Determining the effective sample size of a parametric prior.

Area of Science:

• Statistics
• Bayesian Inference
• Computational Statistics

Background:

• Determining the influence of a parametric prior distribution in Bayesian modeling is crucial for model assessment.
• Existing methods for quantifying prior impact may not be universally applicable or easily interpretable.
• A robust measure of effective sample size (ESS) for priors is needed to standardize Bayesian analyses.

Purpose of the Study:

• To introduce a novel definition for the effective sample size of a parametric prior distribution within Bayesian models.
• To develop and present methods for calculating this effective sample size across diverse settings.
• To provide practical guidelines and demonstrate the application of the proposed method.

Main Methods:

• Constructing a vague prior and updating it to generate a sequence of posteriors for varying sample sizes.
• Defining a distance metric based on the curvature of the log-distributions to compare posteriors with the prior.
• Identifying the posterior that minimizes this distance to determine the effective sample size of the prior.
• Employing Monte Carlo simulation for numerical approximation when analytical computation is infeasible.

Main Results:

• A clear definition and computational framework for the effective sample size of parametric priors are established.
• The method is illustrated in standard Bayesian models, yielding intuitive results.
• Applications to non-standard settings demonstrate the method's versatility and robustness.

Conclusions:

• The proposed effective sample size provides a quantitative measure of a parametric prior's influence in Bayesian analysis.
• The developed methods, including numerical approximations, offer practical tools for researchers.
• This work facilitates a deeper understanding and more rigorous assessment of prior specifications in Bayesian modeling.