Multilinear and Linear Programs for Partially Identifiable Queries in Quasi-Markovian Structural Causal Models

João Pedro Arroyo, Denis Mauá, João Gabriel Barreto Rodrigues, Daniel A. E. Lawand, Junkyu Lee, Radu Marinescu, Alexander G. Gray, Eduardo Rocha Laurentino, Fabio Cozman

Keywords: Quasi-Markovian Structural Causal Model, Partial Identifiability, Linear Programming

TL;DR: We propose new results and techniques concerning the computation of tight probability bounds for partially identifiable queries in quasi- Markovian Structural Causal Models (where ob- servable variables are connected with at most one confounder).

Abstract: We investigate partially identifiable queries in a class of causal models. We focus on acyclic Structural Causal Models that are quasi-Markovian (that is, each endogenous variable is connected with at most one exogenous confounder). We look into scenarios where endogenous variables are observed (and a distribution over them is known), while exogenous variables are not fully specified. This leads to a representation that is in essence a Bayesian network where the distribution of root variables is not uniquely determined. In such circumstances, it may not be possible to precisely compute a probability value of interest. We thus study the computation of tight probability bounds, a problem that has been solved by multilinear programming in general, and by linear programming when a single confounded component is intervened upon. We present a new algorithm to simplify the construction of such programs by exploiting input probabilities over endogenous variables. For scenarios with a single intervention, we apply column generation to compute a probability bound through a sequence of auxiliary linear integer programs, thus showing that a representation with polynomial cardinality for exogenous variables is possible. Experiments show column generation techniques to be superior to existing methods.

Submission Number: 8