Parameter identifiability of discrete Bayesian networks with hidden variables

TL;DR

This paper investigates the parameter identifiability of discrete Bayesian networks with hidden variables, providing algebraic proofs for small models and exploring the impact of Markov equivalence classes on generic identifiability.

Contribution

It introduces algebraic methods to establish identifiability in small Bayesian network models and shows that identifiability depends only on the Markov equivalence class for fixed state spaces.

Findings

Some models have 4-to-one parameterizations, complicating causal interpretation.

Identifiability depends solely on the Markov equivalence class of the DAG.

Binary networks with up to five variables exhibit surprising non-identifiability patterns.

Abstract

Identifiability of parameters is an essential property for a statistical model to be useful in most settings. However, establishing parameter identifiability for Bayesian networks with hidden variables remains challenging. In the context of finite state spaces, we give algebraic arguments establishing identifiability of some special models on small DAGs. We also establish that, for fixed state spaces, generic identifiability of parameters depends only on the Markov equivalence class of the DAG. To illustrate the use of these results, we investigate identifiability for all binary Bayesian networks with up to five variables, one of which is hidden and parental to all observable ones. Surprisingly, some of these models have parameterizations that are generically 4-to-one, and not 2-to-one as label swapping of the hidden states would suggest. This leads to interesting difficulties in interpreting causal effects.