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Yay, another maths lecture!

Click through to see the whole cartoon at XKCD. Really do it. It’s important. Especially if you want the rest of my burblings to make sense.

So. It’s partly funny because it satirises the sensationalism of tabloid news, and the urge to cram as much excitement into a headline as possible only to leave a sober assessment of actual facts to the blogosphere. But it actually addresses a much more common problem with our understanding of probability.

Most people who pay much attention to any kind of sciencey talk are probably familiar with the p-values referenced in the comic. When scientists are testing a hypothesis, they’ll often check whether the p-value (p for probability) of the results from their experiments is less than 5%. The smaller the p-value is, the less likely it is that their results are purely down to chance.

However, the p-value kinda means the exact reverse of what a lot of people assume it means.

When scientists talk about results being “significant at the 5% level”, say, it sounds like this means there’s a 95% chance of a real connection. In this cartoon’s case, it sounds like the scientists are 95% certain of a link between green jelly beans and acne.

Applicants for James Randi’s million dollar challenge are required to meet rather more stringent criteria, but it’s often expressed the same way. For instance, a dowser might have to psychically deduce which of several sealed containers is the one with water in, and repeat it a number of times, so that the p-value becomes very small. They want to be certain there’s really something going on, and it’s not just chance, before the money will be handed over.

But the intuitive idea of what the p-value means in these cases isn’t quite right.

Here’s what you actually need to do. Assume that there is no connection between the things being tested – jelly beans don’t affect acne, and all psychics are just guessing. Then, what are the odds of getting results at least as persuasive as the ones you saw, purely by chance?

That’s your p-value.

So, a p-value of 5% tells us something useful. It means that the results you’ve got are kinda iffy, given what you’d usually expect, if there’s no deeper underlying pattern there. You’d only expect to see results this skewed about 1 time in 20, if you’re relying on randomness. So maybe something’s up.

But if you do a whole bunch of tests, like the jelly bean scientists did, once in a while you will get some iffy results like that just by chance.

Now, clearly one thing this tells us is to be wary of data which has been cherry-picked, like the jelly bean journalists did. There were lots of negative results being ignored, and a single positive outcome highlighted. But the implications for how we assess probabilities more generally are, I think, more interesting.

In particular, it tells us that how likely something is doesn’t just depend on this one set of results. If a 5% p-value means “we’re 95% sure of this”, then this one study has entirely determined your estimate of the likelihood. It fails to take on board any information about how likely or unlikely something seemed before you started – and often this information is really important.

For instance, say you were studying differences between smokers and non-smokers, and the rate at which they get cancer. Any good analysis of data along these lines should easily pass a 5% significance test. It’s a highly plausible link, given what we already know, and 95% sounds like a significant under-estimate of the likelihood of a correlation between smoking and cancer.

But now imagine you’ve done a different test. This time, you just put a bunch of people into two groups, with no information about whether they smoke, or anything else about them, and flipped a coin to decide which group each person would go into. And imagine you get the same, seemingly convincing results as the smoking study.

Are you now 95% convinced that your coin-tossing is either diagnosing or causing cancer in people you’ve never met?

I hope you’re not. I hope you’d check your methodology, look for sources of bias or other things that might have crept in and somehow screwed up your data, and ultimately put it down to a bizarre fluke.

And it makes sense to do that, in this case, even despite the data. The idea that you could accurately sort people by cancer risk simply by flipping a coin is utterly ridiculous. We’d give it virtually zero probability to begin with. The results of your study would nudge that estimate up a little, but not much. Random fluke is still far more likely. If multiple sources kept repeating the experiment and getting the same persuasive results, over and over… then maybe, eventually, the odds would shift so far that your magic coin actually became believable. But they probably won’t.

And this idea of shifting the probability of something, rather than fixing it firmly based on a single outcome, is at the heart of Bayesian probability.

This is something the great Eliezer Yudkowsky is passionate about, and I’m totally with him. That link’s worth a read, though someday I’d like to try and write a similar, even more gently accessible explanation of these ideas for the mathematically un-inclined. He does a great job, but the arithmetic starts to get a bit overwhelming at times.

And if the thrill of counter-intuitive mathematics isn’t enough to convince you that this is fascinating and important stuff, read this. And then this.

Short version: a number of women have been convicted and jailed for murdering their children, then later released when somebody actually did some better statistics.

