A Bayesian nonparametric test for conditional independence

Abstract

This article introduces a Bayesian nonparametric method for quantifying the relative evidence in a dataset in favour of the dependence or independence of two variables conditional on a third. The approach uses Pólya tree priors on spaces of conditional probability densities, accounting for uncertainty in the form of the underlying distributions in a nonparametric way. The Bayesian perspective provides an inherently symmetric probability measure of conditional dependence or independence, a feature particularly advantageous in causal discovery and not employed in existing procedures of this type.

A Bayesian Nonparametric Conditional Two-sample Test with an Application to Local Causal Discovery

“…We now propose a novel conditional independence test of the type C |= X|Z, where X and Z are continuous one-dimensional random variables and C is a binary variable. We combine elements of both the two-sample test proposed by Holmes et al (2015), and a conditional independence test from Teymur and Filippi (2019). The conditional independence test from Teymur and Filippi (2019) Given n draws {(C 1 , X 1 , Z 1 ), ..., (C n , X n , Z n )} from binary variable C and continuous onedimensional random variables X and Z, define X (0) := {X i : C i = 0, i ∈ {1, .., n}} and X (1) := {X i : C i = 1, i ∈ {1, .., n}}.…”

“…(5) Teymur and Filippi (2019) utilise conditional optional Pólya tree (cond-OPT) priors (Ma, 2017) for modelling conditional densities of the form f X|Z (x|z). As we require an expression for the marginal likelihoods of X|Z, X (0) |Z and X (1) |Z, we briefly review the construction of the cond-OPT.…”

“…A default choice for the (conditional) independence test is the (partial) correlation test, which can fail in presence of nonlinear relations between the variables. Recent research proposes a Bayesian nonparametric two-sample test (Holmes et al, 2015), an independence test between continuous variables (Filippi and Holmes, 2017), and a conditional independence test between continuous variables (Teymur and Filippi, 2019). We extend this work by proposing a novel Bayesian nonparametric conditional two-sample test.…”

“…Among the independence tests that are based on Pólya tree priors is a recently proposed conditional independence test (Teymur and Filippi, 2019) which extends a continuous marginal independence test (Filippi and Holmes, 2017) by utilising conditional optional Pólya trees (Ma, 2017). Although…”

“…E(P(C κ )) = Cκ G (x)dx. As argued by Lavine (1994) we will only consider partitions up to a pre-determined level J, making P into a truncated Pólya Tree (Teymur and Filippi, 2019). Hanson and Johnson (2002) provide a rule of thumb J = log 2 (n) , which corresponds to on average finding one observation in each element of the partition.…”