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Lognormal and Gamma Mixed Negative Binomial Regression.

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    Summary
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    We developed a new Bayesian regression model for count data using a mixed negative binomial (NB) distribution. This approach offers efficient computation and flexibility for incorporating prior information, improving count data analysis.

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    Area of Science:

    • Statistics
    • Computational Biology
    • Biostatistics

    Background:

    • Bayesian regression for count data is underdeveloped due to computational challenges.
    • Existing Poisson models lack flexibility for random effects and prior incorporation.

    Purpose of the Study:

    • To propose an efficient Bayesian inference method for count regression.
    • To introduce a lognormal and gamma mixed negative binomial (NB) regression model.
    • To enable incorporation of prior information, such as sparsity in regression coefficients.

    Main Methods:

    • Developed a lognormal and gamma mixed negative binomial (NB) regression model.
    • Implemented efficient closed-form Bayesian inference using Gibbs sampling and variational Bayes.
    • Utilized conditional conjugacy via compound Poisson representation and Polya-Gamma data augmentation.
    • Placed gamma prior on NB dispersion parameter (r) and lognormal prior on NB probability parameter (p).

    Main Results:

    • Achieved efficient closed-form Bayesian inference for NB regression.
    • The proposed model accommodates two types of random effects.
    • Demonstrated routine implementation and generalizability to complex settings.
    • Illustrated algorithms with real-world examples.

    Conclusions:

    • The proposed Bayesian approach enhances count data regression analysis.
    • Efficient algorithms make Bayesian methods more attractive and practical.
    • The model's flexibility supports advanced statistical modeling and prior incorporation.