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

Abstract

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 |= 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.Preprint. Under review.

Causality and independence in perfectly adapted dynamical systems

“…We also found LCD triples that imply a directed path from Raf to Erk in the Markov ordering graph, if the system was at equilibrium under saturation conditions. Such LCD triples were also reported by Boeken and Mooij [6], but not by Mooij et al [34]. The detected LCD triples agree with the direction of edges in the Markov ordering graph for the saturated equilibrium model in Figure 10f.…”

“…However, the results of LCD seem to strongly depend on the implementation details. For example, both Mooij et al [34] and Boeken and Mooij [6] report LCD triples in this dataset that imply a directed path from Mek to Raf, as was predicted by the saturated equilibrium model. Furthermore, the assumption that the system was saturated and at equilibrium may have been violated.…”

“…where we assume that I(t) is an external stimulus or perturbation. 6 Roughly speaking, there is a signalling pathway that goes from I(t) to X s (t) to X r (t) to X m (t) to X e (t) with negative feedback from X e (t) on X s (t). As we did for the reaction network with negative feedback in Section 3, we consider the system under saturation conditions.…”