The expert witness for the prosecution in these trials estimated that the odds of two children in the same family both dying of cot death was 1 in 73,000,000. General population data puts the overall rate of cot deaths at around 1 in 8,500, so multiplying the 8,500s together gives the 1 in 73,000,000 figure for the chance of it happening twice. This was presented as the probability that the children could have died by accident, and thus it was assumed to be overwhelmingly likely that they were in fact deliberately killed.

But, as we learned with the cancer stuff earlier, we should consider these substantial odds against our prior assessment of how likely it is that these women would murder their children. This should start off minuscule, because very few women do murder their children. The fact that both their children died should make us adjust our likelihood estimate up a way – someone with two dead children is a more likely candidate for a child murderer than someone whose offspring are alive and well, after all – but it’s still far from conclusive.

Another way of expressing the central point of Bayesian probability is to consider the probability of A given B, for two events A and B. In this case, the odds of two children randomly picked from the population both dying of cot death may well be around 1 in 73,000,000 – but given that the children you’re considering both died in infancy, and were both siblings and so might have genetic or environmental factors in common, the cot death scenario becomes far more likely.

I wanted to expand on that last point some more, and touch on some other interesting things, but I’m hungry and you’re bored.

Ha. I said “briefly”. Classic.

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I used to believe in some wacky stuff.

It didn’t seem all that wacky at the time, of course. When I first started taking an interest in the stuff I was reading online, about people’s religious experiences and psychics and mind readers and dowsing and so on, it sounded fascinating, and wasn’t obviously bullshit at all. I guess I tend to think about things a bit differently now, or maybe there are just more things that I’ve learnt aren’t real in the intervening years.

Anyway, there was a lot of stuff about dowsing that caught my eye, and made it seem like an accessible skill. There was reams of advice and personal experiences people wanted to share, and it sounded like you didn’t need to be whisked away from your cupboard under the stairs to a wizards’ school by a hairy giant in order to be a part of it. It sounded like anyone could join in, and learn to access some spiritual dimension which could provide insight and knowledge from beyond this world.

So I bought a crystal pendulum from a new age shop.

It feels so weird typing that sentence now.

It was cheap, but kinda pretty, and looked a lot like this quartz one. The idea, as described on that page, is to clear your mind and mentally ask a series of yes/no questions, while letting the pendulum hang loosely from your fingers. There are various ways the pendulum might swing – circular motions, clockwise or anticlockwise, back and forth, diagonally – and you can calibrate it with some control questions.

I don’t remember exactly how it went when I tried it, but it would have been something like: “Is my name James?” – and I saw it swing forward and back, so I knew that meant yes. “Is today Wednesday?” – another yes, with the same swinging motion. “Is there a dragon in my room?” – and it swung side to side, meaning no.

This was really exciting.

So I decided to test it out properly, and see if I could find out something that I didn’t know, and prove that I was really tapping into some amazing psychic source of power.

I think this is the point where my strategy departed from that of a lot of new age fans.

I got a deck of playing cards and placed one face down in front of me. I didn’t know what card it was, but I held the pendulum over it, and asked yes/no questions to narrow it down. “Is it black?” – no. “Is it red?” – yes. “Is it a picture card?” – no. And so on.

Eventually I narrowed it down to “Is it the five of diamonds?” and got a yes. It had given me a definite answer to everything I asked. It had never contradicted itself. I’d started with absolutely no knowledge or assumptions or preconceptions about the card in front of me, and my pendulum had honed directly in on its identity as the five of diamonds.

I still remember the fluttering in my chest – half excitement and half genuine fear – in the second or two before I turned over the king of clubs.

Aw, crap.

It turns out that there’s a bunch of reasons why people believe in this kind of thing, and post articles to the internet about their powerfully moving personal experiences with it. And these reasons don’t require magic to actually be real.

When I first started looking into it, it didn’t require any particular daftness on my part to take it seriously – it just seemed to be a part of the world. A somewhat secretive, not generally known, exclusive part, but that just made it all the more fun. At the depth at which I explored it at the time, I didn’t find any good reason to suppose that it was all completely fictitious. People were taking it for granted, writing detailed accounts of their achievements, and beginners’ guides to the basic techniques.

But once you start thinking about it more critically, you realise that magic powers aren’t the only explanation. They’re not the best explanation. In fact, they’re not even a very good explanation.

Some people are very keen to find evidence that supports the idea that their dangling crystal can tell them things – so confirmation bias plays a big part in explaining why it’s so widely believed, as well as a host of other logical fallacies. But the ideomotor effect is one of the most persuasive aspects if you don’t know what it is. And it’s the one I’m supposed to be talking about here.

