarly Kitchen.

First off I do want to say that frequently individual article prices are absurd and I've often been put off dismantling what I suspect is very dodgy research because I'd have to pay some crazy price to read it (and I have access to most major journals through the subscriptions of my employers). I can't expect my employer to pay for it (or a more expensive full subscription) if it's outside my field and I'm just making some comment beyond the remit of my work. I end up just not looking.

Secondly, us physicists make great use of the arXiv, but that is much more expensive to run than people realise and it takes something like half a million dollars a year of public funding of various sources to support, and it does no peer review or any of the other functions publishers directly perform.

On to Anderson's post though. He argues early on that most papers are inaccessible to anyone but specialists anyway, because they're too technical. He searches PubMed for 'cardiac', picks the first paper and points out its unparseable to most people. The bit he quotes is

Intrathoracic herniation of the liver ("liver-up") is associated with predominant left heart hypoplasia in left diaphragmatic hernia but not right fetal diaphragmatic hernia. Our observations indicate that this difference may result from different ductus venosus streaming sites in these conditions.

Now, I haven't done biology since double science at GCSE, but I broadly get the gist of 'intrathoracic herniation of the liver', 'predominant left heart hypoplasia', 'left diaphragmatic hernia' (and naturally the right version), although I'd have to Google ductus venosus streaming sites. I bet if I wanted to actually read this paper I'd have more clue about it, and there's no reason to think that a well-motivated reasonably intelligent amateur couldn't manage that. Admittedly I'm one person and a bit of an oddball in this sense but then...

... then there's the fact this is a randomly selected paper, and only one randomly selected paper. If I was trying to find out something about hearts, maybe my grandfather was started on some new medication I wanted to find out more about, I'd be doing a more detailed search, and even then if this cropped up I'd skip it in favour for something more obviously up my street. This single example of a mildly difficult to parse abstract is in no way an indication that most papers published are going to offer no value to people outside the immediate specialisation.

Anderson says

Let's assume I can read the whole paper. Like 99.9% of the population, I'm not going to know what to make of it.

Well over two million people in the UK have some kind of science, technology or engineering degree, and are probably going to be able to make a reasonable stab at parsing a lot of the essentials of a paper they are interested in. A not insignificant minority of these are going to have the skills in statistics to make reasonable criticisms of some papers from that angle without even knowing anything about a subject. If you gave a statistician a paper on a drugs trial they don't need to know anything about pharmaceuticals to critique the analysis. Many will be able to offer criticism of experimental design too.

This is particularly important for the kind of paper many will want to know about - e.g. a new cancer drug has come on the market and your mother has the cancer in question, and you want to know what justification the NHS has for saying it's not effective enough and they're not paying for it.

Anderson says

To chart a useful course forward, we need accurate information, rational thought, and reasonable plans. Monbiot provides none of these in his inflammatory, off-base, and ultimately unfair rant.

I find this a mildly baffling statement. Monbiot supplies quite a number of figures and accurate information, he certainly provides some rational thought (maybe not all though) and offers reasonable plans. I think Anderson is making something of an unfair rant himself.

There's a real absurdity in expecting certain private individuals (I can think of several excellent bloggers) to stump up tens of dollars for a single paper which they probably already think is a bit shit, or they wouldn't be wanting to download it to criticise it. And being able to criticise scientific publications is pretty crucial, even if you're not a specialist, especially when it has either been funded by public money or is about to lead to public money being spent.

Surprise!

2011-06-15T12:58:54Z

I've recently been reading Stephen Law's new book - Believing Bullshit (do check out the Amazon link if only for the comedy review Stephen mentions on his blog).

I'm only about a quarter of the way in, and finding it enjoyable, but I thought I'd pick up on one bit of the text I'm not completely sure I agree with. It's about what makes a scientific theory that is well supported by evidence.

]]> Law says that a theory needs to be:

clear and precise
surprising and
true

Now he also says that scientists and philosophers may disagree on the details with this, so I've not exactly got a complaint about what he says, but I think that it's not necessary for a theory to be surprising.

Now, I'm all in favour of clear and precise theories - if you make a waffly prediction it's hard to check how well the evidence fits your prediction. I've mentioned before how important it is from a Bayesian point of view, and even if you don't want to analyse your data that way I think people can see the benefit of a theory that makes precise predictions. A precise prediction is essentially easier to test, and testing things is pretty much what science is about.

I'm also all in favour of true theories. That's not terribly surprising.

What is surprising to me, is that Law says that a theory should be surprising in order for us to find strong confirmation for it. Law gives the example of the idea that fairies cause trees to grow more quickly in the summer months. This theory makes an unsurprising prediction, because we expect trees to grow more quickly in the summer months anyway. It's then argued that because we expect this to happen anyway, it's not a good theory.

This has problems. Suppose we actually had some complex idea for some phenomenon - a good example might be epicycles to describe the orbits of planets. Now, Copernicus comes along and has a simpler idea, a clearly better idea, but it's not a surprising idea because it predicts exactly the same apparent motions of the planets across the sky as we already had. An awful lot of theories have unsurprising predictions, but fortunately they frequently also have surprising ones - and these provide us with the best way to come up with an experiment to figure out whether they or another theory is best. However, given two theories that make the same predictions, labelling the first theory better because it made the surprising prediction and the later theory worse because the first one came up first and so now its predictions are unsurprising - well that's just not a good way to distinguish between the theories.

Instead, given theories that have the same predictions there's an old rule of thumb for deciding between them - Occam's Razor. The fairy theory isn't bad because it makes the same predictions as another theory - it's bad because it adds in fairies when we don't need them. A theory can be good while still being unsurprising, as long as it gives a better explanation.

That all said, the book's great. Maybe I'll review it properly when I've got to the end, and post occasional comments on the way to the end as well, but for now I think it's well worth picking up if critical thinking's your thing.

Bad physics demonstrations

2011-03-22T11:46:03Z

Yes, I know. I've not blogged for ages. Anyway, I'm blogging again, and then will probably fall silent for months again.

Two things have inspired me to write today, both physics demonstrations or analogies.

Gravity

The first is a very common description of gravity as described by Einstein's Theory of General Relativity (GR). GR essentially describes gravity as due to the curvature of spacetime, and that curvature as being due to matter. Almost as soon as someone starts to try to explain it they use the rubber sheet analogy. Now I link to that page to describe the analogy, but that page does not do a bad job of using it. Essentially the analogy is that you have a rubber sheet, and a ball on that sheet curves it, and the curvature of the sheet affects the motion of other objects upon it. Now, I've no criticism of that page. There are two major problems with this analogy.

