On the physics of nested Markov models: a generalized probabilistic theory perspective

TL;DR

This paper explores the nested Markov model from a generalized probabilistic theory perspective, revealing which distributions are physically realizable and establishing a connection between algebraic causal models and physical theories.

Contribution

It proves the universal validity of nested Markov constraints and introduces intermediate models that better characterize physically implementable distributions within GPT frameworks.

Findings

Nested Markov constraints are theory-independent.

Not all nested Markov distributions are physically realizable.

Intermediate models tighten the characterization of realizable distributions.

Abstract

Determining potential probability distributions with a given causal graph is vital for causality studies. To bypass the difficulty in characterizing latent variables in a Bayesian network, the nested Markov model provides an elegant algebraic approach by listing exactly all the equality constraints on the observed variables. However, this algebraically motivated causal model comprises distributions outside Bayesian networks, and its physical interpretation remains vague. In this work, we inspect the nested Markov model through the lens of generalized probabilistic theory, an axiomatic framework to describe general physical theories. We prove that all the equality constraints defining the nested Markov model are valid theory-independently. At the same time, not every distribution within the nested Markov model is implementable, not even via exotic physical theories associated with generalized probability theories (GPTs). To interpret the origin of such a gap, we study three causal models standing between the nested Markov model and the set of all distributions admitting some GPT realization. Each of the successive three models gives a strictly tighter characterization of the physically implementable distribution set; that is, each successive model manifests new types of GPT-inviolable constraints. We further demonstrate each gap through a specially chosen illustrative causal structure. We anticipate our results will enlighten further explorations on the unification of algebraic and physical perspectives of causality.