Eventually here, I want to share some thoughts and a question that I have had lately pertaining to de Finetti’s notion of exchangeability. First, however, I want to set the stage for these thoughts by providing some background. If you are already informed on the basic points pertaining to de Finetti’s representation theorem and its epistemological implications, you will probably just want to skip right to the section on “-exchangeability?”.

Background

De Finetti defines exchangeability as follows (p. 123*):

are exchangeable random quantities if they play a symmetrical role in relation to all problems of probability, or, in other words, if the probability that satisfy a given condition is always the same however the distinct indices are chosen.

De Finetti’s representation theorem shows the importance of this concept to subjective probability. Ultimately, exchangeability provides subjectivists with a bridge to the Frequentist notion of “independent events with fixed but unknown probability;” specifically, de Finetti’s theorem states that “the probability distributions corresponding to the case of exchangeable events are linear combinations of the distributions corresponding to the case of independent equiprobable events” (p. 129). The concept of exchangeability, claims de Finetti, is itself rid of all ties to objectivism as it deals purely with “the evaluations of the probabilities of individual events” (p. 142). Moreover, given the representation theorem, exchangeability gives subjectivists the run of several important statistical results:

The law of large numbers and even the strong law of large numbers are valid for exchangeable random quantities , and the probability distribution of the average of of the random quantities tends toward a limiting distribution when increases indefinitely” (p. 124).

One major philosophical consequence of all of this is summarized by de Finetti (which he understood as a solution to at least one version of the problem of induction): “A rich enough experience leads us always to consider as probable future frequencies or distributions close to those which have been observed” (p. 142). This explains why, even from a purely subjective viewpoint, we would expect to see much agreement between various people’s degrees of belief. If two subjects’ respective degrees of belief about the events in a certain sequence are given by the probability functions and , and if both subjects believe that the indicators corresponding to those events () are exchangeable (so that, for both subjects, for all , , and remains the same for every permutation of the values ), then there is an such that, if the subjects share a string of evidence of length from the sequence, they can be as certain as you like that they will agree on the probability of the next event within any specified of each other.

-Exchangeability?

So exchangeability provides us with this theorem, which itself describes a swamping of the priors effect that, of course, has some potentially rich and interesting epistemological implications. But recall now that the theorem requires that, in order for their respective degrees of belief to converge in the face of evidence, subjects must believe that the events in question are exchangeable. That is, regarding the indicator functions corresponding to these events, they must all believe that for all and that remains constant for every permutation of the values . But this would seem to be rarely the case in real life.

This brings me to my question. Aside from the notion of “partial exchangeability,” which de Finetti defines in order to make these results more readily and generally applicable, does anyone know of any attempts to weaken the notion of exchangeability while retaining its epistemic implications? Here is the sort of weakening that I have in mind, and about which I am curious as to whether it has been done before. I would think that a worthwhile project would be to define a sort of measure of proximity to exchangeability (a measure of “-exchangeability”). In many cases, it might be that we don’t believe that certain events are exchangeable but that we do almost believe this. Perhaps we don’t believe that remains constant for every permutation of the values , but we do believe that this is true for all but a small number of those permutations. Or perhaps values of shift around with distinct permutations of , but only very slightly. In either of these cases, it could be that people come sufficiently close to believing that the events are exchangeable so that swamping / convergence results still follow.

The project that I am envisioning here then would look at various cases of -exchangeability in order to test the robustness of de Finetti’s theorem and its epistemological implications. The driving question here would be, “In what ways, and to what degree, can we stray from the belief that events are fully exchangeable while still getting convergence; i.e., while still being able to demonstrate the existence of a limiting probability distribution as evidence comes in?”

Thoughts?? Has this all been done before? If so, I’d still like to know where. If not, do you think it is worth attempting??

* All quotes are lifted from de Finetti, B. (1937). Foresight: Its logical laws, its subjective sources. In Kyburg, H. E. and Smokler, H. E., editors, Studies in Subjective Probability, pp. 93-158. Wiley, New York, 1964.