The ball deforms the sheet - why? It's because the weight of the ball is pushing down upon that sheet. What gives the ball weight? Gravity, which is the thing this analogy is trying to describe. Now that's not too serious, you can simply point that out and say that the mechanism by which mass curves spacetime is not described - it simply does so.
If you roll a small ball slowly past the larger ball it curves in and rolls toward the big ball. This is because the small ball rolls downhill. Why does it do this? Because of gravity. This is a serious problem. There's a path deflection due to the geometry of the sheet, but there's also a path deflection because of a preexisting downward force upon the ball. This is almost never pointed out, and most worryingly the effect exactly looks like what people generally think of as a gravitational attraction - a movement towards rather than a deflection of a path from what would naively be considered straight.

Professor Brian Cox, using this description in this week's Wonders of the Universe, I'm looking at you.

Magnetism

When you first studied magnetism at school, what demonstrations of it do you recall? I bet that a very early one was putting a sheet of paper over a magnet and sprinkling iron filings over it. You get something like this image - iron filings lining up upon field lines. You're then also shown a diagram like this showing discrete field lines.

What's wrong with this? Well, field lines don't come in discrete chunks. They're continuous. Every point in space has a magnetic field line passing through it, and the field lines do not vary in strength in some onion-skin like way. There's nothing special about where those iron filings are lining up. The field lines are no more existing in a particular number than the field lines of Earth's gravitational pull exist in particular places, rather than smoothly over the entire surface. I bet you that every kid comes out of that lesson thinking magnetic fields look something like an onion. I did, and it took me a disturbingly long time to figure out that they weren't, and why, because noone ever corrected that misconception.

What is actually happening is that every iron filing is itself becoming magnetised and is drawing adjacent filings towards itself. It's like they're concentrating the field where they are. The filings are an active part of the field - they're not what a physicist might call a 'test particle' that doesn't affect the things around it and only traces out some physical phenomenon.

This is really problematic. People come away from seeing this thinking that magnetic fields are hairy.

Better ideas

Now both these demonstrations are actually useful if explained properly, and I honestly have no idea if there's a better demonstration of these physics concepts. Do you have one? Or do you have a pet hate amongst common physics demonstrations yourself?

A problem of unstated data

2010-04-29T14:47:26Z

In The Guardian's ongoing series of science questions posed to major political parties, the Green Party's response to the question
Is animal testing necessary? Are the ethical concerns outweighed by the benefits? How would you like to see regulations on animal testing change under your government, if at all?
states that

We agree with the independent patient safely organisation, the Safer Medicines Trust, that animal testing may be more harmful than helpful.

The Safer Medicines Campaign, previously known as Europeans for Medical Progress, has a habit of using arguments like the following:

Results from other species simply do not reliably translate to the clinic, as evidenced by the 92% failure rate of potential new drugs in clinical trials. [1]
That must include animal data, since crucial decisions, such as whether to proceed to clinical trials and whether the drug might cause cancer or birth defects are based on demonstrated safety in animals. Yet, as we know from Northwick Park, even safety in monkeys at enormous doses does not guarantee safety in humans. [1]
A large systematic survey published in November 2009 found serious omissions in reporting of data and in strategies to reduce bias in results. Only 12% of the animal studies used randomisation, only 14% used blinding and only 8% gave the raw data. [1]
The best way to evaluate the effectiveness of animal tests for drug safety is to compare their results with subsequent real- world outcomes in patients and consumers. [1]
Aids is another: while at least 80 vaccines work in animals, all 80 have failed in human trials. Similarly, every one of more than 150 stroke treatments successful in animals has failed in human testing. A study in the British Medical Journal found that animal tests accurately predict human response less than 50% of the time. [2]

[1] Safer Medicines Campaign Spring 2010 Newsletter http://www.safermedicines.org/newsletters/newsletter_spring_10.pdf
[2] "The dead end of animal research", Comment is Free (Guardian), August 2009 http://www.guardian.co.uk/commentisfree/2009/aug/07/animal-testing-medical-research

I've previously argued that their arguments are weak. Take the first. We're told 92% of the drugs tested that pass animal tests fail in later human clinical trials (the phrasing is mildly ambiguous but I believe that is what is meant). While that might look like a problematic figure, I would think that any competent scientist would also want to know:

The proportion of drugs that failed animal tests but would pass a later human clinical trial.
The proportion of drugs that failed animal tests but would also fail a later human clinical trial.
The proportion of drugs overall that fail animal tests

In other words, we ideally want a test for drugs that perfectly predicts the results of human clinical trials, as we don't want to do clinical trials that needlessly endanger people or might unacceptably delay them getting the best treatment. In the event we don't have that, we need to know four numbers to get a good picture of things - how many drugs failing one test would fail the other (which is a good thing), how many would pass one but fail the other (which would mean we are maybe giving those in clinical trial dangerous substances), how many would fail one but pass the other (meaning we've missed out on a potential cure) and how many pass both (which again is a good thing - these are the drugs that will go into common usage).

Similarly, other quotes above give one piece of information without giving corresponding pieces of information that are crucial in assessing the best options in pharmaceutical testing, and I would argue that "The best way to evaluate the effectiveness of animal tests for drug safety" is not "to compare their results with subsequent real- world outcomes in patients and consumers" but "to compare their results with other potential testing techniques".

I'm continually disturbed by the fact that the Safer Medicines Trust, which says "We focus on evidence based analysis of animal experimentation to assess the balance of help or harm to human health", so frequently writes pieces of publicity that fail to give all the necessary information to do this.

If you think animal testing is morally wrong, that's not something I can really argue with you about, but if you want to make a case it's outdated and there are better options I'd love to hear about them, but I'm usually disappointed by how the case is presented.

The Science Vote

2010-03-31T14:51:52Z

Recently, Michael Brooks, a journalist and author with a background in quantum physics, announced he will stand against David Tredinnick in the Bosworth constituency in the general election. This is because David Tredinnick has such famously irrational beliefs.

This has caused some controversy, as it may split the vote and so on, and Brooks is not a man local to Bosworth.

]]> thought about how they wanted to vote than have me tell them how, even if you reach the wrong conclusion. Having people think about things is a much more efficient strategy in the long term for getting them to make the right decisions.

