Probabilistic graphical models have been a success: they have sparked conceptual developments in physics (Gibbs, 1902), genetics (Wright, 1921), social sciences (Wold, 1954), (Blalock, 1971), and statistics (Haberman, 1974). Probabilistic methods gained their own widespread acceptance with Pearl’s development of Bayesian networks (Pearl, 1988) (Pearl, 2009) and Lauritzen and Spiegelhalter’s algorithmic foundations (Lauritzen, Spiegelhalter, 1988).
At the same time, graphical models present an obvious limitation: most of the innovation in machine learning, artificial intelligence, and statistics comes from probabilistic methods far beyond what is representable using graphical models. These are described in a mixture of code, natural language, and mathematics that has been translated — informally, and to a certain extent — to probabilistic programming languages that allow us to declaratively specify probability distributions. Early artificial intelligence largely diverged from the naive probabilistic approach and produced alternative formalisms largely lacking any formal semantics (with few exceptions). As of today, probabilistic graphical methods are reserved mostly for high-level descriptions and coarse specifications that lack the scalability of general probabilistic modelling: we have missed a compositional theory of probabilistic graphical methods.
Fritz’s Markov categories (Fritz, 2020), are a compositional formalization and generalization of Bayesian networks that has catalyzed a recent trend of axiomatic probability theory (Cho, Jacobs, 2017), (Cho, Jacobs, 2022), (Fritz, Liang, 2023), (Perrone, 2023). Their promise is a single, transparent mathematical formalism unifying graphical methods with probabilistic programming. Their current limitation is the lack of realistic fully-fledged probabilistic programming languages based on them: these should mix traditional programming with the causality reasoning and algorithmic analysis of probabilistic graphical models — as recent work in categorical probability highlights. Markov categories still lack good internal languages.
Literature.
- Probabilistic Graphical Models, Principles and Techniques (Koller, Friedman, 2014)
- Introduction to Probabilistic Graphical Models (Pernkopf, Peharz, Tschiatchek)
References
- Elementary Principles in Statistical Mechanics (Gibbs, 1902)
- Systems of Mating II, the Effects of Inbreeding on the Genetic Composition of a Population (Wright, 1921)
- Causality and Econometrics (Wold, 1954)
- Causal Models in the Social Sciences (Blalock, 1971)
- The Analysis of Frequency Data (Haberman, 1974)
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