Subjective degree of belief

We have seen in The logical view: probability as objective degree of belief that "logical" probability, defined as a degree of belief which it is rational and objective to hold given the available evidence, can give rise to some contradictory results. However, in the 1920s Frank Ramsey in Cambridge Ramsey1926 and Bruno de Finetti in Italy DeFinetti1931 independently proposed a radically different interpretation of probability which avoids these paradoxes by asserting that probability is a subjective degree of belief.

This view takes probability as representing your private assessment of how likely an event is, based on the evidence available to you. This idea has faced some hostility, because superficially it seems to mean that we can think what we want regardless of the evidence , and places us in a relativism in which all opinions are equally valid.

But this is not the case, due to an ingenious argument which lays (only) one reasonable condition on the possible degrees of beliefs we can hold: they must 'cohere' in the sense of following the mathematical laws of probability. By this we mean, for example, that our probabilities of mutually exclusive events must add up to 1 (say in horse racing, the probabilities we give the different horses should all add up to one, because one, and only one, horse will win).

Why should our degrees of belief follow the laws of probability? Here the argument is one that appeals to our own interest: if we were prepared to bet on our degrees of belief and they don't follow the laws of probability, then a clever bookmaker can always sell us a book where we lose money, no matter what happens. This so-called Dutch book argument insures that, as long as we are rational people who don't want to loose money, our subjective degrees of belief follow the rules of probability. The so-called 'Dutch book' argument is only one way of showing that any reasonable beliefs should be quantified according to the laws of probability - essentially we derive these laws through appeals to rational behaviour, rather than assuming them as mathematical 'truths'.

From this (relatively) simple argument we can get a lot of surprising results which make the idea of personal degrees of belief a lot more palatable for those who instinctively think probabilities should be objective. Using the laws and theorems of probability theory (specifically Bayes' theorem), we can use subjective probabilities to learn from experience. Incidentally, this essentially solves the problem of induction, which is one of the main problems in the philosophy of science, and so “subjective Bayesianism” has become one of the main contemporary schools within the philosophy of science.

Using Bayes' theorem, we can show that even if initially two people's judgments of the probability of a hypothesis are vastly different, as evidence for (or against) the hypothesis comes in, the people have to revise their degrees of beliefs in order to satisfy the Dutch book requirement, and, amazingly, the degrees of belief will tend to converge to the same probability as more and more evidence comes in. This one example shows how, even though the probabilities are interpreted as personal and subjective degrees of belief, they are still in a sense objective as people tend to come to agreement over time. This is especially true in cases where there is a lot of evidence to support the degree of belief, such as a repeatedly thrown die. The case of a repeatedly thrown die also explains how subjectivism will give the same probabilities to frequently repeated events as does the frequency interpretation of probability, where the probability is defined as the relative frequency of events in the long run (this arises from the idea of exchangeability, to be discussed elsewhere). The advantage of this view over the frequency interpretation is that it can deal with cases where there is no relative frequency to draw on: for example, Gigerenzer mentions the first ever heart transplant patient who was given a 70% chance of survival by the surgeon. Under the frequency interpretation that statement made no sense, because there had never actually been any similar operations by then.

This development of argument follows de Finetti rather than Ramsey, who instead conceded that in some cases, such as coin tossing and dice throwing, there may well be an objective type of probability.

A problem, on the other hand, is how exactly do we assess a subjective degree of belief? If we take the idea of betting further, then we can imagine that there are plenty of situations where we would simply not be inclined to bet on anything, even hypothetically. While there are some ideas around that we can define people's betting quotients exactly (see Gillies 2000 Gillies2000), a focus on betting can remove the intuitive simplicity of the subjective approach.