NOTMAD: Estimating Bayesian Networks with Sample-Specific Structures and Parameters

TL;DR

This paper introduces NOTMAD, a novel method for estimating context-specific Bayesian networks that learns to mix archetypal networks based on sample context, enabling high-resolution, sample-specific structure and parameter inference without prior knowledge.

Contribution

The paper proposes NOTMAD, a method that models context-specific Bayesian networks as mixtures of archetypal networks, sharing information across contexts and allowing single-sample resolution estimation.

Findings

Successfully infers sample-specific gene expression networks in cancer.

Demonstrates improved statistical power over subsampling methods.

Enables estimation of Bayesian networks without prior knowledge.

Abstract

Context-specific Bayesian networks (i.e. directed acyclic graphs, DAGs) identify context-dependent relationships between variables, but the non-convexity induced by the acyclicity requirement makes it difficult to share information between context-specific estimators (e.g. with graph generator functions). For this reason, existing methods for inferring context-specific Bayesian networks have favored breaking datasets into subsamples, limiting statistical power and resolution, and preventing the use of multidimensional and latent contexts. To overcome this challenge, we propose NOTEARS-optimized Mixtures of Archetypal DAGs (NOTMAD). NOTMAD models context-specific Bayesian networks as the output of a function which learns to mix archetypal networks according to sample context. The archetypal networks are estimated jointly with the context-specific networks and do not require any prior knowledge. We encode the acyclicity constraint as a smooth regularization loss which is back-propagated to the mixing function; in this way, NOTMAD shares information between context-specific acyclic graphs, enabling the estimation of Bayesian network structures and parameters at even single-sample resolution. We demonstrate the utility of NOTMAD and sample-specific network inference through analysis and experiments, including patient-specific gene expression networks which correspond to morphological variation in cancer.