ABSTRACT

MAXENT (MAXimum ENTropy principle) is a general method of statistical inference derived from and intrinsic to statistical mechanics. The probabilities it produces are “logical probabilities” – measures of the logical relationship between hypothesis and evidence. We consider the significance and applications of the “logical probability” of such probabilities. The probability of a “logical probability” is shown to be the probability of the evidence used for the “logical probability”. This suggests a hierarchy of logics, with “evidences” defined as sets of probabilities on the preceding “logic”. Applications to reliability theory are described. We also clarify the meaning of MAXENT and examine arguments in a recent article in which temperature fluctuations are introduced in thermal physics.

INTRODUCTION

A method fundamental to statistical physics is the maximization of entropy. In recent years, this method has been recognized as a general procedure for statistical inference based on the fact that “entropy” is essentially a measure of information uncertainty [1]. The probabilities one obtains using MAXENT (as the “Maximum Entropy Principle” is now called) have a natural interpretation which has not been generally recognized, even by advocates of the procedure. This is the “degree of belief” (DOB) interpretation [2] – that “probability” is a measure of the logical relationship between two propositions: p(H | E) expresses a (normalized) “degree of belief” (DOB) in the relationship of hypothesis H to evidence E. Indeed, MAXENT asserts precisely the (statistical) consequences of assumed evidence since it is based on the idea that one should choose as probability one which maximizes “uncertainty” consistent with the evidence.