Bayesian Logic and Statistical Mechanics — Illustrated by a Quantum Spin 1/2 Ensemble

Abstract

The greatest challenge for the appropriate use of Bayesian logic lies in the need for assigning a “prior”[1–13]. In physical applications, this prior is often strongly molded by the physical “boundary conditions”[1,12,13], and thereby begins to take on some aspects of “objective” meaning. Many claim that the prior usually plays only a minor role in the interpretation of the data. We shall show situations in which a prior, objectively justified by the physical constraints, can lead to a very strong molding of our interpretation of the measured data. We shall also show how the concept of a “probability distribution function over probability distribution functions”[8,9] has practical utility in clarifying and taking advantage of the distinction between the “randomness” intrinsic in the preparation of an ensemble of physical systems, and the uncertainty in our limited knowledge of the values of the parameters that characterize this preparation procedure. In this paper we consider three physical situations in which Bayesian concepts lead to a useful, non-conventional interpretation of the information contained in a small number of physical measurements. The first two of these relate to an ensemble of quantum mechanical spin-1/2 systems, each prepared by the same procedure[13]. The third relates to the energies of molecules sampled from a fluid that obeys the classical statistical mechanics of Gibbs[1,8,9]. In the first example, we show how a prior chosen to match the physical constraints on the system leads to a very conservative “rule of succession”[7,10,3] that discounts any significant deviation from “random” frequencies (for measured values) as being nonrepresentational of the distribution being sampled. In the second example, we show how an appropriate, physically motivated prior leads to a rule of succession that is less conservative than that of Laplace[5], and even, in most circumstances, is less conservative than the maximum likelihood inference[1–3] from the observed frequency. This example also shows how a prior that is uniform in one parameter space can yield a rather peaked prior in the space of another related (and physically important) parameter, (in this case, the temperature). In the third example, we show how Bayesian logic together with the knowledge of the results of only a small number of physical measurement events augments equilibrium statistical mechanics with non-Maxwellian distribution functions and probability distribution functions over temperature. These distribution functions are a more faithful representation of our state of knowledge of the system than is a Maxwellian distribution with a temperature inferred by the principle of maximum likelihood [1–3,14,9].