Date: 05 Oct 2006

Revisiting Prior distributions, Part I: Priors based on a physical invariance principle

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Abstract

Determination of uninformative prior distributions is essential in many branches of knowledge integration and system processing. The conceptual difficulties of this determination are due to lack of uniqueness and consequential lack of objectivity associated with the state of complete ignorance. The present work overcomes the above difficulty by considering a class of priors that are consistent with a physical invariance principle, namely, invariance with respect to a change in the system of dimensional units. These priors do not represent total ignorance and they do not suffer from the aforementioned conceptual difficulties. This Dimensional Invariance Requirement (DIR) leads to a class of prior densities, which constitute a generalization of Jeffrey’s proposal concerning priors of inherently positive variables. This generalization possesses certain important features, from a formal as well as an interpretive viewpoint, which involve the notion of a knowledge-based natural reference point of physical random variables (RV). Conceptual difficulties associated with uninformative priors are resolved, whereas well-established results are derived as special cases of the DIR. Application of this requirement to a system of RV yields the familiar result that at the prior knowledge stage these variables should be considered as independent. Prior distributions for non-dimensional physical quantities are obtained by defining these variables in terms of dimensional quantities. A logarithmic transformation carries the physical prior into a uniform (flat) density that is convenient in certain applications. In a companion paper we examine the improvements gained in the maximum entropy context by means of the proposed class of physical prior densities.