A Bayesian nonparametric conditional two-sample test with an application to Local Causal Discovery.

The performance of constraint-based causal discovery algorithms is prominently determined by the performance of the (conditional) independence tests that are being used. 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. We utilise this conditional two-sample test for testing the conditional independence $C \perp \!\!\! \perp Y |X$ where C denotes a Bernoulli random variable, and X and Y are continuous one-dimensional random variables. This enables a nonparametric implementation of the Local Causal Discovery (LCD) algorithm with binary variables in the experimental setup (e.g. an indicator of treatment/control group). We propose a fair performance measure for comparing frequentist and Bayesian tests in the LCD setting. We utilise this performance measure for comparing our Bayesian ensemble with state-of-the-art frequentist tests, and conclude that the Bayesian ensemble has better performance than its frequentist counterparts. We apply our nonparametric implementation of the LCD algorithm to protein expression data.