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Explicit Gaussian Variational Approximation for the Poisson Lognormal Mixed Model

Author

Listed:
  • Xiaoping Shi

    (Department of Computer Science, Mathematics, Physics and Statistics, University of British Columbia, Kelowna, BC V1V 1V7, Canada)

  • Xiang-Sheng Wang

    (Department of Mathematics, University of Louisiana at Lafayette, Lafayette, LA 70503, USA)

  • Augustine Wong

    (Department of Mathematics and Statistics, York University, Toronto, ON M3J 1P3, Canada)

Abstract

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.

Suggested Citation

  • Xiaoping Shi & Xiang-Sheng Wang & Augustine Wong, 2022. "Explicit Gaussian Variational Approximation for the Poisson Lognormal Mixed Model," Mathematics, MDPI, vol. 10(23), pages 1-18, December.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:23:p:4542-:d:990049
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    References listed on IDEAS

    as
    1. Håvard Rue & Sara Martino & Nicolas Chopin, 2009. "Approximate Bayesian inference for latent Gaussian models by using integrated nested Laplace approximations," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 71(2), pages 319-392, April.
    2. Ormerod, J. T. & Wand, M. P., 2010. "Explaining Variational Approximations," The American Statistician, American Statistical Association, vol. 64(2), pages 140-153.
    3. Yixin Wang & David M. Blei, 2019. "Frequentist Consistency of Variational Bayes," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 114(527), pages 1147-1161, July.
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