Vol. 11, No. 2, 2020

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Compatibility of distributions in probabilistic models: an algebraic frame and some characterizations

Luigi Burigana and Michele Vicovaro

Vol. 11 (2020), No. 2, 213–246
Abstract

A probabilistic model may be formed of distinct distributional assumptions, and these may specify admissible distributions on distinct (not necessarily disjoint) subsets of the whole set of random variables of concern in the model. Such distributions on subsets of variables are said to be mutually compatible if there exists a distribution on the whole set of variables that precisely subsumes all of them. In Section 2 of this paper, an algebraic frame for this compatibility concept is constructed, by first observing that all marginal and/or conditional distributions (also called “probability kernels”) that are implicit in a global distribution form a lattice, and then by highlighting the properties of useful operations that are internal to this algebraic structure. In Sections 3, 4, and 5, characterizations of the concept of compatibility are presented; first a characterization that depends only on set-theoretic relations between the variables involved in the distributions under judgment; then characterizations that are applicable only to pairs of candidate distributions; and then a characterization that is applicable to any set of candidate distributions when the variables involved in each of these are exhaustive of the set of variables in the model. Lastly, in Section 6, different categories of models are mentioned (a model of classical statistics, a corresponding hierarchical Bayesian model, Bayesian networks, Markov random fields, and the Gibbs sampler) to illustrate why the compatibility problem may have different levels of saliency and solutions in different kinds of probabilistic models.

Keywords
probability kernel, conditional distribution, compatibility, lattice, graphical model
Mathematical Subject Classification
Primary: 62H10, 62H22, 62R01, 60E99
Milestones
Received: 25 January 2020
Revised: 11 June 2020
Accepted: 6 July 2020
Published: 28 December 2020
Authors
Luigi Burigana
Department of General Psychology
University of Padua
I-35131 Padova
Italy
Michele Vicovaro
Department of General Psychology
University of Padua
I-35131 Padova
Italy