On the Complexity of Identification in Linear Structural Causal Models

Julian Dörfler, Benito van der Zander, Markus Bläser, Maciej Liskiewicz

Keywords: causal inference, structural causal models, structural equational models, generic identifiability, existential theory over the reals

TL;DR: We prove a PSPACE upper bound for deciding generic identifiability and give the first hardness result for variants of identification.

Abstract:

Learning the unknown causal parameters of a linear structural causal
model is a fundamental task in causal analysis. The task, known as the
problem of identification, asks to estimate the parameters of the model from a
combination of assumptions on the graphical structure of the model and
observational data, represented as a non-causal covariance matrix.
In this paper, we give a new sound and complete algorithm for generic
identification which runs in polynomial space. By a standard simulation
result, namely $\mathsf{PSPACE} \subseteq \mathsf{EXP}$,
this algorithm has exponential running time which vastly improves
the state-of-the-art double exponential time method using a Gröbner basis
approach. The paper also presents evidence that parameter identification
is computationally hard in general. In particular, we prove, that the task
asking whether, for a given feasible correlation matrix, there
are exactly one or two or more parameter sets explaining the observed
matrix, is hard for $\forall \mathbb{R}$, the co-class of the existential theory
of the reals. In particular, this problem is $\mathsf{coNP}$-hard.
To our best knowledge, this is the first hardness result for some notion
of identifiability.

Primary Area: Causal inference

Submission Number: 10241