Efficient inference of interventional distributions

• URL: http://arxiv.org/abs/2107.11712v2
• Date: Tue, 27 Jul 2021 15:14:09 GMT
• Title: Efficient inference of interventional distributions
• Authors: Arnab Bhattacharyya, Sutanu Gayen, Saravanan Kandasamy, Vedant Raval, N. V. Vinodchandran
• Abstract summary: We consider the problem of efficiently inferring interventional distributions in a causal Bayesian network from a finite number of observations. We show that when $mathbfY$ is an arbitrary set, there is no efficient algorithm that outputs an evaluator of a distribution that is $varepsilon$-close to $P_bf x(mathbfY)$ unless all problems that have statistical zero-knowledge, including the Graph Isomorphism problem, have efficient randomized algorithms.
• Score: 13.31079561447385
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We consider the problem of efficiently inferring interventional distributions in a causal Bayesian network from a finite number of observations. Let $\mathcal{P}$ be a causal model on a set $\mathbf{V}$ of observable variables on a given causal graph $G$. For sets $\mathbf{X},\mathbf{Y}\subseteq \mathbf{V}$, and setting ${\bf x}$ to $\mathbf{X}$, let $P_{\bf x}(\mathbf{Y})$ denote the interventional distribution on $\mathbf{Y}$ with respect to an intervention ${\bf x}$ to variables ${\bf x}$. Shpitser and Pearl (AAAI 2006), building on the work of Tian and Pearl (AAAI 2001), gave an exact characterization of the class of causal graphs for which the interventional distribution $P_{\bf x}({\mathbf{Y}})$ can be uniquely determined. We give the first efficient version of the Shpitser-Pearl algorithm. In particular, under natural assumptions, we give a polynomial-time algorithm that on input a causal graph $G$ on observable variables $\mathbf{V}$, a setting ${\bf x}$ of a set $\mathbf{X} \subseteq \mathbf{V}$ of bounded size, outputs succinct descriptions of both an evaluator and a generator for a distribution $\hat{P}$ that is $\varepsilon$-close (in total variation distance) to $P_{\bf x}({\mathbf{Y}})$ where $Y=\mathbf{V}\setminus \mathbf{X}$, if $P_{\bf x}(\mathbf{Y})$ is identifiable. We also show that when $\mathbf{Y}$ is an arbitrary set, there is no efficient algorithm that outputs an evaluator of a distribution that is $\varepsilon$-close to $P_{\bf x}({\mathbf{Y}})$ unless all problems that have statistical zero-knowledge proofs, including the Graph Isomorphism problem, have efficient randomized algorithms.

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