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We study the sample complexity of causal structure learning on a two-variable system with observational and experimental data. Specifically, for two variables X and Y, we consider the classical scenario where either X causes Y , Y causes X, or there is an unmeasured confounder between X and Y. We show that if X and Y are over a finite domain of size k and are significantly correlated, the minimum number of interventional samples needed is sublinear in k. We give a tight characterization of the tradeoff between observational and interventional data when the number of observational samples is sufficiently large. We build upon techniques for closeness testing and for non-parametric density estimation in different regimes of observational data. Our hardness results are based on carefully constructing causal models whose marginal and interventional distributions form hard instances of canonical results on property testing. 

We study the problems of identity and closeness testing of n-dimensional product distributions. Prior works of Canonne et al. (2017) and Daskalakis and Pan (2017) have established tight sample complexity bounds for non-tolerant testing over a binary alphabet: given two product distributions P and Q over a binary alphabet, distinguish between the cases P = Q and dTV(P;Q) > epsilon . We build on this prior work to give a more comprehensive map of the complexity of testing of product distributions by investigating tolerant testing with respect to several natural distance measures and over an arbitrary alphabet. Our study gives a fine-grained understanding of how the sample complexity of tolerant testing varies with the distance measures for product distributions. In addition, we also extend one of our upper bounds on product distributions to bounded-degree Bayes nets. 

We study the problem of efficiently estimating the effect of an intervention on a single variable using observational samples. Our goal is to give algorithms with polynomial time and sample complexity in a non-parametric setting. Tian and Pearl (AAAI ’02) have exactly characterized the class of causal graphs for which causal effects of atomic interventions can be identified from observational data. We make their result quantitative. Suppose 𝒫 is a causal model on a set V of n observable variables with respect to a given causal graph G, and let do(x) be an identifiable intervention on a variable X. We show that assuming that G has bounded in-degree and bounded c-components (k) and that the observational distribution satisfies a strong positivity condition: (i) [Evaluation] There is an algorithm that outputs with probability 2/3 an evaluator for a distribution P^ that satisfies TV(P(V | do(x)), P^(V)) < eps using m=O (n/eps^2) samples from P and O(mn) time. The evaluator can return in O(n) time the probability P^(v) for any assignment v to V. (ii) [Sampling] There is an algorithm that outputs with probability 2/3 a sampler for a distribution P^ that satisfies TV(P(V | do(x)), P^(V)) < eps using m=O (n/eps^2) samples from P and O(mn) time. The sampler returns an iid sample from P^ with probability 1 in O(n) time. We extend our techniques to estimate P(Y | do(x)) for a subset Y of variables of interest. We also show lower bounds for the sample complexity, demonstrating that our sample complexity has optimal dependence on the parameters n and eps, as well as if k=1 on the strong positivity parameter. 

Eliciting causal effects from interventions and observations is one of the central concerns of science, and increasingly, artificial intelligence. We provide an algorithm that, given a causal graph G, determines MIC(G), a minimum intervention cover of G, i.e., a minimum set of interventions that suffices for identifying every causal effect that is identifiable in a causal model characterized by G. We establish the completeness of do-calculus for computing MIC(G). MIC(G) effectively offers an efficient compilation of all of the information obtainable from all possible interventions in a causal model characterized by G. Minimum intervention cover finds applications in a variety of contexts including counterfactual inference, and generalizing causal effects across experimental settings. We analyze the computational complexity of minimum intervention cover and identify some special cases of practical interest in which MIC(G) can be computed in time that is polynomial in the size of G. 

We consider testing and learning problems on causal Bayesian networks as defined by Pearl (Pearl, 2009). Given a causal Bayesian network  on a graph with n discrete variables and bounded in-degree and bounded `confounded components', we show that O(logn) interventions on an unknown causal Bayesian network  on the same graph, and Õ (n/ϵ2) samples per intervention, suffice to efficiently distinguish whether = or whether there exists some intervention under which  and  are farther than ϵ in total variation distance. We also obtain sample/time/intervention efficient algorithms for: (i) testing the identity of two unknown causal Bayesian networks on the same graph; and (ii) learning a causal Bayesian network on a given graph. Although our algorithms are non-adaptive, we show that adaptivity does not help in general: Ω(logn) interventions are necessary for testing the identity of two unknown causal Bayesian networks on the same graph, even adaptively. Our algorithms are enabled by a new subadditivity inequality for the squared Hellinger distance between two causal Bayesian networks. 

We consider testing and learning problems on causal Bayesian networks as defined by Pearl (Pearl, 2009). Given a causal Bayesian network  on a graph with n discrete variables and bounded in-degree and bounded `confounded components', we show that O(logn) interventions on an unknown causal Bayesian network  on the same graph, and Õ (n/ϵ2) samples per intervention, suffice to efficiently distinguish whether = or whether there exists some intervention under which  and  are farther than ϵ in total variation distance. We also obtain sample/time/intervention efficient algorithms for: (i) testing the identity of two unknown causal Bayesian networks on the same graph; and (ii) learning a causal Bayesian network on a given graph. Although our algorithms are non-adaptive, we show that adaptivity does not help in general: Ω(logn) interventions are necessary for testing the identity of two unknown causal Bayesian networks on the same graph, even adaptively. Our algorithms are enabled by a new subadditivity inequality for the squared Hellinger distance between two causal Bayesian networks.