It is essential to approach the interpretation of our algorithm's results with caution and subject them to critical evaluation. In this section, we provide the definition of partial ancestral graphs (P AGs). A P AG shares the same adjacencies as any MAG in the observational equivalence class of MAGs. Section 2. For any v W, let G In this section, we derive the causal effect for the SMCM in Figure 3(top), i.e., (6), as well as prove D.1 Proof of (6) First, using the law of total probability, we have P(y |do (t = t)) = null Rule 3a, (c) follows from Rule 1, and (g) follows from Rule 2. D.2 Proof of Theorem 3.1 Lemma 1. Suppose Assumptions 1 to 3 hold. Given this claim, Theorem 3.1 follows from Tian and Pearl [2002, Theorem 4].

However, the exact formula for front-door adjustment depends on the structure of the graph, which is difficult to learn in practice.

It is essential to approach the interpretation of our algorithm's results with caution and subject them to critical evaluation. In this section, we provide the definition of partial ancestral graphs (P AGs). A P AG shares the same adjacencies as any MAG in the observational equivalence class of MAGs. Section 2. For any v W, let G In this section, we derive the causal effect for the SMCM in Figure 3(top), i.e., (6), as well as prove D.1 Proof of (6) First, using the law of total probability, we have P(y |do (t = t)) = null Rule 3a, (c) follows from Rule 1, and (g) follows from Rule 2. D.2 Proof of Theorem 3.1 Lemma 1. Suppose Assumptions 1 to 3 hold. Given this claim, Theorem 3.1 follows from Tian and Pearl [2002, Theorem 4].

Causal effect estimation from data typically requires assumptions about the cause-effect relations either explicitly in the form of a causal graph structure within the Pearlian framework, or implicitly in terms of (conditional) independence statements between counterfactual variables within the potential outcomes framework. When the treatment variable and the outcome variable are confounded, front-door adjustment is an important special case where, given the graph, causal effect of the treatment on the target can be estimated using post-treatment variables. However, the exact formula for front-door adjustment depends on the structure of the graph, which is difficult to learn in practice. In this work, we provide testable conditional independence statements to compute the causal effect using front-door-like adjustment without knowing the graph under limited structural side information. We show that our method is applicable in scenarios where knowing the Markov equivalence class is not sufficient for causal effect estimation. We demonstrate the effectiveness of our method on a class of random graphs as well as real causal fairness benchmarks.

Modern machine learning approaches excel in static settings where a large amount of i.i.d. training data are available for a given task. In a dynamic environment, though, an intelligent agent needs to be able to transfer knowledge and re-use learned components across domains. It has been argued that this may be possible through causal models, aiming to mirror the modularity of the real world in terms of independent causal mechanisms. However, the true causal structure underlying a given set of data is generally not identifiable, so it is desirable to have means to quantify differences between models (e.g., between the ground truth and an estimate), on both the observational and interventional level. In the present work, we introduce the Interventional Kullback-Leibler (IKL) divergence to quantify both structural and distributional differences between models based on a finite set of multi-environment distributions generated by interventions from the ground truth. Since we generally cannot quantify all differences between causal models for every finite set of interventional distributions, we propose a sufficient condition on the intervention targets to identify subsets of observed variables on which the models provably agree or disagree.

It is inconceivable how chaotic the world would look to humans, faced with innumerable decisions a day to be made under uncertainty, had they been lacking the capacity to distinguish the relevant from the irrelevant---a capacity which computationally amounts to handling probabilistic independence relations. The highly parallel and distributed computational machinery of the brain suggests that a satisfying process-level account of human independence judgment should also mimic these features. In this work, we present the first rational, distributed, message-passing, process-level account of independence judgment, called $\mathcal{D}^\ast$. Interestingly, $\mathcal{D}^\ast$ shows a curious, but normatively-justified tendency for quick detection of dependencies, whenever they hold. Furthermore, $\mathcal{D}^\ast$ outperforms all the previously proposed algorithms in the AI literature in terms of worst-case running time, and a salient aspect of it is supported by recent work in neuroscience investigating possible implementations of Bayes nets at the neural level. $\mathcal{D}^\ast$ nicely exemplifies how the pursuit of cognitive plausibility can lead to the discovery of state-of-the-art algorithms with appealing properties, and its simplicity makes $\mathcal{D}^\ast$ potentially a good candidate for pedagogical purposes.

In this paper, we extend graph-based identification methods by allowing background knowledge in the form of non-zero parameter values. Such information could be obtained, for example, from a previously conducted randomized experiment, from substantive understanding of the domain, or even an identification technique. To incorporate such information systematically, we propose the addition of auxiliary variables to the model, which are constructed so that certain paths will be conveniently cancelled. This cancellation allows the auxiliary variables to help conventional methods of identification (e.g., single-door criterion, instrumental variables, half-trek criterion), as well as model testing (e.g., d-separation, over-identification). Moreover, by iteratively alternating steps of identification and adding auxiliary variables, we can improve the power of existing identification methods via a bootstrapping approach that does not require external knowledge. We operationalize this method for simple instrumental sets (a generalization of instrumental variables) and show that the resulting method is able to identify at least as many models as the most general identification method for linear systems known to date. We further discuss the application of auxiliary variables to the tasks of model testing and z-identification.