Bayesian causal inference in probit graphical models

TL;DR

This paper introduces a Bayesian approach for causal inference in probit graphical models, leveraging DAGs and MCMC to estimate causal effects from observational data with uncertainty quantification.

Contribution

It develops a novel DAG-probit model with Bayesian Model Averaging for causal effect estimation, accounting for DAG and parameter uncertainty.

Findings

Effective MCMC algorithm for posterior inference on DAGs and parameters

Simulation studies validate the model's accuracy in causal effect estimation

Application to gene expression data demonstrates practical utility

Abstract

We consider a binary response which is potentially affected by a set of continuous variables. Of special interest is the causal effect on the response due to an intervention on a specific variable. The latter can be meaningfully determined on the basis of observational data through suitable assumptions on the data generating mechanism. In particular we assume that the joint distribution obeys the conditional independencies (Markov properties) inherent in a Directed Acyclic Graph (DAG), and the DAG is given a causal interpretation through the notion of interventional distribution. We propose a DAG-probit model where the response is generated by discretization through a random threshold of a continuous latent variable and the latter, jointly with the remaining continuous variables, has a distribution belonging to a zero-mean Gaussian model whose covariance matrix is constrained to satisfy the Markov properties of the DAG. Our model leads to a natural definition of causal effect conditionally on a given DAG. Since the DAG which generates the observations is unknown, we present an efficient MCMC algorithm whose target is the posterior distribution on the space of DAGs, the Cholesky parameters of the concentration matrix, and the threshold linking the response to the latent. Our end result is a Bayesian Model Averaging estimate of the causal effect which incorporates parameter, as well as model, uncertainty. The methodology is assessed using simulation experiments and applied to a gene expression data set originating from breast cancer stem cells.