Efficient Symbolic Computations for Identifying Causal Effects

Identification conditions describe the computability of a target query or parameter of interest as a function of the type and amount of information available. In causal identification, this information is often expressed in the form of a causal graph, and data are observed or collected for some subset of variables in the graph. Target queries may be for a single effect alone or for a class of effects in a given model. The derivation of an identification algorithm then defines mathematically the process by which the desired causal effect(s) can be uniquely determined, theoretically, in expectation. Identifiability in expectation, or 'theoretical identifiability,' generally assumes asymptotic properties, infinite data, or other mathematically idealized conditions. In this paper, we explore a fundamental distinction between this theoretical, idealized notion of identifiability and a proposed alternative that is computation-bound. The framework we propose - 'computational identifiability' - is to instead define a finite computational search procedure for an empirical estimator. If this process finds an estimator empirically, within a desired error tolerance, then identifiability is satisfied, conditional on the specified assumptions of the search (i.e., a prior distribution over the parameters) and conditional on the search procedure itself. Through several experiments, we demonstrate how this framework allows us to answer fine-grained, practical identification questions, such as identification with small finite samples, with ambiguous graphical criteria, with mixed observational-interventional data, and across counterfactual data and estimands. Code is available at https://github.com/lbynum/metadentify.

We study identifiability in continuous-time linear stationary stochastic differential equations with a known causal structure. Unlike existing approaches, we relax the assumption of a known diffusion matrix, thereby respecting the model's intrinsic scale invariance. Therefore, rather than recovering drift coefficients themselves, we introduce edge-sign identifiability: for a given causal structure, we ask whether the sign of a given drift entry is uniquely determined across all observational covariance matrices induced by parametrisations compatible with that structure. This leads to a trichotomy of edge-sign identifiability: identifiable, non-identifiable, and partially identifiable. This trichotomy introduces the new notion of partial identifiability to the literature, which we show is a genuine category in our setting. Under a notion of faithfulness, we derive criteria to identify membership of each category for general graphs. Applying our criteria to specific causal structures, both analogous to classical causal settings (e.g., instrumental variables) and novel cyclic settings, we determine their edge-sign identifiability and, in some cases, obtain explicit expressions for the sign of a target edge in terms of the observational covariance matrix.

Causal discovery aims to recover causal relationships from observed data. In various fields, exploring causal relationships among variables remains an important topic, but this task becomes challenging due to the existence of latent confounders. Ignoring such confounders can lead to false associations and incorrect edge directions. In this paper, we study the linear structural equation model with latent confounders. We propose an algorithm that iteratively identifies terminal (observed) nodes and reconstructs the directed acyclic graph of the observed variables. To do this, we recover the precision matrix of the observed variables as a sparse plus low-rank matrix: a sparse matrix captures the conditional dependencies among observed variables, while a low-rank matrix captures the combined influence of a few latent confounders. We establish that for $p$ observed variables, $r$ latent confounders and $s$ edges, our procedure correctly identifies the directed causal relationship among observed variables, for $n \gtrsim \max\{s\log p,\ r p\}$ samples. Experimental results validate our theoretical contributions.