# Recursive Pathways to Marginal Likelihood Estimation with Prior-Sensitivity Analysis

arXiv: https://arxiv.org/abs/1301.6450 · HTML: https://mlsanity.com/paper/1301.6450/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 28 Jan 2013.

## Built on by (most cited first, 5 of 5)

- [Fast and robust Bayesian Inference using Gaussian Processes with GPry](https://mlsanity.com/paper/2211.02045/) (2022)
- [On the safe use of prior densities for Bayesian model selection](https://mlsanity.com/paper/2206.05210/) (2022)
- [Parallelized Acquisition for Active Learning using Monte Carlo Sampling](https://mlsanity.com/paper/2305.19267/) (2023)
- [A note on the area under the likelihood and the fake evidence for model selection](https://mlsanity.com/paper/2602.22965/) (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)

We investigate the utility to computational Bayesian analyses of a particular family of recursive marginal likelihood estimators characterized by the (equivalent) algorithms known as "biased sampling" or "reverse logistic regression" in the statistics literature and "the density of states" in physics. Through a pair of numerical examples (including mixture modeling of the well-known galaxy data set) we highlight the remarkable diversity of sampling schemes amenable to such recursive normalization, as well as the notable efficiency of the resulting pseudo-mixture distributions for gauging prior sensitivity in the Bayesian model selection context. Our key theoretical contributions are to introduce a novel heuristic ("thermodynamic integration via importance sampling") for qualifying the role of the bridging sequence in this procedure and to reveal various connections between these recursive estimators and the nested sampling technique.