Cake prior

by W. B. Meitei, PhD

A cake prior is a special type of prior distribution introduced for Bayesian hypothesis testing to address issues like the Bartlett-Lindley-Jeffreys paradox, which arise when using highly diffuse (non-informative) or improper priors in Bayes factor calculations. The term was coined in recent research to describe priors that allow analysts to use "diffuse" priors, preserving the practical advantages (having one's "cake"), while still producing theoretically sound inference (and "eating it too").

Key characteristics:

• Pseudo-normal form: Cake priors are constructed similarly to multivariate normal distributions, but with special features, such as a covariance structure that depends on the statistical model (e.g., through the Fisher information matrix), and additional scaling factors.
• Resolution of paradoxes: When applied, cake priors allow Bayes factors to avoid the distortions and inconsistencies (e.g., overwhelming support for the null regardless of data) that commonly arise with conventional diffuse or improper priors. This makes hypothesis testing Chernoff-consistent: both Type I and Type II error probabilities approach zero as the sample size increases.
• Penalisation: Bayesian test statistics under cake priors often resemble penalised likelihood ratio tests, aligning Bayesian and frequentist approaches in large samples.

The name "cake prior" reflects the goal: to gain the convenience of diffuse priors ("have your cake") while retaining reliable inference ("eat it too").

Cake priors were formally introduced by Ormerod, Stewart, Yu, and Romanes in 2017.

Suggested Readings:

1. Ormerod, J. T., Stewart, M., Yu, W., & Romanes, S. E. (2024). Bayesian hypothesis tests with diffuse priors: Can we have our cake and eat it too?. Australian & New Zealand Journal of Statistics, 66(2), 204-227.

Suggested Citation: Meitei, W. B. (2025). Cake prior. WBM STATS.