# Problems with Bayesian epistemology

The hotness in philosophy of science is currently Bayesian epistemology. The idea behind Bayesian epistemology involves Bayes’ theorem, which sez that

$P(a | b) = \frac{P(b | a) P(a)}{P(b)},$

where P(a | b) is the probability of a given b.

Philosophers who practice Bayesian epistemology like to say that b is some evidence e and a is a hypothesis h and write:

$P(h | e) = \frac{P(e | h) P(h)}{P(e)}.$

Philosophers then like to imagine that it is possible to use this equation to increase the probability of a hypothesis h by finding suitable evidence.

Popper spent a lot of space on pointing out that the rules of probability can’t be used like this, see Section VI of the introduction to Realism and the Aim of Science, as well as Part II Chapters I and II.

But there is another objection. The rules of probability apply to a space of events. Where does this event space come from? It can only come from an explanatory theory. That explanatory theory can’t be put into Bayes’ formula: it is an assumption required to apply the formula at all. There is no such thing as the probability of a theory.

An associated problem is that explanatory theories apply to the whole of physical reality. Some theories are explicitly universal: it applies to everything that happens anywhere in reality. The general theory of relativity isn’t the theory of gravity on Earth at 2.30 pm on Friday 25 August 2017. Rather, it is supposed to apply to everything happening everywhere all the time. But probabilities are tested using relative frequencies. So to test general relativity you would need multiple copies of the whole of physical reality. But let’s suppose that you’re talking about an explanation of some particular event. For example, there are some instances in which a sustained nuclear fission reaction occurred naturally. The explanation for that event is that wherever circumstances occur that some set of properties a fission reaction will take place. So every instance in the universe where those properties occur will give rise to fission. So to test the probability of that theory you would need to have access to multiple copies of the whole of physical reality.

The last problem I’ll raise is that the point of assessing a theory is that you will either use the theory to solve some problem or reject it. So any epistemology should give you a yes or no answer to the question, as explained here.