Exchangeable Bernoulli Random Variables And Bayes’ Postulate

We discuss the implications of Bayes’ postulate in the setting of exchangeable Bernoulli random variables. Bayes’ postulate, here, stipulates a uniform distribution on the total number of successes in any number of trials. For an infinite sequence of exchangeable Bernoulli variables the conditions of Bayes’ postulate are equivalent to a uniform (prior) distribution on the underlying mixing variable arising from the De Finnetti representation theorem. We show that in the presence of exchangeability, the conditions of Bayes’ postulate are implied by a considerably weaker assumption (denoted henceforth as ‘the weak assumption’) which only specifies the distribution of the total number of successes at a point. The equivalence of the Bayes’ postulate and the weak assumption holds for both finite and infinite sequences. We also explore characterizations of the joint distribution of finitely many exchangeable Bernoulli variables in terms of probability statements similar to the weak assumption.