Margins of discrete Bayesian networks

Abstract

Bayesian network models with latent variables are widely used in statistics and machine learning. In this paper we provide a complete algebraic characterization of Bayesian network models with latent variables when the observed variables are discrete and no assumption is made about the state-space of the latent variables. We show that it is algebraically equivalent to the so-called nested Markov model, meaning that the two are the same up to inequality constraints on the joint probabilities. In particular these two models have the same dimension. The nested Markov model is therefore the best possible description of the latent variable model that avoids consideration of inequalities, which are extremely complicated in general. A consequence of this is that the constraint finding algorithm of Tian and Pearl (UAI 2002, pp519-527) is complete for finding equality constraints. Latent variable models suffer from difficulties of unidentifiable parameters and non-regular asymptotics; in contrast the nested Markov model is fully identifiable, represents a curved exponential family of known dimension, and can easily be fitted using an explicit parameterization.

41 pages

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• The Inflation Technique Completely Solves the Causal Compatibility Problem
• Quantum violations in the Instrumental scenario and their relations to the Bell scenario
• Markov Properties for Graphical Models with Cycles and Latent Variables
• Causal Structure Learning: a Combinatorial Perspective
• Constraint-based Causal Discovery for Non-Linear Structural Causal Models with Cycles and Latent Confounders
• Differentiable Causal Discovery Under Unmeasured Confounding
• Estimating causal structure using conditional DAG models
• Semiparametric Inference For Causal Effects In Graphical Models With Hidden Variables
• Partial Counterfactual Identification from Observational and Experimental Data
• Estimation of causal effects with small data in the presence of trapdoor variables
• Algebraic Problems in Structural Equation Modeling
• Algebraic Equivalence of Linear Structural Equation Models
• Smooth, identifiable supermodels of discrete DAG models with latent variables
• Partial Identifiability in Discrete Data With Measurement Error
• Distinguishing quantum causal scenarios with indistinguishable classical analogs: The significance of intermediate latents
• Latent-free equivalent mDAGs
• Learning latent causal graphs via mixture oracles
• A Class of Algorithms for General Instrumental Variable Models
• Variable elimination, graph reduction and efficient g-formula
• Towards Characterising Bayesian Network Models under Selection
• A decompositional framework for process theories in spacetime
• Dependency in DAG models with Hidden Variables