What is the “Dutch book argument” in Bayesian probability?

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What is the “Dutch book argument” in Bayesian probability?

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Basically, probability is a pretty difficult concept to define and understand when you get right down to it, and the Dutch book argument, according to some people, helps resolve some of these issues.

When we say that the probability that something will happen is, say, 1/6, what do we actually mean? There are some settings where this is pretty straightforward. If I roll a perfectly symmetric 6-sided die, I know that there are 6 identical outcomes, one of which must happen, so clearly the probability of each is 1/6 (this is the classical interpretation of probability). Or, if I have an experiment that I can repeat over and over again, and a certain outcome happens approximately 1 in 6 times, I can conclude that its probability is roughly 1/6 (this is the frequentist interpretation of probability).

But what if someone claims that, say, the probability of a certain candidate winning an election is 1/6? The candidates are not identical like the sides of a die, and there is no way we can run the election repeatedly under identical conditions. The most popular interpretation of probability in a case like this is the Bayesian interpretation, which casts probability as a degree of belief. You start with an initial probability called a prior, and then when you learn new information you use Bayes’ theorem to update the probability. This at least allows you to calculate some probabilities, but what they actually mean is another matter, and there are various different views on how Bayesian probability should be understood.

A starting point that most people can agree on is that people will not have contradictory beliefs. For example, they will not simultaneously believe that the probability that A happens is 1%, the probability that B happens is 0.1%, and the probability that A and B both happen is 100%. The Dutch book argument, according to some, helps to clarify this. The idea is that if somebody has inconsistent beliefs, you can construct a combination of bets which according to their beliefs they should make, and yet which is guaranteed to lose them money overall (a “Dutch book” is a situation where a bookmaker has sold bets such that they are guaranteed to make a profit no matter what happens).

There are lots of different versions of these arguments, and lots of different views on whether they are correct, convincing, or helpful.