# Marginal likelihood computation for model selection and hypothesis testing: an extensive review

arXiv: https://arxiv.org/abs/2005.08334 · HTML: https://mlsanity.com/paper/2005.08334/

## Key facts

- Cited by 33 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 17 May 2020.

## Built on by (most cited first, 30 of 33)

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- [Priors in Bayesian Deep Learning: A Review](https://mlsanity.com/paper/2105.06868/) (2021)
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- [Bayesian Model Selection, the Marginal Likelihood, and Generalization](https://mlsanity.com/paper/2202.11678/) (2022)
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- [BNNpriors: A library for Bayesian neural network inference with different prior distributions](https://mlsanity.com/paper/2105.06964/) (2021)
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- [A survey of Monte Carlo methods for noisy and costly densities with application to reinforcement learning and ABC](https://mlsanity.com/paper/2108.00490/) (2021)
- [Generalized Bayesian Additive Regression Trees Models: Beyond Conditional Conjugacy](https://mlsanity.com/paper/2202.09924/) (2022)
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- [Dynamic Online Ensembles of Basis Expansions](https://mlsanity.com/paper/2405.01365/) (2024)
- [Improving the Accuracy of Amortized Model Comparison with Self-Consistency](https://mlsanity.com/paper/2508.20614/) (2025)
- [On the safe use of prior densities for Bayesian model selection](https://mlsanity.com/paper/2206.05210/) (2022)
- [MCMC-driven importance samplers](https://mlsanity.com/paper/2105.02579/) (2021)
- [Marginal and Joint Cross-Entropies & Predictives for Online Bayesian Inference, Active Learning, and Active Sampling](https://mlsanity.com/paper/2205.08766/) (2022)
- [Adaptive importance sampling for heavy-tailed distributions via α-divergence minimization](https://mlsanity.com/paper/2310.16653/) (2023)
- [Optimality in Noisy Importance Sampling](https://mlsanity.com/paper/2201.02432/) (2022)
- [Spectral information criterion for automatic elbow detection](https://mlsanity.com/paper/2308.09108/) (2023)
- [FlowVAT: Normalizing Flow Variational Inference with Affine-Invariant Tempering](https://mlsanity.com/paper/2505.10466/) (2025)
- [Inference over radiative transfer models using variational and expectation maximization methods](https://mlsanity.com/paper/2204.03346/) (2022)
- [Divide, Conquer, Combine Bayesian Decision Tree Sampling](https://mlsanity.com/paper/2403.18147/) (2024)
- [Variational Neural Stochastic Differential Equations with Change Points](https://mlsanity.com/paper/2411.00635/) (2024)
- [Adaptive posterior distributions for uncertainty analysis of covariance matrices in Bayesian inversion problems for multioutput signals](https://mlsanity.com/paper/2501.01148/) (2025)
- [Fenchel-Young Variational Learning](https://mlsanity.com/paper/2502.10295/) (2025)
- [Enhancing Graphical Lasso: A Robust Scheme for Non-Stationary Mean Data](https://mlsanity.com/paper/2503.19651/) (2025)
- [On Equivariant Model Selection through the Lens of Uncertainty](https://mlsanity.com/paper/2506.18629/) (2025)
- [A note on the area under the likelihood and the fake evidence for model selection](https://mlsanity.com/paper/2602.22965/) (2026)
- [Automatic Laplace Collapsed Sampling: Scalable Marginalisation of Latent Parameters via Automatic Differentiation](https://mlsanity.com/paper/2603.26644/) (2026)
- [A unifying view of contrastive learning, importance sampling, and bridge sampling for energy-based models](https://mlsanity.com/paper/2604.08116/) (2026)

## Abstract (arXiv)

This is an up-to-date introduction to, and overview of, marginal likelihood computation for model selection and hypothesis testing. Computing normalizing constants of probability models (or ratio of constants) is a fundamental issue in many applications in statistics, applied mathematics, signal processing and machine learning. This article provides a comprehensive study of the state-of-the-art of the topic. We highlight limitations, benefits, connections and differences among the different techniques. Problems and possible solutions with the use of improper priors are also described. Some of the most relevant methodologies are compared through theoretical comparisons and numerical experiments.