Bayesian model selection for latent variable causal networks by sequential Monte Carlo

Abstract

Inferring the causal structure of several random variables is a challenging task when interventions are not feasible. Presence of latent confounders further increases the difficulty of the problem, and therefore is neglected in the majority of the causal discovery literature. In this thesis, we adopt a Bayesian approach to causal structure learning by building on the assumption of the independence of cause and effect mecha nisms. Without any additional assumptions, we reformulate causal structure learning as a Bayesian model selection problem where we compare appropriate graph structures using the marginal likelihood of associated graphs. In the presence of confounders, marginal likelihood computation is equivalent to scoring Bayesian networks with latent variables, which is known to be computationally intractable. In order to approximate this quantity, we develop a sequential Monte Carlo algorithm that provides an asymptotically unbiased estimator, along with a Variational Bayes algorithm that provides a variational lower bound for the marginal likelihood. We particularly analyze the mixture of linear basis functions model with Gaussian noise, which is a frequently encountered modelling choice in the empirical literature. In this model, statistical independence of parameters renders Markov equivalent graphs distinguishable, and allows the identification of a unique causal graph. We illustrate the performance of our framework in both synthetic and real data sets, focusing on the bivariate case. Our direct approach seems to perform at the level of state of the art causal discovery methods. The generalizability of our approach makes it a promising framework for large scale causal structure learning.