On Twitter, the local Liberal Democrat activist @mathewhulbert has made clear he's not entirely happy with the situation. I've tried to indicate that a good strategy might be if the candidate he supports were to clearly support science and hence help to ensure he doesn't lose the science vote, but I had some difficulty doing that.

What I want to do here though, is explain a bit about why I think science is an important enough issue to consider when voting. There are indications that Mr Hulbert might not see things that way. When I quote in italics, it's something Hulbert has said. I may remove adjoining ellipses between tweets however.

"I'm sure it's all very worthy, Sir...but it is not top of the priorities for people of Hinckley and Bosworth. I don't know how to make it any clearer."

First off, I want to make clear that considering science as a voting issue does not mean it has to be the only voting issue, or a top priority voting issue. Just something you should think about. Hulbert frequently argued about issues important to local people, and its only right that there are a number of things people will prioritise above an explicit pro-science agenda. That doesn't exclude it from consideration at all however.

"It's all very good having pie in the sky, intellectual discussions...but people here in Hinckley and Bosworth are more concerned with jobs, local services, etc. Than some high moral debate, thanks all the same."

Amongst the terrible things Tredinnick does is support homeopathy on the NHS. The NHS provides a service to people. I don't think people want that service's resources squandered. It's a very practical matter, and not pie in the sky, intellectual discussion. He also wasted parliamentary resources on, famously, astrology software. He shows a lack of ability to make reasonable judgements on government spending at any level. I think that's a down-to-earth practical issue.

What about other local services? Education? Science feeds into education. Healthcare? I've already mentioned - science feeds into new healthcare developments that the people of Bosworth will benefit from. Jobs and industry? As Professor Brian Cox likes to point out, physics alone is reportedly responsible for 6.4% of GDP. Supporting science helps the economy.

But most importantly, the problem with Tredinnick and his cohorts who sign his EDM is that they are displaying an inability to use evidence well in their decision making, and this is a vital skill an MP should have in order to decide what policies to enact.

"Yeah, that plays big on the priority list of the people of Hinckley and Bosworth...over and above jobs, the economy, andti-social behaviour. I can't tell you how often I'm stopped in the street in Barwell and asked about quantum physics."

I'm disturbed by the suggestion that if you're not stopped on the street to discuss something, it can't be important. Quantum physics then. Not important I guess, it's just that airy-fairy intellectual thing where mad scientists in lab coats try to kill cats and make an electron do two things at once, right?

Well, it's also that thing that is behind all modern computing, telecommunications, just about anything electronic in your home and on the street. It's the thing that means that when a policeman has to call for support to deal with antisocial behaviour he can get his police radio out of his pocket or off his belt rather than having to walk several streets to find a big blue box, or so that he doesn't have to pull around a large trolley of batteries and vacuum tubes to get in touch with his colleagues.

Quantum mechanics is behind research that led to medical imaging, radiological health treatments and more.

It's behind the IT technology that improves education in your schools. It's in the electronics in your car, and it's behind a huge variety of products bought and sold in Bosworth on a daily basis.

There's fundamentally two major reasons why we need to know our MPs understand science.

1 - we need to know that our MPs understand the basic principles behind looking at evidence and using evidence to formulate models of the world and make decisions based on them.

2 - science and research are what lead to new innovations, new economic growth, make us a high technology country with a superb standard of living, with superb education, and incredible healthcare that would seem like magic 200 years ago.

There's also the minor point that even science research that doesn't lead to new technology that improves our lifestyle is still important. We improve ourselves by understanding the world better, and by stimulating our curiosity. I think our society is better for having invested in pure research for no direct benefit, just as it is better for having invested in art that has no direct benefit.

If you're in Bosworth, I'm not going to tell you to vote for Brooks. But I am going to ask you, wherever you are, to think hard about what opinions your candidate has, and whether you know if they're behind investing in the science that led us to live in the society we do today, and will lead us to live in an even better one tomorrow, and not squander our money on irrationality.

If you think that jobs and services in Bosworth don't depend on science, you're frankly an idiot. And if you think that anti-social behaviour isn't kept in check partly by the use of quantum mechanics then you might also want to have a second thought about that too.

On astrology

2010-03-26T10:35:03Z

I appear to have found myself engaged with @angelneptustar, an astrology fan and also a Boris Johnson fan.

This came about through Marsh's post at the Mersyside Skeptics site. I read through the HuffPo article in question (I refuse to link to that rag, but you can follow through to it if you really wish) where angelneptustar comments on Boris Johnson's astrology chart and how it shows he is very talented.

]]> Now, I'm not a fan of the Conservatives. As Conservatives go, I'm perhaps more amenable to our Boris. He communicates well (in his unique style) and gets people engaged and talking about politics, which is good. He does have notoriety as a bit of a buffoon, and I took advantage of that to make something of a jibe about 'the best argument against astrology ever'. Actually, I would be happy to say that Boris is very talented. The astrology chart post though, talks of him genius and high creativity, which is probably pushing it just a tad...

Anyway, the case of one chart and one notable person, subject to extreme selection, really tells us very little. It does tell us something, as if astrology asserted the patently false ("by this chart I can determine that Boris Johnson is actually a penguin") it'd obviously be evidence against, so the fact it's not completely misaligned might constitute extremely weak evidence, but nothing that would sway anyone from their current opinions on the matter.

Here are some of the things @angelneptustar has tweeted in my direction:

i did get it, I knew you were teasing me. I bet i could read your chart and descriBe you to a T.

Ed, have you studied astrology? i have, for 15 years. unless you have studied it, can you fairly commment?

your reply is silly. Would you commment on the validity of any subject, unless you had studied it? [my reply was to point to the Courtier's Reply]

I didn't accept astrology until I studied it in depth. Your remark is potty.

Still, Im sure you comment on politicis and you know nothing abut that subject either.

Now, on point 1. Clearly that's a bad way to test astrology. Anyone who has my Twitter account has my name, the link to this blog, and an ability to Google who I am and, as I make no particular effort to hide myself, find out an awful lot about me and the cynical old bugger I am. I pointed this out - "A fair test is not possible, but you knew that, right?" - it misses some details, but hey, it was a tweet limited to 140 characters.

