Explicit Gaussian Variational Approximation for the Poisson Lognormal Mixed Model

• Published in: Mathematics (Basel) Vol. 10; no. 23; p. 4542
• Main Authors: Shi, Xiaoping, Wang, Xiang-Sheng, Wong, Augustine
• Format: Journal Article
• Language: English
• Published: Basel MDPI AG 01.12.2022
• Subjects: Accuracy; Analysis; Approximation; Approximation theory; asymptotic distribution; Asymptotic properties; Calculus of variations; Estimators; exponential family model; Food science; Gaussian processes; Gaussian variational approximation; Inequality; Inference; Information theory; Kullback–Leibler divergence; Lognormal distribution; Mathematical models; maximum likelihood estimation; Methods; Parameters; Poisson distribution; Poisson lognormal mixed model; Canada
• ISSN: 2227-7390; 2227-7390
• DOI: 10.3390/math10234542

In recent years, the Poisson lognormal mixed model has been frequently used in modeling count data because it can accommodate both the over-dispersion of the data and the existence of within-subject correlation. Since the likelihood function of this model is expressed in terms of an intractable integral, estimating the parameters and obtaining inference for the parameters are challenging problems. Some approximation procedures have been proposed in the literature; however, they are computationally intensive. Moreover, the existing studies of approximate parameter inference using the Gaussian variational approximation method are usually restricted to models with only one predictor. In this paper, we consider the Poisson lognormal mixed model with more than one predictor. By extending the Gaussian variational approximation method, we derive explicit forms for the estimators of the parameters and examine their properties, including the asymptotic distributions of the estimators of the parameters. Accurate inference for the parameters is also obtained. A real-life example demonstrates the applicability of the proposed method, and simulation studies illustrate the accuracy of the proposed method.