Let $Y$ be a random variable in $[0,1]$, and let $X_1, X_2, \ldots$ be a sequence of random variables in $[0,1]$. Suppose that the $X_i$'s are conditionally i.i.d given $Y$ ; in other words, I'd like to think of each $X_i$ as a "measurement" or "signal" regarding $Y$, and the signals are i.i.d.
Let $Z_k = Y | X_1,\ldots,X_k$ be the posterior on $Y$ given the first $k$ observations. I believe that the Bernstein von-Mises theorem implies that the sequence $Z_k$ converges to some limit posterior as $k$ grows large. However, BVM does not provide any quantitative guarantees on the rate of convergence as far as I'm aware.
I'm interested in quantitative statements of the form: "$Z_k$ is close to $Z_{k-1}$", for some suitable choice of distance metric on distributions. More precisely, what I'd love to know is whether, for some suitable distance function $d(.,.)$, there exists a bound of the form $d(Z_k, Z_{k-1}) \leq f(k)$ for some decreasing function $f$. Note that I'm asking that the bound on $d(Z_k,Z_{k-1})$ not depend on the particulars of $Y$ and the $X_i$'s, which may or may not be asking too much... If that is asking too much, bounds on $d(Z_k,Z_{k-1})$ which depend as minimally as possible on $Y$ and the $X_i$'s would be appreciated as well.
I've seen a few somewhat similar questions on Mathoverflow and stackexchange, but didn't find anything that really answered this question. Any literature references would be appreciated.
