  A causal scenario is a graph that describes the cause and effect
relationships between all relevant variables in an experiment. A scenario is
deemed `not interesting' if there is no device-independent way to distinguish
the predictions of classical physics from any generalised probabilistic theory
(including quantum mechanics). Conversely, an interesting scenario is one in
which there exists a gap between the predictions of different operational
probabilistic theories, as occurs for example in Bell-type experiments. Henson,
Lal and Pusey (HLP) recently proposed a sufficient condition for a causal
scenario to not be interesting. In this paper we supplement their analysis with
some new techniques and results. We first show that existing graphical
techniques due to Evans can be used to confirm by inspection 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. In doing so, we find that existing methods of entropic
inequalities can be greatly enhanced by conditioning on the specific values of
certain variables.
