Optimal structure learning and conditional independence testing

A new theoretical paper establishes that the minimax optimal rate for structure learning problems is fundamentally determined by the minimax rate for conditional independence testing, providing a unified statistical framework for the field. The finding, posted on arXiv and last revised in June 2026, demonstrates a general reduction between structure learning and conditional independence testing specifically for poly-forests, a class of graphical models [1][2]. The work derives optimal rates for several model types, including Bernoulli, Gaussian, and nonparametric settings [2]. Graphical models are probabilistic models where a graph expresses the conditional dependence structure between random variables, and they are commonly used in statistics and machine learning [3]. Structure learning refers to the task of recovering this graph from observed data. The new result shows that the statistical difficulty of learning the graph is precisely characterized by the difficulty of testing whether certain variables are conditionally independent. The authors show that a suitable modification of the PC algorithm, a classic constraint-based method for learning Bayesian networks, achieves the optimal rate in these settings [2]. Bayesian networks represent variables and their conditional dependencies via a directed acyclic graph and are widely used for reasoning under uncertainty, such as modeling probabilistic relationships between diseases and symptoms [5]. The paper, authored by Ming Gao, was first submitted in July 2025 and underwent two revisions, growing from 68 KB to 75 KB in its third version [1]. The theoretical contribution offers a lens through which the statistical complexity of structure learning can be analyzed using the established tools of minimax hypothesis testing [2]. Statistical inference, the broader field underpinning this work, involves using data analysis to infer properties of an underlying probability distribution [6]. By linking structure learning directly to conditional independence testing, the framework may simplify the analysis of algorithms designed to uncover dependence relationships in high-dimensional data. Copulas, for instance, are another statistical tool used to model dependence between random variables, often employed in quantitative finance to describe tail risk [7]. The arXiv paper provides a mathematical foundation that connects these testing procedures to the guarantees that can be offered when learning the entire dependence graph.