Suppose we consider the log-ratio of the pullback of the target to the reference
where for some constant and proper pdf . Let be a functional such that . Then, note that
where we recall by construction, and the variance comes from the definition of variance. Both and come from linearizations of the and functions, respectively. If (where exactly), then these approximations are quite close to accurate. There is some connection here to the divergence, but I forget what.
Functional derivative
I'm now supposing we have something where we have two distributions and and we somehow interpolate between them, and so we make steps according to minimizing the variance diagnostic, assume , and and or something of that ilk.