Finding Optimal Bayesian Networks

TL;DR

This paper extends the theoretical understanding of greedy Bayesian network search algorithms, showing they can identify inclusion-optimal models under realistic assumptions like the composition property, even with hidden variables and selection bias.

Contribution

It relaxes previous assumptions to demonstrate that greedy search algorithms can find inclusion-optimal models under broader, more realistic conditions.

Findings

Guarantees identification of inclusion-optimal models under the composition property.

Shows the composition property holds with various generative models including unobserved variables.

Extends optimality results to scenarios with selection bias.

Abstract

In this paper, we derive optimality results for greedy Bayesian-network search algorithms that perform single-edge modifications at each step and use asymptotically consistent scoring criteria. Our results extend those of Meek (1997) and Chickering (2002), who demonstrate that in the limit of large datasets, if the generative distribution is perfect with respect to a DAG defined over the observable variables, such search algorithms will identify this optimal (i.e. generative) DAG model. We relax their assumption about the generative distribution, and assume only that this distribution satisfies the {em composition property} over the observable variables, which is a more realistic assumption for real domains. Under this assumption, we guarantee that the search algorithms identify an {em inclusion-optimal} model; that is, a model that (1) contains the generative distribution and (2) has no sub-model that contains this distribution. In addition, we show that the composition property is guaranteed to hold whenever the dependence relationships in the generative distribution can be characterized by paths between singleton elements in some generative graphical model (e.g. a DAG, a chain graph, or a Markov network) even when the generative model includes unobserved variables, and even when the observed data is subject to selection bias.