Bayesian networks are among the most prominent graphical models for probability distributions.

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.

Why the level of dependency grows with the number of conditions in M. PolyTree decomposition is similar to

Why the level of dependency grows with the number of conditions in M. PolyTree decomposition is similar to

Ancestral graphs are a prevalent mathematical tool to take into account latent (hidden) variables in a probabilistic graphical model.

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. 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.

We develop optimal algorithms for learning undirected Gaussian trees and directed Gaussian polytrees from data. We consider both problems of distribution learning (i.e. in KL distance) and structure learning (i.e. exact recovery). The first approach is based on the Chow-Liu algorithm, and learns an optimal tree-structured distribution efficiently. The second approach is a modification of the PC algorithm for polytrees that uses partial correlation as a conditional independence tester for constraint-based structure learning. We derive explicit finite-sample guarantees for both approaches, and show that both approaches are optimal by deriving matching lower bounds. Additionally, we conduct numerical experiments to compare the performance of various algorithms, providing further insights and empirical evidence.