It is known fact for random variables $(X,Y) \sim p(x,y)=p(x)p(y|x)$ the mutual information is concave function of $p(x)$ for fixed $p(y|x)$.
I have two confusions in interpreting the above fact:
1) when it says "....concave function of $p(x)$", does that mean function of the probability of one particular realization of $X$ i-e $x$ ?
2) If yes, let us suppose $p(x)$ is dependent on certain parameter $\theta$ i-e $p(x;\theta)$. When $\theta$ increases probability of one particular realization $x$ gets decreased i-e $p(X=x)$ decreases as $\theta$ increases. How the concavity of mutual information between $X$ and $Y$ relates to $\theta$?