Phase-Space Priors in Information Entropy

Abstract

It is possible to demand that in estimating probabilities of events in the real world, one should include physical measures by taking into account the phase-space belonging to each event. In many cases this appears absurd, as in pattern recognition, etc. or in economic events and indeed in most situations that involve personal choices. Still I suggest (in consonance with several previous, unheeded suggestions made by others) that the demand should not be jettisoned, and for the following reasons:

• (a) There are macroscopic events for which the physical phase-space can be estimated with greater or lesser difficulty at one or other level of sophistication. Even for these cases predictive procedures (like MaxEnt) are currently practised without bothering about the physical probability measure, as ought to be done.

• (b) A consistent formulation of MaxEnt, encompassing both the information and statistical-physical domains, is possible by starting with a Boltzmann-Gibbs entropy in which macroscopic events are included as marginals. The physical measure (estimated on the basis of microscopic considerations) emerges naturally as the prior probability for the macro-events.