Assume some joint distribution $P(X,Y) = P(Y|X)P(X)$.
It is well know that, for fixed $P(Y|X)$, mutual information is a concave function of $P(X)$ and, for fixed $P(X)$, a convex function of $P(Y|X)$ (e.g. see Theorem 2.7.4 in Cover & Thomas).
Is the mutual information function convex in the joint distribution $P(X,Y)$, given the marginals are not fixed? I have read conflicting statements. For example,
M Mihm, K Ozbek, "Decision Making with Rational Inattention" (pdf) says it is convex (see statement right after eq.3)
AG Stefani et al, "A Tight Lower Bound on the Mutual Information of a Binary and an Arbitrary Finite Random Variable in Dependence of the Variational Distance" (pdf) says it is not convex (first paragraph).
I suspect it is neither concave nor convex but am looking for a definitive statement.