# Can a likelihood's relative entropy be related to its predictive accuracy?

Suppose I have some prior $$\pi(\theta)$$, from which I draw $$N$$ samples, each having parameter $$\theta_i$$. These $$\theta_i$$'s are known to me. Suppose that one of these samples (unknown to me which) generates some data $$y$$, following some likelihood $$\mathcal{L}(y|\theta)$$. Now, I can try to identify which $$i$$ generated the $$y$$ by calculating evidences of each $$i$$ generating the data $$y$$. The true $$i$$ will then usually have the highest evidence.

The question that I am interested in is: how predictive is this data $$y$$ to getting the right $$i$$? How often can I expect to get it right, and with what certainty? I am essentially asking: what the information content that $$y$$ gives me for identification?

One way (possibly not a good one of looking at this) is with some information theory-like arguments, so a possible avenue I thought of:

I could calculate the relative entropy (KL-divergence) of the prior to the posterior, $$D_{KL}(p(\theta|y)||\pi(\theta))$$. This can be converted to a factor, for example, if the information is a factor of 50, I would say that I can expect to identify the right $$i$$ out of a sample of $$N=50$$. Is that right? What is the actual relation between accuracy and $$N$$ given some information as quantified by the relative entropy? Is there a way of quantifying this?

Another way to go about this might be by looking at the distributions of the evidences given a wrong datapoint and a right datapoint, but how to go about this, I am also unsure of.

I feel like stuff this must have been done before, so relevant references would also be appreciated!