I had the great pleasure a few nights ago to see Derren Brown‘s new illusionist / mentalist, hypnotist show Svengali. It’s fantastic, and highly recommended. If you’ve seen any of Derren’s previous shows on TV, then some of the routines will be familiar. This fails to make them any less baffling. If you’re unfamiliar with his work, here’s a sample:
(Here’s a bit more). One of the main themes of much of Brown’s work is his ability to recreate the “powers” of psychics, mind-readers and spiritualists without the pretence of supernatural intervention or paranormal activity. For example, in 2004 he performed a seance “live” on channel 4, and in 2011 trained a member of the British public to become a faith healer.
There is an important and quite general lesson to be learned from Brown’s abilities. In the course of last night’s performance, Brown did a number of things which, if they had been performed by someone claiming psychic powers, would seem, if not totally convincing, at least on the way to suggesting psychic powers. I remain at a complete loss as to how Brown seems to read the minds of audience members and anticipate their seemingly free choices.
Suppose that Connie claims to be a witch – a real, proper, supernatural witch – and as proof of her powers, performs a great feat of mind-reading. Being the mathematical nerds that we are, we decide to formalise our inference that Connie is a witch (and should thus be burned). Help us, Rev. Bayes!
Let:
Now, what is the probability that Connie is a witch, given that she just read that girl’s mind:
Three of these factors are fairly straightforward.
is the prior probability that Connie is a witch, that is, how likely was it that Connie was a witch before we saw her totally really that girl’s mind. We’d probably set that number to be small.
is the probability that Connie could read that girl’s mind, given that she really is a witch. We’ll assume that witches can do that kind of thing, so this number is fairly high.
is “not-W” i.e. Connie is not a witch. Thus,
was set when we set the prior, since
.
Here’s the important bit: the other term! 
Here, then, is the Bayesian utility of Derren Brown. If I had to guess what 
I’m not a fan of “skepticism” as a rational slogan. Of course, these terms are open to interpretation, but skepticism struggles to distinguish itself from denialism, the idea that not-believing is somehow the superior intellectual pose. I do, however, support the following principle – call it a form of skepticism if you like. Given the claim “evidence E supports/proves the hypothesis H”, one must consider alternatives to H. One must be skeptical of H, assume for a moment that it is false, and see if some other way of explaining the evidence E is available and plausible. The fact that all the evidence fits with my favourite theory H is not sufficient to show that the evidence makes my theory probable.
(There is a loophole, you may have noticed. Perhaps Brown really is a witch, but is just pretending to be an illusionist. We’ll need some scales, and a duck …)
Letters to Nature 
Luke, I agree with you that “one must consider alternatives to H”. So surely there are a couple more possibilities, both for Connie and for Brown.
1. Supernatural power (Brown could be using it, perhaps unaware of it, as you note).
2. Trickery (the obvious choice, but we need to consider whether Connie has the sophistication necessary to trick people like Brown does).
3. Genuinely natural, but as yet not fully understood, psychic powers (again, both Brown & Connie could be using this without being fully aware).
4. Amazing coincidence. (We may think coincidence is too far-fetched, but some people seem to think the universe’s fine-tuning can be explained by coincidence, and the odds against that are way longer than what you are describing here.)
I don’t think the addition of these extra possibilities changes your Bayesian analysis of Connie’s supernatural powers much, but it makes outright scepticism a little less likely I would think. Which should make us all a little more humble.
Derren Brown is a good mentalist. Interestingly, his “act” is that he’s (of course) not psychic, just really good at psychology. Which is just as much BS, but some people believe it.
I’m glad you enjoy Derren’s work. One of his main creative partners is a good family friends of ours and I can assure you witch craft is the explanation.
Seriously Derren is a superb mentalist and a brilliant close up card magician, there are few rivals in my opinion. But one is Banacheck you should look him up here:
http://en.wikipedia.org/wiki/Banachek
He’s considered something of a legend in the art of mental magic. He convinced physicists that he had real Pk powers. Banacheck administrates a prize for anyone that claims super natural powers for real. $1m is on offer and at a conference I recently attended they did a live challenge (the claimant, a manufacturer of those silly balance wrist bands lost but I admired Banachecks respectful way of administrating the test). I thought what Banacheck and friends (including Derren Brown) have shown is that magicians are often far better able to assess the truth of psychic claim than scientists. If it were my $1m Id sooner trust it to people that know the dececption techniques and how to spot them, than someone who knows Bayes theorem.