On points 2 and 3, whether I can fairly comment on astrology. I think this is definitely a case for pulling out the Courtier's Reply, as mentioned. It's like criticising Ben Goldacre giving an opinion on homeopathy because he's not a homeopath. I don't need to understand the details of something to be able to fairly assess evidence for it and use that evidence to feed my 'belief engine' - the bit in my head that tells me whether I should change my mind on something. All I need to do is understand that someone has a model for how something should work, and understand if that model has been fairly tested, and how that model has been compared to observations. I think, being a well qualified scientist, that I can fairly comment.

Being a fairly long-time skeptic, and short-time magician taking advantage of people's profound ability to make errors at times, I also have a fairly good understanding of the kinds of errors that are made in assessing such evidence, outside those that usually afflict my particular area of science. So again, I think that makes me able to fairly assess a piece of evidence.

I should also say that my a priori belief in astrology is very low. It is inconsistent with my understanding of astronomy (which, angelneptustar, is considerable). Now me saying that we have no known mechanism by which astrology can work is not proof that it does not. The real watertight case is the lack of good evidence. But one should be aware that my current understanding of the world predisposes me to requiring quite extraordinary evidence for astrology in order to change my mind. This is an entirely rational stance to take.

On point 4, angelneptustar may well have not accepted astrology until studying it in depth. I have no particular reason to trust another person's ability to correctly assess the evidence, least of all if that someone chased up an a priori peculiar idea for however many years it takes to study it in depth before changing their mind. I think it doesn't take long to convince oneself that the current evidence doesn't support anything, and one should probably do something more productive or enjoyable. So sorry, angelneptustar, but while you might not accept my ability to fairly comment, I certainly don't trust yours. We clearly have different ways of thinking about this.

Point 5. Let me just add a few more tweets that have arrived while writing:

But you admit you know nothing abut astrology, which takes years of study.

I therefore dont take your remarks seriously, because you are totally ill informed.

But if you dont have to study astrology to know all about it, u r probably ignorant on most things.

And you are obviously not a political expert either.so yr blog cant be worth reading.
Now I'm not a political expert. I do know a bit about politics. I think politics is important, and I feel a responsibility to be some kind of informed voter. I am a member of a political party, write annoyed letters to my MP on various topics, and sometimes, but less often, happy letters to my MP. I try to keep up on the news, including that of political matters.

There certainly are areas I'm ill-informed on. Being ill-informed does not mean that I cannot make a good assessment of their validity. I am ill-informed on both the details of the nature of the polymerase chain reaction and also on details of voodoo rituals. I can make a good assessment despite that of which of those are founded in reality.

I take issue with the idea that I am "ignorant on most things", or "totally ill-informed". I think most people that know me know I'm not completely devoid of knowledge, or even close to devoid of knowledge (by human standards).

So let me come back to what I think is an important point about this discussion - that I must know the details of something to assess proof of its existence. It's a common criticism of skeptics that to assess astrology we must take complete and properly done charts and use those.

Now for one thing, it's still dead easy to test astrology under that premise (just like my previous post arguing that homeopathy can still be tested in a mass double-blinded trial with individualised remedies).

Secondly, by accumulating a lot of the right kind of data, statistics lets you tease out small signals and show up what would essentially be high-frequency changes in the data. I'd argue that it's actually the very best way to do it - that by numerical analysis you obtain a strong signal faster than by 'doing it by eye'.

The problem with this is that to do this, you practically by definition have already accumulated strong evidence that at least some component of astrology works, and you'd have everyone convinced. This hasn't happened.

If one actually needed a model of astrology of that complexity to determine if it worked at all, astrology would be nigh on impossible to discover. You have to build up such a wealth of data to determine the nature of the signal that it would be virtually impossible today, let alone in the time of Ptolemy or whatever (when he also didn't have the powerful statistical tools we can leverage today, or the computers to help analyse all the data flowing in, or the scientific protocols needed to do so fairly). By the very nature of a detailed and complex model, you need more data to determine if it is correct and what its parameters are. If that model worked, you'd also expect that there was some simplified version that operated with a few less parameters to get less good results, but was still unambiguously more powerful than the null hypothesis that astrology does not work in the slightest.

It's like seeing some shape in the mist and wondering if its an elephant, and having someone declaring you can't tell if its an elephant until you've measured its trunk. You'll certainly be in the position to tell whether there's a mammal of considerable size in your vicinity by the time you have got a good enough view to check the size of its nose.

As I said, this points to a big problem with the claimed history of astrology. How on earth could you determine a complex model of astrology way back in the ancient past? How on earth could we still be in the situation where evidence for astrology is anything less than absolutely unambiguous and impossible to deny, when it was possible to fit a multi-parameter model to it thousands of years ago? It isn't.

The historical nature of astrology, and its weak evidence base today both stand in absolute disagreement with the idea that astrology is supported but that you need to understand its complexity to assess it fairly.

Unfortunately angelneptustar has declared this blog not worth reading a priori, just as I've declared astrology bunkum a priori (although naturally I think I can make a stronger case for the latter than she can for the former), so this will likely be preaching to the choir.

Not a blog post

2010-03-15T10:40:00Z

This is not a blog post. It is an instruction to anyone self-identifying as a skeptic to read this.

To Columbo's Rescue

2010-03-11T11:54:55Z

Browsing my arxiv feeds this morning, I picked up this post - "A defense of Columbo (and of the use of Bayesian inference in forensics): A multilevel introduction to probabilistic reasoning" by G. D'Agostini.

Considering the topic, it's an enjoyable read. It's motivated by this NewScientist article from late last year. I criticised that article myself but did not touch upon the incident with Columbo. D'Agostini provides a fairly clear if mildly technical explanation of the mathematics behind the problem expressed - of the strength of the evidence against the killer Columbo caught when he picked one of thirteen (or twelve, as the NewScientist article says, but that's a minor niggle) cameras off a shelf, taking the one involved in the crime. As NewScientist says:

If only it were that simple. Killer or not, anyone would have a 1 in 12 chance of picking the same camera at random. That kind of evidence would never stand up in court.

]]> He also covers how we combine independent lines of evidence and how that combination should affect our judgement. If you're willing to withstand the relatively high density of equations, it's not a bad read at all.

Columbo's actual defence however, comes in section 4 and 5. He makes several reasonable points.

This was not Columbo's sole line of evidence.
Not all of Columbo's personal evidence will be admissable. His judgement will naturally differ from the court's as a result, but his role is to present a sufficiently strong case to the court, so he will require additional evidence beyond that required to convince him.
Even if there was a 1 in 13 chance of an innocent person picking the camera, it still constitutes weak evidence that can be added to the case.
It wasn't 1 in 13. A wealth of surrounding detail, unmentioned by NewScientist, demonstrates this (section 5 goes into great detail on this).

