The propensity approach

As of the 23rd May 2022 this website is archived and will receive no further updates. was produced by the Winton programme for the public understanding of risk based in the Statistical Laboratory in the University of Cambridge. The aim was to help improve the way that uncertainty and risk are discussed in society, and show how probability and statistics can be both useful and entertaining.

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Some people may find all the other interpretations and definitions of probability are somewhat unintuitive: surely the probability of something happening is not really a "degree of belief", but something objective about the world that would be the same even if there were no human (or alien) beings around to believe it? The degree of belief or measure of uncertainty interpretations may seem divorced from the real world, while the alternative frequency theory leaves us with no way of making sense of single case probabilities, even though many people feel there are objective probabilities of single events, such as the probability that I will get that dream job, or that Elvis is still alive.

This leaves the idea that there is something about the experimental set-up that determines the 'true' probability. Let's take the example of a coin again. The probability of throwing heads is determined by the composition of the coin, the way it is thrown and other surrounding factors (wind, gravity, etc). This has been termed the "propensity" for things to happen, which is determined by the specific set-up of the situation.

A famous instance of such an interpretation of probability is the later philosophical work of Karl Popper Popper1983. Earlier on in his career Popper1934, Popper was a follower of the frequency interpretation, and his particular version of propensity theory was influenced by this. For him, the main problem with the frequency theory was that it doesn't take account of the probabilities of single events. Popper's solution lay in a modification of the frequentist concept of 'collective', i.e. the sequence of events on which the relative frequency is based. Instead of having a collective of repeating the same thing (e.g. throwing a die), Popper talked about repeatable conditions: we don't throw the same die in an approaching-infinite amount of time, instead we look at the conditions inherent in the set up so that we can recreate the throwing of the die. Of course this is hypothetical, since in more complicated scenarios it is impossible to actually recreate a set up. However, this is about a philosophical justification rather than arriving at workable numbers, so we are only concerned about the principle, and about finding intuitive conceptions of probability. In any case, the frequentist alternative of throwing a die an infinite number of times might be considered equally imaginary.

How does this construction of an alternative to the frequentist collective look when we consider a probability which a frequentist denies can objectively exist, such as the (single event) probability that, say Labour will lose the next election? Here in effect we are imagining alternative worlds: consider an infinite amount of possible future worlds that are at the same stage as ours is at the moment. If in 70% of these worlds Labour will go on to lose the election, while in 30% it will win, then we can say that the probability of Labour losing the next election is 0.7.

This alternative-world thought-experiment should not be taken to mean that the philosopher believes that these worlds exist; they are merely a way of constructing a collective, and are just as fictional as infinite throws of dice. However, through this invocation of possible worlds, probability in the propensity interpretation has become a metaphysical entity, which seems to be even more removed from empirical evidence than frequentist probability, which at least constrains us to look only at cases where we can make some sort of approximate guess, based on the relative frequency of what we have observed so far. The metaphysical probability has therefore failed to satisfy a lot of philosophers.

The propensity theory does not have one canonical defining writer who can be said to define the essence of the theory. Although Popper is the most famous philosopher to have developed a propensity theory, he has many rivals. See Gillies Gillies2000 for an introduction to Popper and some other versions of the theory, including Gillies' own version.