Partition Weighted Approach for Estimating the Marginal Posterior Density with Applications

Area of Science:

• Bayesian statistical analysis
• Computational statistics
• Statistical modeling

Background:

• Marginal posterior density is crucial for Bayesian inference, providing complete parameter information.
• It's essential for Bayes factors, posterior model probabilities, and diagnostic measures.
• The ideal Conditional Marginal Density Estimator (CMDE) often requires unavailable closed-form conditional posterior densities.

Purpose of the Study:

• To develop a practical method for estimating marginal posterior density when closed-form expressions are unavailable.
• To introduce the Partition Weighted Marginal Density Estimator (PWMDE) as a realization of the CMDE.
• To demonstrate the utility and performance of the PWMDE in real-world applications.

Main Methods:

• Developed the Partition Weighted Marginal Density Estimator (PWMDE).
• The PWMDE is an unbiased estimator requiring only single MCMC output and the known unnormalized posterior density.
• Evaluated performance through simulation studies and analysis of two real datasets (dissociative identity disorder, prostate cancer).

Main Results:

• The PWMDE successfully realizes the theoretically optimal CMDE.
• The estimator is unbiased and requires minimal input (single MCMC output, unnormalized posterior).
• Simulation studies and real data analyses confirm the desirable properties and applicability of the PWMDE.

Conclusions:

• The PWMDE offers a robust and practical solution for marginal posterior density estimation in Bayesian analysis.
• This method enhances parameter inference, model comparison, and diagnostic capabilities, especially when closed-form densities are absent.
• The PWMDE is a valuable tool for researchers in various fields, including clinical and medical studies.