Calibrating the prior distribution for a normal model with conjugate prior

Area of Science:

• Statistics
• Bayesian Inference
• Frequentist Properties

Background:

• Hyperparameters in Bayesian models influence posterior estimates.
• Understanding hyperparameter effects on frequentist properties is essential for robust statistical inference.

Purpose of the Study:

• To examine the impact of hyperparameters on the long-run frequentist properties of posterior estimates.
• To define and develop MSE optimal and credible interval optimal hyperparameters.
• To assess the consequences of hyperparameter choices on mean squared error (MSE) and credible interval coverage.

Main Methods:

• Analysis of a normal model with a conjugate prior.
• Evaluation of hyperparameter effects on the MSE of posterior means.
• Assessment of hyperparameter effects on the true coverage of credible intervals.
• Development of criteria for MSE optimality and credible interval optimality.

Main Results:

• Hyperparameter choices have a more detrimental effect on credible interval coverage than on the MSE of posterior means.
• MSE optimal hyperparameters minimize the MSE of posterior point estimates.
• Credible interval optimal hyperparameters yield minimum length credible intervals with nominal coverage.

Conclusions:

• Careful hyperparameter selection is critical for accurate statistical modeling.
• The choice of hyperparameters has distinct impacts on point estimate accuracy and interval estimate reliability.
• The study provides a framework for evaluating hyperparameter choices in Bayesian analysis.