Causally Informative Entropic Inequalities within Families of Distributions with Shared Marginals

The joint probability distribution of observable variables from a system is constrained by the underlying causal structure. In the presence of hidden variables, untestable independencies that involve hidden variables lead to testable causally-imposed inequality constraints for observable variables, whose violation can reject the compatibility of a causal structure with data. One type of causally informative inequalities is entropic inequalities, which appear in the space of entropic terms associated with the distribution of observable variables. We derive a new type of minimum information (minInf) entropic inequalities that substantially increases causal inference power. These new entropic inequalities appear when considering the constraints that the causal structure imposes on entropic terms determined by information minimization within families of distributions that preserve sets of marginals shared with the original distribution. We introduce a new family of minInf data processing inequalities and a procedure to recursively combine different types of data processing inequalities to create tighter testable entropic inequalities. We extensively illustrate the applicability of this procedure in the instrumental causal scenario, integrating the new inequalities with standard instrumental entropic inequalities constructed with multivariate instrumental sets. We also provide additional examples with other types of entropic inequalities, such as the Information Causality and Groups-Decomposition inequalities.