We show there is only one consistent way to update a probability assignment, that given by Bayes's rule. The price of inconsistent updating is a loss of efficiency. The implications of this for the problem of induction are discussed.

It has been alleged that Bayesian usage of prior probabilities allows one to obtain empirical statements on the basis of no evidence whatever. We examine this charge with reference to several examples from the literature, arguing, first, that the difference between probabilities based on weighty evidence and those based on little evidence can be drawn in terms of the variance of a distribution. Moreover, qua summaries of vague prior knowledge, prior distributions only transmit the empirical information therein contained and, therefore, (...) their consequences for long-run frequency behavior are "a priori" in at best a Pickwickian sense. (shrink)

The pre-designationist, anti-inductivist and operationalist tenor of Neyman-Pearson theory give that theory an obvious affinity to several currently influential philosophies of science, most particularly, the Popperian. In fact, one might fairly regard Neyman-Pearson theory as the statistical embodiment of Popperian methodology. The difficulties raised in this paper have, then, wider purport, and should serve as something of a touchstone for those who would construct a theory of evidence adequate to statistics without recourse to the notion of inductive probability.

An important contribution to the foundations of probability theory, statistics and statistical physics has been made by E. T. Jaynes. The recent publication of his collected works provides an appropriate opportunity to attempt an assessment of this contribution.