Which causal scenarios are interesting?

A causal scenario is a graph that describes the cause and effect relationships between all relevant variables in an experiment. A scenario is deemed interesting if different operational probabilistic theories give rise to different marginal distributions of the observed variables -- a Bell-type experiment is one example. Graphs that are not interesting are boring in a precise sense: the predictions of an arbitrary generalised probabilistic theory (quantum or otherwise) can be reproduced by local hidden variables respecting the causal structure of the graph. Henson, Lal and Pusey (HLP) [New J. Phys. 16, 113043] recently proposed a sufficient condition for a causal scenario to be boring. In this paper we review and propose several sufficient conditions for a graph to be interesting. We first show that existing graphical techniques due to Evans can be used to confirm that many graphs are interesting without having to explicitly search for inequality violations. For three exceptional cases -- the graphs numbered 15,16,20 in HLP -- we show that there exist non-Shannon-type entropic inequalities that imply these graphs are interesting.