Semidefinite tests for latent causal structures

TL;DR

This paper introduces a semidefinite programming approach to test whether observed data is compatible with latent causal structures in Bayesian networks, offering a computationally efficient alternative to algebraic methods.

Contribution

The authors develop a novel semidefinite test for latent causal models that is more computationally efficient than traditional algebraic geometric techniques.

Findings

Semidefinite tests can efficiently identify compatible latent structures.

The method outperforms entropic inequality-based tests in certain scenarios.

The approach simplifies testing causal compatibility in complex models.

Abstract

Testing whether a probability distribution is compatible with a given Bayesian network is a fundamental task in the field of causal inference, where Bayesian networks model causal relations. Here we consider the class of causal structures where all correlations between observed quantities are solely due to the influence from latent variables. We show that each model of this type imposes a certain signature on the observable covariance matrix in terms of a particular decomposition into positive semidefinite components. This signature, and thus the underlying hypothetical latent structure, can be tested in a computationally efficient manner via semidefinite programming. This stands in stark contrast with the algebraic geometric tools required if the full observable probability distribution is taken into account. The semidefinite test is compared with tests based on entropic inequalities.