The Science of Discworld IV

pluck from a hat for illustrative purposes), but you also have a hunch that this year they are playing a lot better than usual. Put the two together, and you will assess their chances as being higher.
    Bayesian inference can put numbers to all of this, and provide a rational system for calculating the probabilities concerned – except for prior probabilities, which are plugged into the formulas but are not consequences of them. So the method is a ‘worlds of if’ approach:
if
the prior probability is such and such,
then
the consequences of new data will be so and so. The formula does not justify any particular prior probability; however, its consequences may let us test the accuracy of the prior probability, by comparison with observations. Bayesian inference often outperforms more ‘rational’methods. Although we may not be certain that we’ve assessed the prior probabilities correctly, it may still be better to make a guess, rather than ignoring such influences altogether.
    In conventional statistics, a statement being tested – a hypothesis – should be accepted (or at least not rejected) if the evidence agrees with it. In the Bayesian approach, however, the hypothesis should be rejected, despite the evidence, if its prior probability is very low. Indeed, it may be reasonable to reject the alleged
evidence
, on the same grounds.
    For example, suppose the hypothesis is the existence of UFOs, and the evidence is a photograph of one. The photo supports the hypothesis, but if you believe that the chance of UFOs existing is extremely small, then the evidence is not convincing. The photo might be a fake, for example; but even if you don’t know whether it is genuine, you are justified in rejecting the hypothesis … unless, of course, it turns out that your prior probability is wrong. So Bayesian inference does not disprove the existence of UFOs: instead, it quantifies the view that ‘extraordinary claims require extraordinary evidence’. And a photo isn’t extraordinary enough.
    Anyway, the neuroscience theory holds that the brain operates by generating beliefs about the world. Here a belief is defined to be what the brain decides about some event or phenomenon, so it is hard to deny that the brain operates by generating such things. The theory says something less tautologous, however: it asserts that the brain combines two distinct sources of information: memory and data. It does not just
assess
incoming sensory data as such; it
compares
them to what’s already stored in memory.
    Experiments carried out by Daniel Wolpert and his team support the view that the results of these comparisons correspond very closely to Bayes’s formula. The brain seems to have evolved an effective and fairly accurate way to combine its existing knowledge with new information, thereby modifying what it holds in its memory.The experiments look at how we move our limbs to perform some task. Suppose we want to pick up a cup of coffee. There are many ways to do this, and most end in disaster. If we tip the cup too far, for example, the coffee will spill. The response of our muscles is affected by inherent random fluctuations in the motor system, and some strategies for picking up the cup are less error-prone than others. Optimal choices, determined by Bayesian decision theory, generally agree with the actual motions observed.
    We repeat, this doesn’t imply that the brain carries out Bayesian calculations the way a mathematician would consciously do using pencil and paper. On the contrary, the brain has evolved neural networks that produce the same general results. The choices indicated by Bayesian decision theory are the choices that best fit reality, assuming that memory and data are being combined. This fit provides an evolutionary advantage – on the whole, those choices work better. So the neural networks that control how we walk, run, hold or throw objects, have been selected to mimic the results of Bayesian decision theory – our way to formalise mathematical rules that describe whatever nature is actually doing.
    More generally, we can speculate that similar neural networks control our snap judgements about social or political matters. Again, there are two ingredients: prior beliefs already in memory, and new data. Crucially, the Bayesian model shows why beliefs may override data. If you are certain that global warming is a hoax – for whatever reasons, good or bad – the Bayesian decision machine in your head will