ABSTRACT

Unknown prior probabilities can be treated as intervening variables in the determination of a posterior distribution. In essence this involves determining the minimally informative information system with a given likelihood matrix.

Some of the consequences of this approach are non-intuitive. In particular, the computed prior is not invariant for different sample sizes in random sampling with unknown prior.

GENERALITIES

The role of prior probabilities in inductive inference has been a lively issue since the posthumous publication of the works of Thomas Bayes at the close of the 18th century. Attitudes on the topic have ranged all the way from complete rejection of the notion of prior probabilities (Fisher, 1949) to an insistence by contemporary Bayesians that they are essential (de Finetti, 1975). A careful examination of some of the basics is contained in a seminal paper by E.T. Jaynes, the title of which in part suggested the title of the present essay (Jaynes, 1968).

The theorem of Bayes, around which the controversy swirls, is itself non-controversial. It is, in fact, hardly more than a statement of the law of the product for probabilities, plus the commutativity of the logical product. Equally straightforward is the fact that situations can be found for which representation by Bayes theorem is unassailable. The classic classroom two-urn experiment is neatly tailored for this purpose. Thus, the issue is not so much a conceptual one, involving the “epistemological status of prior probabilities, as it is a practical One.