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Causal effect identification is concerned with determining whether a causal effect is computable from a combination of qualitative assumptions about the underlying system (e.g., a causal graph) and distributions collected from this system. Many identification algorithms exclusively rely on graphical criteria made of a non-trivial combination of probability axioms, do-calculus, and refined c-factorization (e.g., Lee & Bareinboim, 2020). In a sequence of increasingly sophisticated results, it has been shown how proxy variables can be used to identify certain effects that would not be otherwise recoverable in challenging scenarios through solving matrix equations (e.g., Kuroki & Pearl, 2014; Miao et al., 2018). In this paper, we develop a new causal identification algorithm which utilizes both graphical criteria and matrix equations. Specifically, we first characterize the relationships between certain graphically-driven formulae and matrix multiplications. With such characterizations, we broaden the spectrum of proxy variable based identification conditions and further propose novel intermediary criteria based on the pseudoinverse of a matrix. Finally, we devise a causal effect identification algorithm, which accepts as input a collection of marginal, conditional, and interventional distributions, integrating enriched matrix-based criteria into a graphical identification approach. 

The Ladder of Causation describes three qualitatively different types of activities an agent may be interested in engaging in, namely, seeing (observational), doing (interventional), and imagining (counterfactual) (Pearl and Mackenzie, 2018). The inferential challenge imposed by the causal hierarchy is that data is collected by an agent observing or intervening in a system (layers 1 and 2), while its goal may be to understand what would have happened had it taken a different course of action, contrary to what factually ended up happening (layer 3). While there exists a solid understanding of the conditions under which cross-layer inferences are allowed from observations to interventions, the results are somewhat scarcer when targeting counterfactual quantities. In this paper, we study the identification of nested counterfactuals from an arbitrary combination of observations and experiments. Specifically, building on a more explicit definition of nested counterfactuals, we prove the counterfactual unnesting theorem (CUT), which allows one to map arbitrary nested counterfactuals to unnested ones. For instance, applications in mediation and fairness analysis usually evoke notions of direct, indirect, and spurious effects, which naturally require nesting. Second, we introduce a sufficient and necessary graphical condition for counterfactual identification from an arbitrary combination of observational and experimental distributions. Lastly, we develop an efficient and complete algorithm for identifying nested counterfactuals; failure of the algorithm returning an expression for a query implies it is not identifiable. 

Causal knowledge is sought after throughout data-driven fields due to its explanatory power and potential value to inform decision-making. If the targeted system is well-understood in terms of its causal components, one is able to design more precise and surgical interventions so as to bring certain desired outcomes about. The idea of leveraging the causal understand- ing of a system to improve decision-making has been studied in the literature under the rubric of structural causal bandits (Lee and Bareinboim, 2018). In this setting, (1) pulling an arm corresponds to performing a causal intervention on a set of variables, while (2) the associated rewards are governed by the underlying causal mechanisms. One key assumption of this work is that any observed variable (X) in the system is manipulable, which means that intervening and making do(X = x) is always realizable. In many real-world scenarios, however, this is a too stringent requirement. For instance, while scientific evidence may support that obesity shortens life, it’s not feasible to manipulate obesity directly, but, for example, by decreasing the amount of soda consumption (Pearl, 2018). In this paper, we study a relaxed version of the structural causal bandit problem when not all variables are manipulable. Specifically, we develop a procedure that takes as argument partially specified causal knowledge and identifies the possibly-optimal arms in structural bandits with non-manipulable variables. We further introduce an algorithm that uncovers non-trivial dependence structure among the possibly-optimal arms. Finally, we corroborate our findings with simulations, which shows that MAB solvers enhanced with causal knowledge and leveraging the newly discovered dependence structure among arms consistently outperform their causal-insensitive counterparts. 

We consider the problem of learning causal re- lationships from relational data. Existing ap- proaches rely on queries to a relational condi- tional independence (RCI) oracle to establish and orient causal relations in such a setting. In practice, queries to a RCI oracle have to be replaced by reliable tests for RCI against available data. Relational data present several unique challenges in testing for RCI. We study the conditions under which traditional iid-based CI tests yield reliable answers to RCI queries against relational data. We show how to conduct CI tests against relational data to robustly recover the underlying relational causal struc- ture. Results of our experiments demonstrate the effectiveness of our proposed approach. 

We study the problem of identifying the best action in a sequential decision-making setting when the reward distributions of the arms exhibit a non-trivial dependence structure, which is governed by the underlying causal model of the domain where the agent is deployed. In this setting, playing an arm corresponds to intervening on a set of variables and setting them to specific values. In this paper, we show that whenever the underlying causal model is not taken into account during the decision-making process, the standard strategies of simultaneously intervening on all variables or on all the subsets of the variables may, in general, lead to suboptimal policies, regardless of the number of interventions performed by the agent in the environment. We formally acknowledge this phenomenon and investigate structural properties implied by the underlying causal model, which lead to a complete characterization of the relationships between the arms’ distributions. We leverage this characterization to build a new algorithm that takes as input a causal structure and finds a minimal, sound, and complete set of qualified arms that an agent should play to maximize its expected reward. We empirically demonstrate that the new strategy learns an optimal policy and leads to orders of magnitude faster convergence rates when compared with its causal-insensitive counterparts. 

Tests of conditional independence (CI) of ran- dom variables play an important role in ma- chine learning and causal inference. Of partic- ular interest are kernel-based CI tests which allow us to test for independence among ran- dom variables with complex distribution func- tions. The efficacy of a CI test is measured in terms of its power and its calibratedness. We show that the Kernel CI Permutation Test (KCIPT) suffers from a loss of calibratedness as its power is increased by increasing the number of bootstraps. To address this limita- tion, we propose a novel CI test, called Self- Discrepancy Conditional Independence Test (SDCIT). SDCIT uses a test statistic that is a modified unbiased estimate of maximum mean discrepancy (MMD), the largest difference in the means of features of the given sample and its permuted counterpart in the kernel-induced Hilbert space. We present results of experi- ments that demonstrate SDCIT is, relative to the other methods: (i) competitive in terms of its power and calibratedness, outperforming other methods when the number of condition- ing variables is large; (ii) more robust with re- spect to the choice of the kernel function; and (iii) competitive in run time. 

Conditional independence (CI) tests play a central role in statistical inference, machine learning, and causal discovery. Most existing CI tests assume that the samples are indepen- dently and identically distributed (i.i.d.). How- ever, this assumption often does not hold in the case of relational data. We define Relational Conditional Independence (RCI), a generaliza- tion of CI to the relational setting. We show how, under a set of structural assumptions, we can test for RCI by reducing the task of test- ing for RCI on non-i.i.d. data to the problem of testing for CI on several data sets each of which consists of i.i.d. samples. We develop Kernel Relational CI test (KRCIT), a nonpara- metric test as a practical approach to testing for RCI by relaxing the structural assumptions used in our analysis of RCI. We describe re- sults of experiments with synthetic relational data that show the benefits of KRCIT relative to traditional CI tests that don’t account for the non-i.i.d. nature of relational data.