Nonparametric Bayes inference on conditional independence
Tsuyoshi Kunihama, David B. Dunson · Biometrika · 2016
In many application areas, a primary focus is on assessing evidence in the data refuting the assumption of independence of |$Y$| and |$X$| conditionally on |$Z$|, with |$Y$| response variables, |$X$| predictors of interest, and |$Z$| covariates. Ideally, one would have methods available that avoid parametric assumptions, allow |$Y, X, Z$| to be random variables on arbitrary spaces with arbitrary dimension, and accommodate rapid consideration of different candidate predictors. As a formal decision-theoretic approach has clear disadvantages in this context, we instead rely on an encompassing nonparametric Bayes model for the joint distribution of |$Y$|, |$X$| and |$Z$|, with conditional mutual information used as a summary of the strength of conditional dependence. We construct a functional of the encompassing model and empirical measure for estimation of conditional mutual information. The implementation relies on a single Markov chain Monte Carlo run under the encompassing model, with conditional mutual information for candidate models calculated as a byproduct. We provide an asymptotic theory supporting the approach, and apply the method to variable selection. The methods are illustrated through simulations and criminology applications.