# On the safe use of prior densities for Bayesian model selection

arXiv: https://arxiv.org/abs/2206.05210 · HTML: https://mlsanity.com/paper/2206.05210/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 76 references that are in the map.
- First posted on arXiv 10 Jun 2022.

## Builds on (most cited first, 10 of 10)

- [Practical Bayesian model evaluation using leave-one-out cross-validation and WAIC](https://mlsanity.com/paper/1507.04544/) (2015)
- [Marginal likelihood computation for model selection and hypothesis testing: an extensive review](https://mlsanity.com/paper/2005.08334/) (2020)
- [Prior knowledge elicitation: The past, present, and future](https://mlsanity.com/paper/2112.01380/) (2021)
- [Bayesian Evidence and Model Selection](https://mlsanity.com/paper/1411.3013/) (2014)
- [Comparison of Bayesian predictive methods for model selection](https://mlsanity.com/paper/1503.08650/) (2015)
- [A Joint introduction to Gaussian Processes and Relevance Vector Machines with Connections to Kalman filtering and other Kernel Smoothers](https://mlsanity.com/paper/2009.09217/) (2020)
- [Deep Importance Sampling based on Regression for Model Inversion and Emulation](https://mlsanity.com/paper/2010.10346/) (2020)
- [Automatic tempered posterior distributions for Bayesian inversion problems](https://mlsanity.com/paper/2107.11614/) (2021)
- [Recursive Pathways to Marginal Likelihood Estimation with Prior-Sensitivity Analysis](https://mlsanity.com/paper/1301.6450/) (2013)
- [Compressed particle methods for expensive models with application in Astronomy and Remote Sensing](https://mlsanity.com/paper/2107.08465/) (2021)

## Built on by (most cited first, 4 of 4)

- [Spectral information criterion for automatic elbow detection](https://mlsanity.com/paper/2308.09108/) (2023)
- [Adaptive posterior distributions for uncertainty analysis of covariance matrices in Bayesian inversion problems for multioutput signals](https://mlsanity.com/paper/2501.01148/) (2025)
- [Data-driven informative priors for Bayesian inference with quasi-periodic data](https://mlsanity.com/paper/2511.22296/) (2025)
- [A note on the area under the likelihood and the fake evidence for model selection](https://mlsanity.com/paper/2602.22965/) (2026)

## Abstract (arXiv)

The application of Bayesian inference for the purpose of model selection is very popular nowadays. In this framework, models are compared through their marginal likelihoods, or their quotients, called Bayes factors. However, marginal likelihoods depends on the prior choice. For model selection, even diffuse priors can be actually very informative, unlike for the parameter estimation problem. Furthermore, when the prior is improper, the marginal likelihood of the corresponding model is undetermined. In this work, we discuss the issue of prior sensitivity of the marginal likelihood and its role in model selection. We also comment on the use of uninformative priors, which are very common choices in practice. Several practical suggestions are discussed and many possible solutions, proposed in the literature, to design objective priors for model selection are described. Some of them also allow the use of improper priors. The connection between the marginal likelihood approach and the well-known information criteria is also presented. We describe the main issues and possible solutions by illustrative numerical examples, providing also some related code. One of them involving a real-world application on exoplanet detection.