The Inflation Technique Completely Solves the Causal Compatibility Problem

TL;DR

This paper introduces a hierarchy of inflation-based relaxations that asymptotically and efficiently determine causal compatibility, solving a longstanding problem in causal inference with proven convergence and practical effectiveness.

Contribution

It develops a formal hierarchy of inflation tests that are proven to be asymptotically tight, providing a complete solution to the causal compatibility problem.

Findings

Hierarchy converges to a zero-error causal compatibility test

Any distribution passing the n-th order test is close to a compatible distribution

First or second order tests suffice for many causal structures

Abstract

The causal compatibility question asks whether a given causal structure graph -- possibly involving latent variables -- constitutes a genuinely plausible causal explanation for a given probability distribution over the graph's observed variables. Algorithms predicated on merely necessary constraints for causal compatibility typically suffer from false negatives, i.e. they admit incompatible distributions as apparently compatible with the given graph. In [arXiv:1609.00672], one of us introduced the inflation technique for formulating useful relaxations of the causal compatibility problem in terms of linear programming. In this work, we develop a formal hierarchy of such causal compatibility relaxations. We prove that inflation is asymptotically tight, i.e., that the hierarchy converges to a zero-error test for causal compatibility. In this sense, the inflation technique fulfills a longstanding desideratum in the field of causal inference. We quantify the rate of convergence by showing that any distribution which passes the $n^{th}$-order inflation test must be $O\left(n^{-1/2}\right)$-close in Euclidean norm to some distribution genuinely compatible with the given causal structure. Furthermore, we show that for many causal structures, the (unrelaxed) causal compatibility problem is faithfully formulated already by either the first or second order inflation test.