# Automatic tempered posterior distributions for Bayesian inversion problems

arXiv: https://arxiv.org/abs/2107.11614 · HTML: https://mlsanity.com/paper/2107.11614/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 2 of its 26 references that are in the map.
- First posted on arXiv 24 Jul 2021.

## Builds on (most cited first, 2 of 2)

- [Marginal likelihood computation for model selection and hypothesis testing: an extensive review](https://mlsanity.com/paper/2005.08334/) (2020)
- [Layered Adaptive Importance Sampling](https://mlsanity.com/paper/1505.04732/) (2015)

## Built on by (most cited first, 6 of 6)

- [On the safe use of prior densities for Bayesian model selection](https://mlsanity.com/paper/2206.05210/) (2022)
- [Adaptive posterior distributions for uncertainty analysis of covariance matrices in Bayesian inversion problems for multioutput signals](https://mlsanity.com/paper/2501.01148/) (2025)
- [Enhancing Graphical Lasso: A Robust Scheme for Non-Stationary Mean Data](https://mlsanity.com/paper/2503.19651/) (2025)
- [Impact Study of Numerical Discretization Accuracy on Parameter Reconstructions and Model Parameter Distributions](https://mlsanity.com/paper/2305.02663/) (2023)
- [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)

We propose a novel adaptive importance sampling scheme for Bayesian inversion problems where the inference of the variables of interest and the power of the data noise is split. More specifically, we consider a Bayesian analysis for the variables of interest (i.e., the parameters of the model to invert), whereas we employ a maximum likelihood approach for the estimation of the noise power. The whole technique is implemented by means of an iterative procedure, alternating sampling and optimization steps. Moreover, the noise power is also used as a tempered parameter for the posterior distribution of the the variables of interest. Therefore, a sequence of tempered posterior densities is generated, where the tempered parameter is automatically selected according to the actual estimation of the noise power. A complete Bayesian study over the model parameters and the scale parameter can be also performed. Numerical experiments show the benefits of the proposed approach.