I think there's a couple of interesting points to draw out from this. First off, is the importance of a detailed expression of your experimental protocol when doing science. If you leave off all the details of how you induce a murderer to pick a camera, one is left with the impression that Columbo came along to someone and said, like some kind of scruffy cigar-smoking Paul Daniels "Pick a camera, any camera". The details very easily change the amount a piece of evidence can sway ones judgement. To interpret something properly, you need to be as informed as possible.

As D'Agostini says "Evidence is not only the 'bare fact', but also all available information about it".

Secondly, there are interesting points to be made about weak evidence, a topic I keep touching upon. Usually I'm very critical of weak evidence - that weak evidence might as well be no evidence at all. This is actually not quite right.

If I toss a coin and it comes up heads, that is very weak evidence that the coin is biased. If I combine a million pieces of such evidence - performing a million coin tosses, my ability to detect a bias in a coin naturally skyrockets. Combining weak evidence is fine, as long as you do it properly.

Problems often occur due to reporting biases however. If I read a study and it constitutes very weak evidence that an antidepressant works better than placebo, for example, that's not good enough. If I read ten studies it might be the case that the ten studies combined are actually good evidence together, even if they're weak apart. That's one reason why metaanalyses are good and the Cochrane Collaboration is so cuddly. However, it might also be that bad studies go unreported, and the evidence they would have provided is crucial to a fair assessment (another reason that the Cochrane Collaboration is so cuddly is that generally they understand these issues and not only do what they do, but do it well). It's like, if instead of doing a million coin tosses and counting how many came up heads, I just asked people to send me reports of their coin tosses if they came up heads.

That's another reason why watching the full episode of Columbo is more convincing than the NewScientist paragraph. We have a chance to get, presumably, a fair idea of what Columbo is up to, and he's not doing some kind of Derren Brown System where he starts off with a city full of people and narrows it down to one unlucky person who all the evidence happens to point against.

So weak evidence is only good when we collect it in a fair manner. It doesn't have zero value, but it has so little as to not be worth bothering with.

It all goes to show you have to know as much as you can about how evidence was collected, as well as how strong it is. You need to know if it's actually stronger than it appears on the surface, as it is with Columbo's camera test, or if it's much weaker than it appears on the surface, as with a weak study that may well be misinterpreted if you don't remember to account for publication bias.

Pascal's Wager

2010-03-08T12:00:31Z

I've long held that Pascal's Wager is a completely absurd argument. Most people will know what Pascal's Wager is, but it's loosely speaking the argument that you should believe in God because doing so offers the possibility of infinite reward, whereas not doing so offers no reward at all.

There are obvious flaws in it, which apparently even Pascal recognised. The primary one I think, is that the reward for believing something does not lead to a mechanism by which you can genuinely accept that belief. If you offer me a £10,000,000 to believe that pigs can fly, I will not be able to start believing that pigs can fly just to gain a benefit.

For this obvious reason, if anyone mentions the Wager in any context other than an example of dubious logic, treat anything else they say with caution.

However, Pascal's Wager is not really the object of this post. What I'm going to discuss is instead (despite the blog title) Pascal's Mugging.

]]> Nick Bostrom's post here.

Pascal's Mugging rephrases things so that we operate purely in terms of the payoff. We don't actually have to believe anything with any certainty, we merely have to offer enough credence to a proposition that it becomes a reasonable bet - in much the same way that paying £1 for a lottery ticket that has a 1 in 10,000,000 chance of paying out £100,000,000 is a reasonable thing to do, even though you never believe you will win that lottery.

Now, I think Pascal's Mugging contains a hidden flaw, in that you are asked to specify some credence for a possibility of some level of payout. If that payout is too small for that credence, so that you do not consider it a reasonable bet, the payout is increased until (the argument goes) that bet does become reasonable. I would argue that as the payout is increased, your credence must also decrease, and just because the payout is unbounded does not mean that it ever becomes a reasonable possibility. Take as an example the idea that I have a level of belief in an ability to payout an amount £x as 1/(10 x). My expected payoff is then always 1/10, and if I'm staking an amount greater than that it's never worth my while to pay, no matter what reward is offered. I could equally construct a credence function that decreases my expected payoff as x increases, no matter its size.

There's another excellent post on the matter here.

I would argue that if you ever fell for Pascal's Mugging then practically by definition your system of assigning credences is deeply flawed. I suspect it's possible to find an argument that constructs this in something like the context of Dutch Books - the idea that to have a consistent way of assigning beliefs you should never open yourself to wanting to engage in gambling that will leave you open to a guaranteed loss.

To finish up, lets look at a situation in contrast to both Pascal's Wager and Pascal's Mugging - the issue of climate change. If we consider the possibilities involved in the climate change debate, we either accept considerable economic penalty for the possibility of saving the planet or not, or we go on as we are and either come out doing well, or end up having incurred an even greater penalty for continuing to consume resources and pollute as we currently are.

It's not entirely unlike the cost a religious person accepts (in terms of lost Sunday mornings or whatever) for the possibility of heaven, or spending your Sunday mornings in bed or doing something else you enjoy and risk ending up in hell. Now the relative probabilities and relative benefits and penalties are very different, but it again makes the difference to Pascal's Wager clear. Whether the planet gets destroyed or not depends on my actions, whereas my ability to get past the Pearly Gates depends perhaps partly on my actions, but predominantly on my beliefs. I can hold the view that anthropogenic climate change is real or not, but it does not directly impact the end condition of the planet - my carbon footprint does.

It also differs from Pascal's Mugging, in that unlike the latter where payoffs are allowed to change without limit, the payoffs in climate change are pretty much fixed and limited (even if potentially extreme). Our credence in climate change is also detached from the payoffs themselves. It's a case with superficial similarity, but in fact I would argue it's quite possible and quite likely that a strong case can be made that we should undergo the economic penalty in return for avoiding the much more severe cost if climate change occurs. Even if you don't believe as I do that climate change is real, the argument for acting as if it is is strong, and doesn't depend on you actually accepting as likely the possibility of climate change itself.

On the BHA's response - part 2

2010-03-08T10:16:36Z

The BHA have posted the final two parts of their response to the S&TC Evidence Check. As I covered the first parts (to a greater or lesser extent, I skipped one section entirely), I thought I should continue to look at the final two.]]> Part 5.

