But what does that mean? Can I say that bigfoot people are cleverer, or not? What Fisher did was suggest that we should choose some arbitrary level at which we say ‘OK, it’s pretty unlikely that we’d see results this extreme given the null hypothesis, so I’m going to behave as though it’s a real effect.’ Fisher himself said that a p-value of 0.05 –

And we’re Bayesian at a deeper level, too. Our brains, our perception, seem to work by predicting the world – prior probabilities – and updating those predictions with information from our senses: new data. Our conscious experience of the world can be best described as our priors. I predict, therefore I am.

What is the chance of seeing this result, given some hypothesis? If you’ve ever read any stories about science in the media, you’ll probably recognise the phrase ‘statistical significance’. You may also have come across ‘p-values’. A p-value is the likelihood of seeing results at least as extreme as those you’ve seen, given the null hypothesis,

Divine Benevolence was a work of theodicy: an attempt to explain why God, if all-powerful and all-benevolent, allows evil in the world. As David Hume put it, apparently quoting Epicurus: ‘Is he willing to prevent evil, but not able? then is he impotent. Is he able, but not willing? then is he malevolent. Is he both able and willing? whence then is

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.

‘Necessarily,’ wrote David Howie, a historian of statistics, ‘these inferences were tentative. They were advanced not with certainty but with degrees of confidence that were updated or modified to account for new information.’95 That is: they were done in a Bayesian fashion. Each time Jeffreys got new information, he updated his prior confidence in

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.’

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

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’.