Open Access
December 2023 The Posterior Predictive Null
Gemma E. Moran, John P. Cunningham, David M. Blei
Author Affiliations +
Bayesian Anal. 18(4): 1071-1097 (December 2023). DOI: 10.1214/22-BA1313


Bayesian model criticism is an important part of the practice of Bayesian statistics. Traditionally, model criticism methods have been based on the predictive check, an adaptation of goodness-of-fit testing to Bayesian modeling and an effective method to understand how well a model captures the distribution of the data. In modern practice, however, researchers iteratively build and develop many models, exploring a space of models to help solve the problem at hand. While classical predictive checks can help assess each one, they cannot help the researcher understand how the models relate to each other. This paper introduces the posterior predictive null check (PPN), a method for Bayesian model criticism that helps characterize the relationships between models. The idea behind the PPN is to check whether data from one model’s predictive distribution can pass a predictive check designed for another model. This form of criticism complements the classical predictive check by providing a comparative tool. A collection of PPNs, which we call a PPN study, can help us understand which models are equivalent and which models provide different perspectives on the data. With mixture models, we demonstrate how a PPN study, along with traditional predictive checks, can help select the number of components by the principle of parsimony. With probabilistic factor models, we demonstrate how a PPN study can help understand relationships between different classes of models, such as linear models and models based on neural networks. Finally, we analyze data from the literature on predictive checks to show how a PPN study can improve the practice of Bayesian model criticism.

Funding Statement

This research was supported by ONR N00014-17-1-2131, ONR N00014-15-1-2209, DARPA SD2 FA8750-18-C-0130, the Simons Foundation, NSF NeuroNex, the Sloan Foundation, the McKnight Endowment, and the Gatsby Charitable Trust.


We thank Scott Linderman for helpful discussions about this work.


Download Citation

Gemma E. Moran. John P. Cunningham. David M. Blei. "The Posterior Predictive Null." Bayesian Anal. 18 (4) 1071 - 1097, December 2023.


Published: December 2023
First available in Project Euclid: 7 December 2023

MathSciNet: MR4675035
Digital Object Identifier: 10.1214/22-BA1313

Keywords: Bayesian workflow , model criticism , predictive checks

Vol.18 • No. 4 • December 2023
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