Efficient Bayesian network structure learning via local Markov boundary search

TL;DR

This paper introduces an efficient, information-theoretic method for learning Bayesian network structures using local Markov boundary search, achieving polynomial sample complexity and broad applicability without distributional assumptions.

Contribution

It proposes a simple greedy algorithm for Markov boundary discovery that improves sample complexity and extends to general graph structures under new identifiability conditions.

Findings

Polynomial sample complexity for Markov boundary recovery.

Successful application to polytrees with explicit identifiability conditions.

Algorithm performs well in simulation studies.

Abstract

We analyze the complexity of learning directed acyclic graphical models from observational data in general settings without specific distributional assumptions. Our approach is information-theoretic and uses a local Markov boundary search procedure in order to recursively construct ancestral sets in the underlying graphical model. Perhaps surprisingly, we show that for certain graph ensembles, a simple forward greedy search algorithm (i.e. without a backward pruning phase) suffices to learn the Markov boundary of each node. This substantially improves the sample complexity, which we show is at most polynomial in the number of nodes. This is then applied to learn the entire graph under a novel identifiability condition that generalizes existing conditions from the literature. As a matter of independent interest, we establish finite-sample guarantees for the problem of recovering Markov boundaries from data. Moreover, we apply our results to the special case of polytrees, for which the assumptions simplify, and provide explicit conditions under which polytrees are identifiable and learnable in polynomial time. We further illustrate the performance of the algorithm, which is easy to implement, in a simulation study. Our approach is general, works for discrete or continuous distributions without distributional assumptions, and as such sheds light on the minimal assumptions required to efficiently learn the structure of directed graphical models from data.