Scalable Inference for Bayesian Proportion Models: Variational Approaches under Contaminated Likelihoods

Bayesian proportion models, such as Binomial and Beta regression, are widely applied in health sciences, social research, and digital analytics. While these models provide coherent uncertainty quantification, they are highly sensitive to contamination and outliers. Furthermore, traditional inference methods, particularly Markov chain Monte Carlo (MCMC), are computationally prohibitive for large datasets. This study develops scalable variational inference approaches for Bayesian proportion models under contaminated likelihoods. Two complementary strategies are considered: (i) mixture-based inference that explicitly models contamination via an ε-contamination likelihood, and (ii) divergence-based inference that employs β-divergence to reduce the influence of outliers. Both methods are implemented in a stochastic variational framework suitable for large-scale applications. Simulation studies demonstrate that the proposed approaches substantially improve estimation accuracy, posterior calibration, and predictive performance under contamination, while maintaining scalability. The results highlight that robustness and efficiency can be achieved simultaneously, providing practical tools for modern Bayesian analysis of proportion data.