Consider a prior $P$ on some finite set $X$, and a posterior $Q$ on $X$ formed via a Bayesian update of $P$ given some data $y$.
I am interested in finding alternative ways to write the following difference of expectations $$E_Q[X]-E_P[X]$$ that use the fact that $Q$ is a Bayesian update of $P$.
I have been looking into Radon-Nikodym derivatives but I am not sure how to use them here.
In general, with RN derivative you only can say:
$$\mathbf{E}_Q[X]-\mathbf{E}_P[X]=\mathbf{E}_P\left[X\left(\frac{dQ}{dP}-1\right)\right]=\mathbf{E}_Q\left[X\left(1-\frac{dP}{dQ}\right)\right]$$
When Q is the Bayesian update of P, $\frac{dQ}{dP}$ is proportional to the likelihood of the sample $y$.