Updating of moments

Consider a statistical model, given by the distribution of the observation X, conditional on the parameter θ, and the prior distribution of the parameter θ. Let H_x denote the function that maps the prior mean and the prior covariance matrix into the posterior mean and the posterior covariance matrix, when X = x is observed. We prove that if the conditional distribution of X belongs to an exponential family, then the function H_x characterizes the distribution of Xθ.