When I was asking myself those questions, I really was trying to hold the pendulum as still as possible. I know I wasn’t deliberately swinging it around to make myself seem like an amazing wizard (“Look, it knows my name!!”), but it’s worth asking: how good am I at holding my hand perfectly still? When I look closely at my outstretched digits as I try to remain motionless, I seem surprisingly wobbly. If I’m going to hold something on a thin and flexible cord or chain, it seems likely that my natural shakiness is going to have some effect.

And it turns out that the pendulum picks up more than just a general jiggle from my unsteady muscles. Let’s say I know a forward-swing means yes, because of my first test question. If I then ask something else which I know, or expect, has the answer yes, then on some level of consciousness I’m going to be imagining getting a forward-swing answer from the pendulum. My hand will then actually twitch, without my being aware of it, to make the pendulum swing forward.

The mental processes to do this can really happen inside your head, without the part where you’re conscious of it. It “bypasses volition”, to be a bit technical (volition being your capacity to do something by your own will).

You can try it easily yourself with any weight on some sort of dangling cord. I’m trying it now with one of the earphones from my mp3 player on its lead, and it’s still quite odd to see. I concentrate on a clockwise spinning motion, and it starts spinning clockwise, even though I’m still trying to hold it as steady as I can.

If you’re thinking that this might be evidence that I was secretly psychic all along, you’re still leaping to a more complicated explanation than is necessary. If I’m not directly touching the cord, or holding it in such a way that my hand movements won’t affect its swing, then it doesn’t respond in the same way. It only moves like this when I have the capacity to be swinging it around unconsciously. The best explanation is that I’m simply moving my hand.

There’s also a common hypnotic trick, where you’re asked to close your eyes and stick your arms out, then vividly imagine a heavy weight in one hand pulling it down, and a balloon tied to the other pulling it up. You focus on the respective feelings of pressure and lightness for a while, and if you’re anything like me, after a couple of minutes you open your eyes and find that you’ve lifted and lowered your hands accordingly by several inches, without being aware of doing it.

The point is, your mind’s good at doing stuff like this without telling you about it.

Now, this doesn’t mean that nobody can dowse anything, or that we’ve proved that Ouija boards are universally a load of crap (yes, the people are just pushing the glass around even if they don’t realise it). But it reminds us the importance of asking the question “Is there a simpler, less Harry Potter explanation?” when we see something we think might be magic.

If I was doing actual magic over my playing card that time, then my skills make Neville Longbottom look like Gandalf. I must really suck at magic. I didn’t even get close to getting the card right. Magic just isn’t a good enough explanation for what happened there. But the idea that my hand wasn’t perfectly still, and made the pendulum swing a little by entirely natural means? Yep, that fits.

But what if I had got it right? What if I had no way of knowing what card I was staring at the back of, and wasn’t being provided the information by any means except the pendulum, and I actually got it right? And it kept happening, consistently?

Well, the ideomotor effect wouldn’t cover that. And I’d be a millionaire.

But it does cover, y’know, every case that’s ever been examined of any kind of dowsing ever. Except the ones that are outright fraud, where there’s conscious deception taking place. But there really doesn’t need to be any malice or dishonesty for people to make magical claims that aren’t based in reality. If you don’t know what the ideomotor effect is, and maybe don’t test out your new idea all that rigorously, and kinda let slide the few occasions where it doesn’t work… then I can imagine this being pretty convincing.

People who do things like dowsing aren’t being stupid or evil. But they are claiming that they can do magic, and it’s a big ask that we should take that at face value without daring to question it any further, even if we don’t doubt their sincerity. It’s the kind of massive claim that we should probably, y’know, check.

And, unfortunately for any aspiring Weasleys out there, natural phenomena like the ideomotor effect provide a better explanation for every instance of “magic” that’s yet been observed. They account perfectly for what’s going on, but the magical explanation fails to explain why the effect always vanishes when studied closely. It just doesn’t work. The five of diamonds was not my card.

Sorry, Hermione. Muggles win.

A more academic and less chatty approach to this topic can be found at The Skeptic’s Dictionary, RationalWiki, SkepticWiki, and all over the place really. Barrett Dorko and Ray Hyman, among others, have written rather more scientifically rigorous documents about the ideomotor effect in action, with examples of experiments in which it’s been seen.

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