Inferential applications of the moments of the logit-normal distribution

2026-06-22

AI Summary

The lack of efficient and stable methods for computing higher-order moments of the logit-normal distribution has hindered its application in inferential statistics. This work proposes an innovative approach based on a logistic function approximation, integrating probability integral transforms with numerical analysis techniques to circumvent direct numerical integration. For the first time, this method enables highly accurate and computationally efficient estimation of moments of any positive integer order. Evaluated on the first eight moments, it substantially outperforms existing Mordell integral-based schemes, eliminating numerical instabilities while achieving significantly faster computation than standard numerical integration in R. As practical demonstrations, the proposed method accelerates expectation propagation in logistic regression and improves integral approximations in logistic mixture models.

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.

Problem

• logit-normal distribution
• moments
• inferential problems
• numerical instability
• logistic regression

Innovation

• logit-normal moments
• logistic function approximation
• Expectation Propagation
• numerical stability
• efficient inference