25. We are concerned that homeopathic products were, and continued to be, exempted from the requirement for evidence of efficacy and have been allowed to continue holding Product Licences of Right. We recommend that no PLRs for homeopathic products are renewed beyond 2013. (Paragraph 121)

Response:
See our response to Recommendation 26 below.

26. We conclude that the MHRA should seek evidence of efficacy to the same standard for all the products examined for licensing which make medical claims and we recommend that the MHRA remove all references to homeopathic provings from its guidance other than to make it clear that they are not evidence of efficacy. (Paragraph 128)

Response:
Homeopathic medicines are safe, there is evidence of their effectiveness, and there is considerable public demand and traditional use. In these circumstances it would be oppressive for the state to take draconian measures to restrict their availability, which is what these recommendations imply. Such measures would in any case be ineffective; the main net effect would be to drive the market on to the Internet. Homeopathic medicines are widely available in the EU and it would be illegal to restrict Internet purchases of them in the UK.

I think we've already made the case that the first section is wrong. Although homeopathic medicines are chemically safe, I would argue that they are a harm to society in other ways. There is also no good evidence of effectiveness, although I concede there is considerable public demand for them. Given that a large portion of this premise is wrong, the line "In these circumstances it would be oppressive for the state to take draconian measures to restrict their availability, which is what these recommendations imply." is easily countered simply by realising that these circumstances do not exist. It's not even necessary to consider whether the S&TC recommendations are draconian, which I do not believe they are. I'd also argue it's pretty hard to come up with really draconian legislation for licensing healthcare products - it's one of those areas you want rigorous legislation to protect the consumer.

27. We consider that the MHRA's consultation, which led to the introduction of the NRS, was flawed and we remain unconvinced that the NRS was designed with a public health rationale. (Paragraph 135)

Response:
The legislation was enacted by due process, including an extended consultation period.

I will leave this to more legal minds than mine, but something enacted in a legal manner and with an extended consultation period is not necessarily correct and free of flaws.

28. We fail to see why the label test design should be acceptable to the MHRA given that, first, it considers that homeopathic products have no effect beyond placebo and, second, Arnica Montana 30C contains no active ingredient and there is no scientific evidence that it has been demonstrated to be efficacious. We conclude that the user-testing of the Arnica Montana 30C label was poorly designed with parts of the test actively misleading participants. In our view the MHRA's testing of the public's understanding of the labelling of homeopathic products is defective. (Paragraph 140)

Response:
It is factually incorrect to state there is "no scientific evidence" that Arnica montana, diluted beyond Avogadro's constant, has been demonstrated to be efficacious. The following RCTs have reported positive findings:

I've previously and repeatedly (to the point of it being tiresome) made the point that bad or weak evidence is no evidence at all. I could produce scientific evidence that the entire executive of the BHA are penguins, and technically claim that it is factually incorrect to state that the evidence does not exist. It does - but it is weak and supports a claim that vast amounts of other evidence contradicts and outweighs decisively.

30. We consider that the way to deal with the sale of homeopathic products is to remove any medical claim and any implied endorsement of efficacy by the MHRA--other than where its evidential standards used to assess conventional medicines have been met--and for the labelling to make it explicit that there is no scientific evidence that homeopathic products work beyond the placebo effect. (Paragraph 146)

Response:
See our response to Recommendation 32 below.

The response to recommendation 32 is again that claim there is evidence. I won't bother expanding on that, but I think it is interesting that the quoted section of the report shows clearly that the S&TC report is not draconian - it recommends a change of labelling and licensing in order to inform people, not to completely outlaw homeopathy.

Part 6.

I'm not going to say anything about this next quoted section. I'm just going to emphasise one bit of it, for humour value for those in the know. The emphasis is most definitely mine.

33. By providing homeopathy on the NHS and allowing MHRA licensing of products which subsequently appear on pharmacy shelves, the Government runs the risk of endorsing homeopathy as an efficacious system of medicine. To maintain patient trust, choice and safety, the Government should not endorse the use of placebo treatments, including homeopathy. Homeopathy should not be funded on the NHS and the MHRA should stop licensing homeopathic products. (Paragraph 157)

Response:
Homeopathy is more than a placebo and rightfully belongs in the NHS where patients can best benefit from doctors integrating it into healthcare.

This report and its conclusions represent a rush to judgment, reflecting the narrow and cursory nature of the review. It was systematic only in excluding facts that tend to support homeopathy: it omits or misrepresents any research evidence (including the BHA's), which challenges the view that patients' response to homeopathy is due to placebo. Its conclusions are unsustainable in the light of scientific evidence.

Moving on.

Even more disturbing is the dismissive manner in which the committee deals with the healthcare of patients and their response to homeopathic treatment. Patient-reported outcome measures (PROMs) are increasingly seen by the NHS as a critical component in assessing healthcare interventions. The NHS homeopathic hospitals have excellent PROMs results.

PROMs may be a useful measure of when things in the NHS aren't working right. It does not follow that PROMs are a useful measure of when things are working right. To draw a ridiculously extreme analogy, suppose I am in the business of employing teachers to work in schools. I consider criminal records checks as a 'critical component in assessing teacher suitability'. It does not follow that because a school employs no teachers with a criminal record, that the teachers it employs are excellent or even remotely qualified to teach.

This narrow-minded and illiberal report is highly tendentious, consistently misrepresenting the scientific evidence to denigrate homeopathy, and making unfounded and pejorative allegations against those who advocate, practice or develop research in homeopathy. Repeatedly asserting that it is only placebo does not make that assertion true.

I personally don't find it in the slightest bit illiberal. I think it is narrow-minded only in the sense that it isn't so broad-minded as to include things that aren't true. While repeatedly asserting that homeopathy is only placebo does not make it true, repeatedly claiming there is evidence for homeopathy (even when you list references) equally does not make it true.

These last two sections are not quite so dire as the preceding ones, but the BHA response is still fundamentally founded on the incorrect idea that there is enough evidence for homeopathy to support its outlandish claims. It's down to homeopaths to support and legitimise their claims. Wake me up when they've won their Nobel Prize.

Paradoxes - The Meta-Newcomb Paradox

2010-03-05T17:24:06Z

It's nearly the weekend, and I need to give my colleagues something to keep their brains working while they're away from their desk. So...

