Causal Inference by String Diagram Surgery

Abstract

Extracting causal relationships from observed correlations is a growing area in probabilistic reasoning, originating with the seminal work of Pearl and others from the early 1990s. This paper develops a new, categorically oriented view based on a clear distinction between syntax (string diagrams) and semantics (stochastic matrices), connected via interpretations as structure-preserving functors.A key notion in the identification of causal effects is that of an intervention, whereby a variable is forcefully set to a particular value independent of any prior dependencies. We represent the effect of such an intervention as an endofunctor which performs 'string diagram surgery' within the syntactic category of string diagrams. This diagram surgery in turn yields a new, interventional distribution via the interpretation functor. While in general there is no way to compute interventional distributions purely from observed data, we show that this is possible in certain special cases using a calculational tool called comb disintegration. We demonstrate the use of this technique on a well-known toy example, where we predict the causal effect of smoking on cancer in the presence of a confounding common cause. After developing this specific example, we show this technique provides simple sufficient conditions for computing interventions which apply to a wide variety of situations considered in the causal inference literature.

On the Equivalence of Causal Models: A Category-Theoretic Approach

“…Due to lack of space, we omit technical details that have no bearing on the following discussion. We refer the reader to Jacobs et al (2019) for the details and to Awodey (2010) or Leinster (2016) for general introductions to category theory. In their approach, a causal graph is reformulated as a string diagram category representing the "syntactical" structure of the graph, while specific causal models are regarded as "semantic" assignments of values and stochastic matrices to each component of the string diagram, i.e., functors from the string diagram category to the category of stochastic matrices Stoch.…”

“…Following Jacobs et al (2019), we first define an intervention as a surgery of a string diagram. An intervention on a variable X ∈ V G is denoted by cut X , which removes the box as well as all the incoming wires of X and replaces them with the "intervened state" x with no input:…”

“…The marginal distribution of X after intervention is given by F G (x) (we thus assume that the model F G already contains the information about how each variable could be manipulated. At this point we depart from the original formulation of Jacobs et al (2019), in which possible post-intervention distributions are restricted to the uniform distribution). Then the whole causal model and joint distribution after the intervention are given by composition of cut X with the causal model functor:…”

“…In this paper we propose a novel criterion and systematic method for determining the equivalence of two causal models, drawing on the category-theoretic formulation of causal models developed by Jacobs et al (2019). In this framework, a causal model is identified with a functor, which is a probabilistic interpretation of a string diagram constructed from the directed acyclic graph (DAG) of the model.…”

Causality analysis and fault ascription in component-based systems

“…In a similar spirit to our definition of counterfactuals, [26] defines the semantics of counterfactual analysis for traces of events of stochastic rule-based models. [23] proposes a general categorical formalization of interventions on a single variable.…”