Burigana, Luigi; Vicovaro, Michele Inferring properties of probability kernels from the pairs of variables they involve. (English) Zbl 1461.62223 Algebr. Stat. 11, No. 1, 79-97 (2020). Summary: 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. MSC: 62R01 Algebraic statistics 62H22 Probabilistic graphical models 62G05 Nonparametric estimation 60E05 Probability distributions: general theory Keywords:probability kernel; conditional probability; compatibility; lattice; Bayesian model PDFBibTeX XMLCite \textit{L. Burigana} and \textit{M. Vicovaro}, Algebr. Stat. 11, No. 1, 79--97 (2020; Zbl 1461.62223) Full Text: DOI References: [1] 10.1214/ss/1009213728 · Zbl 1059.62511 [2] 10.1002/9780470316870 [3] 10.1017/CBO9780511811357 · Zbl 1310.68002 [4] ; Dawid, J. Roy. Statist. Soc. Ser. B, 41, 1 (1979) · Zbl 0408.62004 [5] 10.1146/annurev.psych.55.090902.142005 [6] 10.1007/978-0-387-74101-7 · Zbl 1251.68001 [7] 10.1002/9780470684023 · Zbl 1277.62022 [8] ; Lauritzen, Graphical models. Oxford Statistical Science Series, 17 (1996) · Zbl 0907.62001 [9] 10.1007/978-1-4471-3267-7 [10] ; Pollard, A user’s guide to measure theoretic probability. Cambridge Series in Statistical and Probabilistic Mathematics, 8 (2002) · Zbl 0992.60001 [11] ; Rouder, New handbook of mathematical psychology, 504 (2017) This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.