Abstract

A probabilistic model may involve families of probability functions such that the functions in a family act on a definite (possibly multiple) variable and are indexed by the values of some other (possibly multiple) variable. “Probability kernel” is the term here adopted for referring to any one such family. This study highlights general properties of probability kernels that may be inferred from set-theoretic characteristics of the pairs of variables on which the kernels are defined. In particular, it is shown that any complete set of such pairs of variables has the algebraic form of a lattice, which is then inherited by any complete set of compatible kernels defined on those pairs; that on pairs of variables a criterion may be applied for testing whether corresponding probability kernels are compatible with one another and may thus be the building blocks of a consistent probabilistic model; and that the order between pairs of variables within their lattice provides a general diagnostic about deducibility relations between probability kernels. These results especially relate to models that involve a number of random variables and several interrelated conditional distributions acting on them; for example, hierarchical Bayesian models and graphical models in statistics, Bayesian networks and Markov fields, and Bayesian models in the experimental sciences.

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