Causal Graphical Models: Heterogeneous Data Meet Structure Identification

Abstract

Causal relationship, rather than statistical association, provides the basic understanding of nature. Learning causal structure from observational data is one of the most fundamental ways to explore causality, and the process is often facilitated by causal graphical models. However, most existing approaches ignore the issue of data heterogeneity, and one only considers the case where each measured variable is a scalar. In this dissertation, we provide approaches to address heterogeneous observational data and functional measurements in causal structure learning.

To address the heterogeneity problem, we propose a novel causal Bayesian network model that embeds heterogeneous samples onto a low-dimensional manifold and builds Bayesian networks conditional on the embedding. The new framework allows for more precise network inference by improving the estimation resolution from population level to observation level. Moreover, while causal Bayesian networks are in general not identifiable with purely observational data due to Markov equivalence, with the blessing of causal effect heterogeneity, we prove that the causal structure is uniquely identifiable with our proposed model under mild assumptions. Furthermore, while cycles and unmeasured confounders are inevitable in nature causal systems, we show our general model class that accommodates these structures still allows causal identification.

To address the functional measurements, we develop a novel Bayesian network model for mul-tivariate functional data where the conditional independence and causal structure are represented by a directed acyclic graph. Our model is built on the strategy of adaptive basis expansion. We show a special case where the functional objects are drawn from a mixture of Gaussian processes, which allows unique causal structure identification even when the functional data are purely observational and measured with noise.