Bayesian inference approach for a mixture of normal regression models

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Mixture distributions are widely used to model data with distinct groups, providing a flexible approach to estimating density. However, Bayesian approaches for mixture models pose challenges, such as label switching in the Gibbs sampler output due to the non-identifiability of component parameters. We review advanced methods for Bayesian analysis, including the Markov chain Monte Carlo (MCMC) reversible jump algorithm and model comparison based on joint measures of fit and complexity. We also present a Bayesian regression model based on a two-component mixture model, implemented using the Gibbs sampler algorithm and applied to a dataset of time measurement differences between two clocks. Our theoretical investigation highlights the importance of latent variables in implementing the Bayesian normal mixture model with two components. When applied to the dataset, our model effectively assigned probabilities to the two states of the phenomenon under study and identified two processes with identical slopes, intercepts, and variances. Our findings demonstrate the power of Bayesian mixture models in uncovering hidden structures within complex datasets. In general, our review and application provide insight into the challenges and potential solutions for Bayesian mixture modeling and highlight the usefulness of these methods in various fields.