INTRODUCTION

Belief revision

The most frequently discussed method of revising a subjective probability distribution P to obtain a new distribution P*, based on the occurrence of an event E, is Bayes's rule: P*(A) = P(AE)/P(E). Richard Jeffrey (1965, 1968) has argued persuasively that Bayes's rule is not the only reasonable way to update: use of Bayes's rule presupposes that both P(E) and P(AE) have been previously quantified. In many instances this will clearly not be the case (for example, the event E may not have been anticipated), and it is of interest to consider how one might proceed.

Example. Suppose we are thinking about three trials of a new surgical procedure. Under the usual circumstances a probability assignment is made on the eight possible outcomes Ω = {000, 001, 010, 011, 100, 101, 110, 111}, where 1 denotes a successful outcome, 0 not. Suppose a colleague informs us that another hospital had performed this type of operation 100 times, with 80 successful outcomes. This is clearly relevant information and we obviously want to revise our opinion. The information cannot be put in terms of the occurrence of an event in the original eight-point space Ω, and the Bayes rule is not directly available. Among many possible approaches, four methods of incorporating the information will be discussed: (1) complete reassessment; (2) retrospective conditioning; (3) exchangeability; (4) Jeffrey's Rule.