  Density Ratio Estimation has attracted attention from the machine learning
community due to its ability to compare the underlying distributions of two
datasets. However, in some applications, we want to compare distributions of
random variables that are \emph{inferred} from observations. In this paper, we
study the problem of estimating the ratio between two posterior probability
density functions of a latent variable. Particularly, we assume the posterior
ratio function can be well-approximated by a parametric model, which is then
estimated using observed information and prior samples. We prove the
consistency of our estimator and the asymptotic normality of the estimated
parameters as the number of prior samples tending to infinity. Finally, we
validate our theories using numerical experiments and demonstrate the
usefulness of the proposed method through some real-world applications.
