Characterization and Learning of Causal Graphs with Small Conditioning Sets

Constraint-based causal discovery algorithms learn part of the causal graph structure by systematically testing conditional independences observed in the data. These algorithms, such as the PC algorithm and its variants, rely on graphical characterizations of the so-called equivalence class of causal graphs proposed by Pearl. However, constraint-based causal discovery algorithms struggle when data is limited since conditional independence tests quickly lose their statistical power, especially when the conditioning set is large. To address this, we propose using conditional independence tests where the size of the conditioning set is upper bounded by some integer $k$ for robust causal discovery. The existing graphical characterizations of the equivalence classes of causal graphs are not applicable when we cannot leverage all the conditional independence statements. We first define the notion of $k$-Markov equivalence: Two causal graphs are $k$-Markov equivalent if they entail the same conditional independence constraints where the conditioning set size is upper bounded by $k$. We propose a novel representation that allows us to graphically characterize $k$-Markov equivalence between two causal graphs. We propose a sound constraint-based algorithm called the $k$-PC algorithm for learning this equivalence class. Finally, we conduct synthetic, and semi-synthetic experiments to demonstrate that the $k$-PC algorithm enables more robust causal discovery in the small sample regime compared to the baseline PC algorithm.Comment: 30 page

Approximate Causal Effect Identification under Weak Confounding

Causal effect estimation has been studied by many researchers when only observational data is available. Sound and complete algorithms have been developed for pointwise estimation of identifiable causal queries. For non-identifiable causal queries, researchers developed polynomial programs to estimate tight bounds on causal effect. However, these are computationally difficult to optimize for variables with large support sizes. In this paper, we analyze the effect of "weak confounding" on causal estimands. More specifically, under the assumption that the unobserved confounders that render a query non-identifiable have small entropy, we propose an efficient linear program to derive the upper and lower bounds of the causal effect. We show that our bounds are consistent in the sense that as the entropy of unobserved confounders goes to zero, the gap between the upper and lower bound vanishes. Finally, we conduct synthetic and real data simulations to compare our bounds with the bounds obtained by the existing work that cannot incorporate such entropy constraints and show that our bounds are tighter for the setting with weak confounders.Comment: Published in ICML 202