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The fields of quantum non-locality in physics, and causal discovery inmachine learning, both face the problem of deciding whether observed data iscompatible with a presumed causal relationship between the variables (forexample a local hidden variable model). Traditionally, Bell inequalities havebeen used to describe the restrictions imposed by causal structures on marginaldistributions. However, some structures give rise to non-convex constraints onthe accessible data, and it has recently been noted that linear inequalities onthe observable entropies capture these situations more naturally. In thispaper, we show the versatility of the entropic approach by greatly expandingthe set of scenarios for which entropic constraints are known. For the firsttime, we treat Bell scenarios involving multiple parties and multipleobservables per party. Going beyond the usual Bell setup, we exhibitinequalities for scenarios with extra conditional independence assumptions, aswell as a limited amount of shared randomness between the parties. Many of ourresults are based on a geometric observation: Bell polytopes for two-outcomemeasurements can be naturally imbedded into the convex cone of attainablemarginal entropies. Thus, any entropic inequality can be translated into onevalid for probabilities. In some situations the converse also holds, whichprovides us with a rich source of candidate entropic inequalities.

Causal structures from entropic information: Geometry and novel scenarios