# Adaptive posterior distributions for uncertainty analysis of covariance matrices in Bayesian inversion problems for multioutput signals

arXiv: https://arxiv.org/abs/2501.01148 · HTML: https://mlsanity.com/paper/2501.01148/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 22 of its 65 references that are in the map.
- First posted on arXiv 2 Jan 2025.

## Builds on (most cited first, 22 of 22)

- [Stationary Graph Processes and Spectral Estimation](https://mlsanity.com/paper/1603.04667/) (2016)
- [Marginal likelihood computation for model selection and hypothesis testing: an extensive review](https://mlsanity.com/paper/2005.08334/) (2020)
- [Asymptotically Exact, Embarrassingly Parallel MCMC](https://mlsanity.com/paper/1311.4780/) (2013)
- [A Survey of Monte Carlo Methods for Parameter Estimation](https://mlsanity.com/paper/2107.11820/) (2021)
- [Layered Adaptive Importance Sampling](https://mlsanity.com/paper/1505.04732/) (2015)
- [User-Friendly Covariance Estimation for Heavy-Tailed Distributions](https://mlsanity.com/paper/1811.01520/) (2018)
- [The Impact of the Mini-batch Size on the Variance of Gradients in Stochastic Gradient Descent](https://mlsanity.com/paper/2004.13146/) (2020)
- [Deep Importance Sampling based on Regression for Model Inversion and Emulation](https://mlsanity.com/paper/2010.10346/) (2020)
- [Gradient Importance Sampling](https://mlsanity.com/paper/1507.05781/) (2015)
- [Automatic tempered posterior distributions for Bayesian inversion problems](https://mlsanity.com/paper/2107.11614/) (2021)
- [Langevin Incremental Mixture Importance Sampling](https://mlsanity.com/paper/1611.06874/) (2016)
- [The Recycling Gibbs Sampler for Efficient Learning](https://mlsanity.com/paper/1611.07056/) (2016)
- [Metropolis Sampling](https://mlsanity.com/paper/1704.04629/) (2017)
- [A Review of Multiple Try MCMC algorithms for Signal Processing](https://mlsanity.com/paper/1801.09065/) (2018)
- [Joint Parameter and Parameterization Inference with Uncertainty Quantification through Differentiable Programming](https://mlsanity.com/paper/2403.02215/) (2024)
- [Adaptive Independent Sticky MCMC algorithms](https://mlsanity.com/paper/1308.3779/) (2013)
- [Parallel sequential Monte Carlo for stochastic gradient-free nonconvex optimization](https://mlsanity.com/paper/1811.09469/) (2018)
- [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)
- [Compressed Monte Carlo with application in particle filtering](https://mlsanity.com/paper/2107.08459/) (2021)
- [Optimality in Noisy Importance Sampling](https://mlsanity.com/paper/2201.02432/) (2022)
- [R-NL: Covariance Matrix Estimation for Elliptical Distributions based on Nonlinear Shrinkage](https://mlsanity.com/paper/2210.14854/) (2022)

## Built on by (most cited first, 1 of 1)

- [Enhancing Graphical Lasso: A Robust Scheme for Non-Stationary Mean Data](https://mlsanity.com/paper/2503.19651/) (2025)

## Abstract (arXiv)

In this paper we address the problem of performing Bayesian inference for the parameters of a nonlinear multi-output model and the covariance matrix of the different output signals. We propose an adaptive importance sampling (AIS) scheme for multivariate Bayesian inversion problems, which is based in two main ideas: the variables of interest are split in two blocks and the inference takes advantage of known analytical optimization formulas. We estimate both the unknown parameters of the multivariate non-linear model and the covariance matrix of the noise. In the first part of the proposed inference scheme, a novel AIS technique called adaptive target adaptive importance sampling (ATAIS) is designed, which alternates iteratively between an IS technique over the parameters of the non-linear model and a frequentist approach for the covariance matrix of the noise. In the second part of the proposed inference scheme, a prior density over the covariance matrix is considered and the cloud of samples obtained by ATAIS are recycled and re-weighted to obtain a complete Bayesian study over the model parameters and covariance matrix. ATAIS is the main contribution of the work. Additionally, the inverted layered importance sampling (ILIS) is presented as a possible compelling algorithm (but based on a conceptually simpler idea). Different numerical examples show the benefits of the proposed approaches