Bayesian Approach of Modified Half - Cauchy Chen Distribution

In this article, we explore a Bayesian framework for parameter estimation and model evaluation using a specific probability model, referred to as the MHCC Distribution. The model defines the likelihood of observed data through parameters α, β, and θ. We employ both Gibbs sampling and Stan, a state-of-the-art platform for Bayesian statistical modeling, to estimate the parameters of the model. A key focus is on validating the model through posterior predictive checks. Our analysis also includes a detailed evaluation of model diagnostics, including trace plots, autocorrelation plots, and Gelman-Rubin convergence diagnostics. The goal of this work is to provide a comprehensive approach to model fitting, diagnostics, and validation in Bayesian inference.