The conference was for a sceptic’s organisation. I don’t think scepticism has any similarity to denialism. Scepticism is simply having a high bar for accepting objective claims. A good summary of the modern movement here:
http://en.wikipedia.org/wiki/Scientific_skepticism
Denialism is refusing to accept the evidence once it reaches that high bar.
Having been involved in the sceptic movement for some time I hope I can give a good commentary here. Most sceptics were sceptical of climate change until about 10-15 years ago, and then the movement changed to accept the data and even attack climate change denialists. Going to many sceptics events such as sceptics in the pub, the Amazing meeting etc I don’t think Ive ever met a single climate change there denialist in years. The problem is denialists often call themselves sceptics, but that’s a just a smokescreen; and one that’s pretty easy to see through to a even half trained eye. Just as there are pseudo scientists there also pseudo sceptics. Just as pseudo scientists shouldn’t tarnish real scientists, the same with pseudo sceptics. In my opinion if you are not a (genuine) sceptic (requiring strong evidence before endorsing an objective claim) you shouldn’t be doing science, it’s the high bar for acceptance that gives science its credibility, hence scepticism is a crucial part of science.
I must admit that I am quite sceptical regarding whether Brown actually even does the tricks he claims to do. That is, sceptical of his claims that he does not use actors and just uses psychology. My evidences:
1. In the episode where he apparently uses a ring bound in a voodoo doll to control a woman’s actions, that woman is an actor and her resume on imdb listed that Derren Brown episode (http://www.imdb.com/name/nm2052141/resume – no longer present here, but is on archive – http://web.archive.org/web/20070701115418/http://www.newfacestalent.co.uk/clients.php?n=Magda+Rodriguez&i=97 – asked to remove it, I suspect)
2. In the case where Derren Brown claims to stare down people (http://www.youtube.com/watch?v=qEuKgEjwi6A), the guy who appears at 1:30 featured himself in another video (now removed – http://www.youtube.com/watch?v=pEId-hdjerw) claiming that he was acting
3. Ignoring all the above, consider this ‘trick’ – http://www.youtube.com/watch?v=ACp5nTlpv6c. Ask yourself, in what possible way could this be legal, permitted, or moral? If I was the ‘victim’ I would be outraged (to put it mildly). If it were my particular friends, every one of them would likewise be outraged to be manipulated in such a way, and so I would say something. The whole scenario just beggars belief. The most plausible explanation seems to me to be quite clearly that he uses actors. Anything else would be immoral, illegal, and frankly implausible.
Honestly, if I’m right, and he does use actors, you might wonder why I would post that here anyway. Surely there is a lesson to be learned here, and he does an amazing job at encouraging people to be sceptical of supernatural and other preposterous claims, whether or not he uses actors. And if he does use actors, he has shown that purported ‘sceptics’ can also be fooled into believing grossly overstated powers of psychology. Isn’t that a good thing? I don’t think so. If we want to inform people so they are more discerning regarding purported supernatural powers, surely we want people to be just as discerning about psychological claims. Otherwise we are just replacing gullibility regarding one falsity for another. The second lesson is not being taught. Both types of gullibility irritate me.
I’m guessing some of his tricks are genuinely what they purport to be – mind/manipulation tricks that don’t make use of actors. But I can’t tell which do and which don’t. For me, the illusion of Derren Brown is ruined.
One notational remark on Bayes. (Yeah, I know, I’m commenting on a really old post. Sorry.)
The version of the formula that Luke gives here is indeed standard but I think it’s usually neater to work with odds ratios rather than probabilities as such. Then the theorem looks like this:
Pr(W|R)/Pr(!W|R) = Pr(W)/Pr(!W) . Pr(R|W)/Pr(R|!W).
In other words: the odds ratio after getting new evidence is the odds ratio before, times the ratio of the likelihoods. I find this form much easier to remember correctly and write down correctly. In the present case the things whose odds we’re comparing — W and !W — happen to be exclusive and exhaustive, but the formula looks just the same when they aren’t, or (once you choose a good notation) when you have more than two things to compare. And you can do the whole thing without ever explicitly computing any actual probabilities; all you need care about is ratios of probabilities and ratios of likelihoods.
So then Luke’s observation is that since apparently Pr(R|W) and Pr(R|!W) aren’t very different, the posterior odds are similar to the prior odds. Simple!
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