Posterior calibration of posterior predictive p values

Area of Science:

• Bayesian statistics
• Statistical modeling
• Model checking

Background:

• Accurate control of Type I error rates is crucial in statistical inference.
• The posterior predictive p value (ppp) is commonly used in Bayesian analysis but often fails to maintain a uniform distribution under the null model, hindering model misfit detection.
• Existing calibration methods for ppp ('prior-cppp') are sensitive to prior distributions.

Purpose of the Study:

• To propose an alternative calibration method for the posterior predictive p value (ppp) that addresses the limitations of existing approaches.
• To introduce a 'posterior-cppp' method that ensures a uniform distribution under the null model for reliable Type I error rate control.
• To demonstrate the utility of the 'posterior-cppp' in various statistical testing scenarios.

Main Methods:

• Developed a novel calibration technique for the posterior predictive p value (ppp) by utilizing the posterior distribution under the null model.
• The proposed 'posterior-cppp' method is designed to be independent of prior information.
• Applied the 'posterior-cppp' methodology to diverse testing problems, including independence of dichotomous variables, linear regression model misfit with outliers, and latent class analysis.

Main Results:

• The 'posterior-cppp' method achieves a uniform distribution under the null model, enabling accurate control of the Type I error rate.
• This approach effectively detects model misfit, unlike the standard ppp which can have low power.
• The methodology demonstrated successful application across various statistical testing scenarios.

Conclusions:

• The 'posterior-cppp' offers a robust solution for calibrating posterior predictive p values in Bayesian data analysis.
• This method provides a reliable tool for model checking, particularly when prior information is unavailable or unreliable.
• The 'posterior-cppp' enhances the power to detect model misfit and ensures valid statistical inference.