This is more of a general statistical question, but addressing it may give some insight into Stan models, so I appeal to the statisticians in the forum. I realize that the term “likelihood expectations” that I used in the title is a misnomer since the likelihood is generally not even a pdf, but hopefully you get the idea (maybe a better term would be likelihood importance sampling). Basically, I wonder when the expression below will be a good approximation to posterior expectations:
Basically, this weighs prior draws by the likelihood function. I tested this is in a very simple model with one parameter, and the estimates I get using the expression above are virtually the same as the true posterior estimates (both the mean and SD).
Any thoughts on when this approximation may fall apart? Thanks.