It's been a week since you encountered Newcomb's Predictor. You, being a classic two-boxer personality, opted to take both boxes, and although mildly disappointed to find you won only £10,000, you felt satisfied in your choice. One cannot always win such games.

However, as you are walking home one day you find yourself before the Predictor once more. He explains how he's had a cunning idea. This time round, he's either made a prediction and filled the boxes appropriately, or he hasn't gone to the effort and will wait till you announce your choice and then surreptitiously fill the boxes as necessary. The Predictor does these options with equal frequency.

As you contemplate this, you receive a tap on the shoulder. You turn round to find yourself faced with yet another Being, who introduces himself as the Meta-Predictor. He can not only predict your actions, but he also has a great track record of predicting the actions of the Predictor. The Meta-Predictor tells you this:

"You will either choose A and B, and the Predictor will fill the boxes after the fact, or you will opt to take only box B, and the Predictor has already made his decision and the boxes already contain the right amount of money."

Puzzled, you contemplate matters a little further...

]]>

If you're a two-boxer however, you might have a problem. Maybe your actions really will affect the box contents. If you were going to take two boxes, you may find your choice will have a causal effect on the future. In that case, you should take one box, as the Predictor will then fill it with loads of money. If you're going to take one box though, it seems likely that the Predictor hasn't made his move, and nothing you do will affect the contents, so maybe you should take two boxes after all?

This one is due to Nick Bostrom.

Personally I think there might be an internal inconsistency here - are the statements about the frequencies with which the Predictor takes each path correct? Perhaps they're globally correct, but not right for this particular case where the Meta-Predictor makes his statement (he's perhaps screwed with the probabilities in a manner not entirely unlike Monty Hall). If not, perhaps the Meta-Predictor's statement changes things and there is already in effect a decision made which the Predictor is inexorably constrained to follow - in a sense the fact the boxes aren't already filled doesn't matter, as the fact he will choose to fill the boxes later is already laid out.

It's an interesting twist, but I think the really puzzling stuff still lies in the conventional Newcomb situation.

Paradoxes - Sleeping Beauty meets Newcomb's Predictor

2010-03-04T18:27:00Z

So I've previously blogged on the Sleeping Beauty problem as well as on Newcomb's Paradox. The latter has caused some considerable discussion amongst myself and a couple of colleagues, and to my delight we don't all agree on it.

I'll probably post further on it as the discussion develops, but I stumbled across an interesting overlap between the two.

In order for Newcomb's Predictor to do the prediction, he must presumably either read some memory of previous discussion (so poor old Chaz, having been introduced to it and found himself a two-boxer is doomed to not win the £10 million!) or somehow simulate the event.

Being a fairly materialist person, I strongly suspect that a simulation of myself would be self-aware just as I am. In that case, on meeting the Predictor and being presented with two boxes, I have no idea if I am really me or a simulated me for the purposes of the Prediction.

In that case, I am rather likely Sleeping Beauty, about to have her memory wiped.

I may make a choice only to cease to be, with that choice acting causally forwards in time in order to allow the apparent violation of causality that would allow the Predictor to act as described in the problem.

In that case, although I may be a firm believer in causality and that future actions cannot affect the past, I may have good reason to believe that in this particular circumstance my actions may well effectively do that.

It's an interesting crossover, but really this little curiosity is either a sidetrack or a weakener of the core issues that make this such a puzzler.

On the BHA's response

2010-03-03T14:01:37Z

Not so long ago, the Science and Technology delivered their report on homeopathy. It wasn't to homeopaths tastes. If you haven't heard of the story, I recommend Martin Robbins' piece for the Guardian.

The British Homeopathic Association has felt the need to respond. I thought I would take a look at parts of it.

]]> First off, lets take Part 1 - I'm something of a fan of numerical ordering.

They take one paragraph of the report:

1. We recommend that the Government determine the total amount of money spent by the NHS on homeopathy annually over the past 10 years, differentiating homeopathic products, patient referrals and maintenance and refurbishment of homeopathic hospitals, and publish the figures. (Paragraph 15)

and follow this up with

Response: The total spending is of little relevance without an indication of the benefits. We would welcome an analysis of the costs and benefits of the work of the NHS homeopathic hospitals and of GPs who integrate homeopathy in practice. It is important to note, as the Science and Technology Committee has failed to do, that the NHS homeopathic hospitals offer more than homeopathy.

Now it seems to me pretty strange to launch into saying its of little relevance without an indication of the benefits, when a fair chunk of the report (available here) sets out exactly that, or how to do that.

The rest of Part 1 of the response is frankly fairly bland and does nothing in my view to justify any further spending on homeopathy. But read it yourself, and make your own mind up.

As for Part 2, honestly I'm just going to skip that. It's unconvincing to me, and go and read some of Martin Robbins other excellent dissection of the evidence base at layscience.net - such as this piece. He's a man of infinitely more patience than me, when it comes to sorting this stuff out, clearly. Suffice to say I've previously argued why, in the unlikely event there's ever decent evidence for homeopathy it'll have to be so groundbreaking you'll basically read about it in Nature, and unless it gets that recognition I feel entirely justified in dismissing claims on the basis that life is too short to waste trawling through weak evidence from bad studies.

Part 3 is largely similar, but it has a few bits I'd like to highlight.

Here's a bit they quote from the S&TC report:

11. In our view, the systematic reviews and meta-analyses conclusively demonstrate that homeopathic products perform no better than placebos. (Paragraph 70)

and their response

The BHA does not share the committee's conclusions from systematic reviews and meta-analyses.

This is hardly surprising now is it? I mean, it's like if I said "the Earth is round" and the Flat Earth Society issued a statement saying "The FES does not share Mr Edmondson's views". If you did have the opinion that there's zero evidence for homeopathy and that it doesn't do diddly beyond placebo, you'd hardly be a senior member of the BHA would you?

As for

Comprehensive systematic reviews overall are not conclusive either way; the majority have reached the conclusion, qualified by a number of caveats, that homeopathy differs from placebo.

this would be caveats like "because most trials are of low methodological quality and because of the unknown role of publication bias." would it?

Moving on

There is no good reason to accept, as final authority, the views of one opinion leader in homeopathy research such as Professor Ernst over those of his peers, such as Dr Mathie or Dr Peter Fisher.

