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.
Keywords
probability kernel, conditional probability, compatibility,
lattice, Bayesian model