Everything Is Predictable

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, whi

There’s something wonderful about how counterintuitive the theorem is. What do you mean, a test being 99 per cent accurate isn’t the same as a 99 per cent chance that it’s right?

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.

A ‘derivative’ is the rate of change of a slope on a graph. If you had a graph of time (seconds) and distance (metres), the shape of the line tells you something about the speed (metres per second). If the line is straight, your speed is constant. If the line is curved, your speed is changing. A derivative measures the slope of the curve at an exac

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

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

His insight in probability theory was that probabilities are beliefs; our beliefs, if we act on them, are themselves a kind of bet. As Ramsey put it: ‘All our lives, we are in a sense betting. Whenever we go to the station, we are betting that a train will really run, and if we had not a sufficient degree of belief in this, we should decline the be

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

inverse probability, although it’s really the heart of what we want probability to do. What we want, or at least what science wants, out of statistics is to answer: given the results I’ve seen, what can I say about my hypothesis?