I need to prove that the the KL Divergence $D(\bar{\mu}(\theta)||Y(\lambda))$ between the following variables is convex w.r.t $\lambda$.
The variables are defined as:
$ \bar{\mu}(\theta) =\frac{1}{N}\sum\limits_{j=1}^{N} X_{j}(\theta)$ where $X_{j}(\theta)\sim Bern(\theta)$ for all $j$,
and $Y(\lambda)\sim Bern(\lambda)$.
I was hoping there is a closed expression that I could prove convex by derivating w.r.t $\lambda$ twice and show that it is positive.
In fact - I have read online that the KL divergence is convex w.r.t its arguments when the arguments are probability distributions, but in my case I want to show convexity w.r.t the parameter $\lambda$ that controls the distribution on the right.
Maybe the two cases are the same? I would love an explanation.