Structure Learning for Unfaithful Distributions: The Minimal Dependence Faithfulness
Keywords: Bayesian networks, Causality, Faithfulness, Structure learning
TL;DR: Relaxing faithfulness in causal discovery, we introduce minimal dependence faithfulness to handle unfaithful structures like XOR, modifying the PC algorithm to detect these and output candidate DAGs.
Abstract: Causality detection is to identify the ``true'' directed acyclic graph (DAG) of a causal model from the joint probability distribution of the observed variables.
Algorithms such as PC and its modified versions perform this task under the restrictive faithfulness assumption, that is the DAG encodes all conditional independencies imposed by the distribution.
However, all existing algorithms fail to detect the simple structure where a variable is the XOR of several Bernoulli variables, violating faithfulness. We generalize this type of unfaithfulness that appears in other, non-XOR, examples and define the \emph{minimal dependence} of a given variable $X$ as the set of variables, such that $X$ is independent of each variable in the set but depends on at least one of them, the \emph{dependent member} if conditioned on the remainder of the set.
Minimal dependencies of size at least two violate faithfulness. Consequently, we relax faithfulness to \emph{minimal dependence faithfulness}, restricting the neighbors of a node to its dependent members, and impose \emph{minimal orientation faithfulness} that generalizes the orientation rules under faithfulness.
We then determine the structure of the dependent members of a node $X$ in the true DAG and show that they are connected to $X$ either directly or indirectly by a collider.
Finally, we provide a sound and complete modification of the PC algorithm to detect this kind of unfaithfulness and output all possible candidates for the true DAG.
Primary Area: causal reasoning
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Submission Number: 1968
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