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In constraint-based causal discovery, the existing algorithms systematically use a series of conditional independence (CI) relations observed in the data to recover an equivalence class of causal graphs in the large sample limit. One limitation of these algorithms is that CI tests lose statistical power as conditioning set size increases with finite samples. Recent research proposes to limit the conditioning set size for robust causal discovery. However, the existing algorithms require exhaustive testing of all CI relations with conditioning set sizes up to a certain integer k. This becomes problematic in practice when variables with large support are present, as it makes CI tests less reliable due to near-deterministic relationships, thereby violating the faithfulness assumption. To address this issue, we propose a causal discovery algorithm that only uses CI tests where the conditioning sets are restricted to a given set of conditioning sets including the empty set C. We call such set of CI relations IC conditionally closed. We define the notion of C-Markov equivalence: two causal graphs are C-Markov equivalent if they entail the same set of CI constraints from IC. We propose a graphical representation of C-Markov equivalence and characterize such equivalence between two causal graphs. Our proposed algorithm called the C-PC algorithm is sound for learning the C-Markov equivalence class. We demonstrate the utility of the proposed algorithm via synthetic and real-world experiments in scenarios where variables with large support or high correlation are present in the data. Our source code is available online at github.com/kenneth-lee-ch/cpc. 

Answering counterfactual queries has important applications such as explainability, robustness, and fairness but is challenging when the causal variables are unobserved and the observations are non-linear mixtures of these latent variables, such as pixels in images. One approach is to recover the latent Structural Causal Model (SCM), which may be infeasible in practice due to requiring strong assumptions, e.g., linearity of the causal mechanisms or perfect atomic interventions. Meanwhile, more practical ML-based approaches using naive domain translation models to generate counterfactual samples lack theoretical grounding and may construct invalid counterfactuals. In this work, we strive to strike a balance between practicality and theoretical guarantees by analyzing a specific type of causal query called domain counterfactuals, which hypothesizes what a sample would have looked like if it had been generated in a different domain (or environment). We show that recovering the latent SCM is unnecessary for estimating domain counterfactuals, thereby sidestepping some of the theoretic challenges. By assuming invertibility and sparsity of intervention, we prove domain counterfactual estimation error can be bounded by a data fit term and intervention sparsity term. Building upon our theoretical results, we develop a theoretically grounded practical algorithm that simplifies the modeling process to generative model estimation under autoregressive and shared parameter constraints that enforce intervention sparsity. Finally, we show an improvement in counterfactual estimation over baseline methods through extensive simulated and image-based experiments.