Restricted Hidden Cardinality Constraints in Causal Models

In causal studies, systems of variables are described by causal models [18, 22], which are composed of two elements: (i) the graphical representation of relationships between variables in a model, encoded in a directed acyclic graph, and (ii) the mathematical description of conditional probability distribution of each variable given its causal parents. When a causal model involves hidden (i.e., unobserved) variables, any characterization of the model verifiable by observations should only include observed variables. Therefore, one of the objectives of causal inference is to eliminate all hidden variables from inequalities and equalities that describe the model. In principle, this can be achieved using the Tarski-Seidenberg quantifier elimination method [12]. However, its complexity is such that only models with few variables can be solved using this technique, hence the reason for the many attempts to simplify the problem.