Everything Is Predictable

Bayes’ theorem is the same idea, but taken a bit further. In natural language, it means: the probability of event A, given event B, equals the probability of B given A, times the probability of A on its own, divided by the probability of B on its own.

the probability of some event is the number of ways that event can occur, divided by the total number of things that could occur.

No] testimony is sufficient to establish a miracle,’ wrote Hume, ‘unless the testimony be of such a kind, that its falsehood would be more miraculous, than the fact, which it endeavours to establish.’

More than that. All decision-making under uncertainty is Bayesian – or to put it more accurately, Bayes’ theorem represents ideal decision-making, and the extent to which an agent is obeying Bayes is the extent to which it’s making good decisions. Logic itself, all that stuff you may remember about ‘All men are mortal; Socrates is a man; ergo Socra

Bernoulli proved that for either of those, or any other combination, there is a number of balls that you can draw out of the urn which will give you that level of confidence. He also showed that there is no point at which either you reach certainty or that increasing your sample ceases to give you greater confidence.

Bernoulli’s claim was that the more you flipped the coin, the closer, on average, to the ‘true’ probability your results would be. You might reasonably say that this is pretty obvious. And also, so what? You know you’ll see roughly half of the coin-flips come up heads. You don’t need to flip the coin a million times to prove it.

Now, we’re no longer talking about working out the probability of seeing some result, given certain facts about the world. We’re talking about the exact opposite: what is the probability of the world being a certain way, given the results that we’re seeing? The two ideas are sampling probabilities – what can we predict about a sample of something,

But the point is that when you’re trying to work out how likely something is, what we need to talk about is the number of outcomes – the number of outcomes that result in whatever it is you’re talking about, and the total number of possible outcomes. This was, I think it’s fair to say, the first real formalisation of the idea of ‘probability’.

In order to work out the thing we really want to know, we need more information. In the example of the cancer test, we need to know how common breast cancer is in the population. In medical terms, that’s the prevalence or the background rate, but in Bayes’ theorem in general, it’s known as your prior probability, or ‘prior’. In