Inferential applications of the moments of the logit-normal distribution

Abstract

Despite the implicit appearance of logit-normal random variables in many inferential problems, the logit-normal distribution is poorly studied. Most frustratingly, no default method exists for finding logit-normal moments, which are often assumed analytically unknown. In this paper, we introduce a method for estimating logit-normal moments of any positive integer order, based on approximating the logistic function. We will show our method is highly accurate up to the $8^\text{th}$ moment, avoids the numerical instability observed with Mordell integral based approximations of the first moment, and is faster than numerical integration in R. Focusing on two inferential applications, we will show our approximation methods are sufficiently accurate to enable faster implementation of Expectation Propagation for logistic regression, but is not general enough to directly evaluate the logistic normal integral that appears in some logistic mixed models.