Bayesian Inference in Categorical, Count and Privacy-preserving Latent Variable Models

The efficacy of an intervention, such as a vaccine, can be established through the estimation
of several numerical measures. Mitigated fraction is one of the contemporary numerical measures,
and it serves the purpose of reducing the severity of a specific disease rather than completely
preventing its occurrence. In this paper, an efficient approach to calculating the mitigated fraction
is presented, with Bayesian approach which involves utilizing the values of latent variables within a generalized linear mixed model (GLMM). This proposed Bayesian method works with many link functions efficiently compared to traditional frequentist approach. The concept of the mitigated fraction was
introduced in veterinary medicine to quantify the reduction in the severity of disease occurring in
vaccinated animals as compared to non-vaccinated animals. The USDA's Center for Veterinary
Biologics (CVB) recommends a form of the mitigated fraction when
the disease severity is generally graded by some continuous measure or by some discrete assessment
resulting in unambiguous ranks.
Our Bayesian approach works effectively when observations are ordinal and measured longitudinally. 

Differential privacy (DP) is a mathematical framework for releasing information with formal privacy guarantees. Despite the existence of various DP procedures for
performing a wide range of statistical analysis and machine learning tasks, methods of good utility are still lacking in obtaining valid statistical inference with DP guarantees. We address this gap by proposing PRivacy-loss-Efficient and Consistent Inference based on poSterior quantilEs (PRECISE). PRECISE is a general-purpose Bayesian approach for constructing privacy-preserving posterior intervals. We establish the Mean-Squared-Error consistency for our proposed private posterior quantiles converging to the population posterior quantile as sample size increases or privacy guarantees weaken. We conduct extensive experiments to compare the utilities of PRECISE with common existing privacy-preserving inferential approaches in various inferential tasks, data types
and sizes, privacy loss levels. The results demonstrated a significant advantage of PRECISE with its nominal coverage and substantially narrower intervals than the existing methods, which are prone to either under-coverage or impractically wide intervals. 

This study introduces the NB-INGARCH model, tailored for over-dispersed count data, addressing limitations of traditional Poisson models. Estimating NB-INGARCH parameters poses challenges due to its complex likelihood function. We advocate Bayesian estimation using minimally informative and hierarchical priors, emphasizing data-driven inference. Markov Chain Monte Carlo methods yield numerical results, demonstrating the Bayesian approach's robustness and reliability over maximum likelihood estimation (MLE). Classical methods for estimating the dispersion parameter often rely on a two-step process-fixing the dispersion parameter while estimating others, followed by profile likelihood optimization. In contrast, Bayesian estimation jointly infers all parameters, simplifying the process, reducing uncertainty, and enhancing analysis coherence. Bayesian estimates exhibit smaller standard errors and more stable confidence intervals compared to the classical approach. Persistence parameters and long-term averages remain consistent across prior models, highlighting minimal prior influence. This work advances count time series analysis and provides practical tools for model. 

Missing data is common in scientific studies, including medical research and public health studies. We propose efficient Bayesian methods to analyze categorical data with missing values using the multinomial probit (MNP) model. In particular, we develop a Markov Chain Monte Carlo (MCMC) sampling method based on non-identifiable multinomial probit models and compare its performance with that of identifiable multinomial probit (MNP) models. We conduct our investigation through simulation studies, which show that the proposed methods can handle substantial missing values. The method of marginalizing redundant parameters based on the non-identifiable model outperforms the others in terms of mixing and convergence of the MCMC sampling components. We then apply the proposed methods to Mental Health Client-Level Data (MH-CLD) collected by the State Mental Health Agency (SMHA).