I have a question on (I think?) Bayesian statistics. Consider the following situation: -P is a class of probability measures on some subset A of the real line -q is a probability measure on some subset B of the real line -f is a function on AxB -My prior distribution on the random variable (X,Y) is an independent draw with X~p for some p in P and Y~q. I'm interested in cases where: if I observe the realization of f(X,Y), my posterior distribution on X is still an element of P. ~~~~~~ Ultimately, I'm after natural examples of this with P being a pretty small parametrized class (e.g. Gaussian; exponential). My model example is the case where q and everything in P are both Gaussian and f is linear. Does anybody here know any nice examples, any related general theory, or even in any vaguely related buzzwords that I might search? I'm totally oblivious here, and I don't have peers to ask. Thanks!! EN p.s. This is my first post here. Please forgive me if I've written too much, given an inappropriate title, or committed any other faux pas.