Abstract
AbstractThe 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 categorical 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 10.1515/jci-2017-0020, 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 nth-order inflation test must be $\begin{array}{} \displaystyle {O}{\left(n^{{{-}{1}}/{2}}\right)} \end{array}$-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.
Highlights
A Bayesian network or causal structure is a directed acyclic graph (DAG) where vertices represent random variables, each of which is generated by a non-deterministic function depending on the values of its parents
We introduce a graphical preprocessing which precisely recasts the general causal compatibility problem in terms of causal compatibility with correlation scenarios, such that there is no loss of generality in our approach
This paper is organized as follows: In Section 2 we introduce the concept of a correlation scenario, we define primal and dual notions of the causal compatibility problem and their approximations
Summary
A Bayesian network or causal structure is a directed acyclic graph (DAG) where vertices represent random variables, each of which is generated by a non-deterministic function depending on the values of its parents. Demanding faithfulness can be thought of as a second filtering step, where the fundamental filtering of causal discovery is the exclusion of any causal structure which cannot explain the observed probability distribution, even granting fine-tuning. Our main result here is that this error asymptotically tends to zero when inflation is expressed as a hierarchy of ever-higher-order tests of causal compatibility This implies that the inflation criterion — far from being a relaxation — is meta-equivalent to the causal compatibility problem, and constitutes an alternative way of understanding general causal structures. This preprocessing — fairly useful in its own right — implies the universal applicability of the inflation technique as defined here.
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