Robust Bayesian Inference in the Multilevel Zero-Inflated Generalized Poisson Model.

Outliers, over-dispersion, and zero inflation are issues with count data. Traditional models like Poisson and negative binomial often fail to account for these issues, leading to biased estimates and poor model fit. These frameworks are extended by the Zero-Inflated Generalized Poisson (ZIGP) model, which takes into consideration not only zero inflation but also over-dispersion or under-dispersion. However, in the presence of outliers and hierarchical data structures. This study develops a robust Bayesian inference framework for the multilevel ZIGP model. Standard Bayesian methods often lack robustness under model misspecification and in the presence of outlier data. The framework uses a Robust expectation solution (RES) algorithm and generalized Bayesian inference (GBI) for robust estimation against outliers. These approaches improve estimation accuracy using robust loss functions and scaling parameters to minimize the influence of outliers. Simulation studies confirm that the Robust Expectation Solution (RES) algorithm significantly outperformed the Expectation-Maximization (EM) algorithm in reducing bias and mean squared error (MSE), especially in the presence of outliers. Regular Bayesian and EM algorithms were more sensitive to outliers, leading to potential bias and instability in parameter estimates. Our robust Bayesian framework, specifically the Generalized Bayesian Inference (GBI), demonstrated improved robustness and stability under model misspecification and outlier contamination. The main results show that tuning quantiles and optimizing scaling parameters improved parameter calibration and reduced bias and mean square error (MSE). We applied the framework to neonatal mortality data, identifying key risk factors such as maternal education, wealth status, rural residence, and age at first birth.