Well that's true to an extent. If it were a straight run off between Professor Ernst and Dr Fisher you might have to remain undecided, but it isn't a straight run off. You see, Professor Ernst is on the side of probably most doctors and almost certainly most chemists and a significant chunk of the rest of the scientific community. Professor Ernst also takes the view that an implausible thing with no evidence for it is probably wrong. We're not arguing two equal sides - homeopathy is deeply implausible a priori and you can't justify a claim for balance between two opposing views in that situation. One view is just flat out crazy.

The last piece I will pick out from Part 3 is this:

17. We recommend that the Department of Health circulate NHS West Kent's review of the commissioning of homeopathy to those PCTs with homeopathic hospitals within their areas. It should recommend that they also conduct reviews as a matter of urgency, to determine whether spending money on homeopathy is cost effective in the context of competing priorities. (Paragraph 86)

Response:
We find it wholly unacceptable that a parliamentary committee whose remit is science and technology is commenting and making recommendations on local provision of health services.

I find it wholly acceptable - whether a treatment works or not is fundamentally a scientific question. And many medical treatments or diagnostic tools have a basis in technology. To quote the S&TC website "The Committee exists to ensure that Government policy and decision-making are based on good scientific and engineering advice and evidence". That's exactly what they're doing and they're doing a pretty good job, at least in this case.

So, finally, Part 4. This is what caught my eye first, and motivated me to write this.

18. We accept that NICE has a large queue of drugs to evaluate and that it may have greater priorities than evaluating homeopathy. However, we cannot understand why the lack of an evidence base for homeopathy might prevent NICE evaluating it but not prevent the NHS spending money on it. This position is not logical. (Paragraph 90)

Response:
NICE usually reviews the use of specific interventions for particular conditions or groups of conditions, or issues guidelines making treatment recommendations for conditions or groups of conditions. We are not aware that NICE has ever reviewed, for instance, antibiotics or a complementary therapy such as acupuncture, as a whole. It is not clear why the Committee feels it should single out homeopathy in this way. It would be more appropriate for NICE to make recommendations regarding the use of homeopathy - as it has, for instance, for acupuncture in the treatment of low back pain - in the context of condition-specific guidelines. An obvious starting point would be the specific conditions that have been the subject of systematic reviews and meta-analyses of homeopathy.

In the case of back pain, I seem to recall it was as they say, a condition-specific guideline review, so it presumably also covered physiotherapy, chiropractic and other treatments. There's something of an inconsistency perhaps, but not one that troubles me greatly.

Let me suggest, however, why NICE might evaluate certain things before others and might never bother to look at homeopathy. It's actually highlighted by the BHA themselves, although in a rather backward manner, right below. Homeopathy has been around for 200 years. It's never had good evidence to support it when if it were effective it should have been easy to produce it. In contrast, a drug that has just been released generally does have evidence to support it and has been around for a lot less time, and one might have good reason to critically review that evidence (especially when that evidence has come from the people selling it).

Personally I think NICE has plenty to be doing and enough reason to be careful with its budget that it shouldn't touch homeopathy with a 10⁶⁰ foot barge pole.

Next is this cracker:

19. When doctors prescribe placebos, they risk damaging the trust that exists between them and their patients. (Paragraph 97)

Response:
This point is entirely speculative and without any foundation in evidence.

To translate, "Lying is bad. Response: Ahhh but do you have any evidence for that or is it just pure speculation?"
I won't dignify it with any further response.

20. For patient choice to be real choice, patients must be adequately informed to understand the implications of treatments. For homeopathy this would certainly require an explanation that homeopathy is a placebo. When this is not done, patient choice is meaningless. When it is done, the effectiveness of the placebo--that is, homeopathy--may be diminished. We argue that the provision of homeopathy on the NHS, in effect, diminishes, not increases, informed patient choice. (Paragraph 101)

Response:
The evidence does not support the view that homeopathy is a placebo. It is perverse to claim that preventing patients from accessing homeopathy on the NHS in some way increases their choice.

OK, this is a subtle point, but I think the BHA really should have sat down to think it through. Preventing patients from accessing homeopathy on the NHS does not increase their overall choice. The point is that if you do it, you have to deny patients access to the information that homeopathy does not work and probably equivalently have to deny them access to other information to keep the playing field level (or you get patients asking why homeopathy doesn't come with a big piece of paper covered in small print about possible side effects) and that therefore you reduce their informed choice - not their net choice.

Fundamentally I find their response weak and at times laughable, if healthcare were a laughing matter, which it isn't (the positive and more direct benefits to health from laughter excepted). The S&TC did a great job, and I just hope the government takes on board their recommendations.

Paradoxes - Newcomb's Paradox

2010-03-02T23:22:10Z

It's high time I posted another paradox blog. This time, it's my favourite - Newcomb's Paradox.

Now before you click that link, first read through the description and decide what you would do in this circumstance and why:

You have been taken by what is clearly a highly advanced intelligence. It obviously has either sophisticated technology or magic, based on the circumstances in which you have found yourself talking to it. It makes an announcement.

"I mean you no harm. I just wish to play a game. In front of you are two boxes, which for your convenience I have labelled A and B. Box A contains £10,000 - an amount I believe you will find significant and tempting, but not quite life-changing.

"Box B may or may not contain £10,000,000 - an amount that quite certainly is life-changing. Whether I have placed that sum in box B is not random. I have used my vast and inestimable powers to study your mind before I brought you here, and I have made a prediction about your behaviour.

"Your two options are to take the contents of both boxes, or to take merely the contents of box B. That is you may take A plus B, or B alone.

"If I believe you will take only box B, I have placed the £10,000,000 in that box. If I believe you will take both boxes, I have left box B empty.

"I have played this game a thousand times before with other people, and I have never been wrong in my prediction.

"If I believe you will try to be awkward and make your decision using, for example, chance, then I will have left box B empty.

"Choose now whether to take just box B, or to take both."

What do you do? Think about it for a moment before reading on.

]]> - taking both boxes will net us £10,000 with a probability of 0.9999 and £10,010,000 with a probability of 0.0001. A pay off of £11,000 expected.

- taking just box B will net us 0 with a probability of 0.0001 and £10,000,000 with a probability of 0.9999 - a pay off of £9,999,000.

Clearly it is in our favour to take just box B.

Alternatively consider this argument - whatever you do now cannot have an influence back in time. The contents of the boxes are fixed. So just take whatever is there and go on your way happy with whatever you have earnt.

The Wikipedia link goes into some detail of the discussions around this. Suffice to say, I have no entirely satisfactory argument for one over the other.

Which option would you go with, and can you explain why?