# Dimension-Robust MCMC in Bayesian Inverse Problems

arXiv: https://arxiv.org/abs/1803.03344 · HTML: https://mlsanity.com/paper/1803.03344/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 9 Mar 2018.

## Built on by (most cited first, 8 of 8)

- [Conditional Optimal Transport on Function Spaces](https://mlsanity.com/paper/2311.05672/) (2023)
- [Surrogate Model for Geological CO2 Storage and Its Use in Hierarchical MCMC History Matching](https://mlsanity.com/paper/2308.06341/) (2023)
- [Bayesian Learning via Q-Exponential Process](https://mlsanity.com/paper/2210.07987/) (2022)
- [Spatiotemporal Besov Priors for Bayesian Inverse Problems](https://mlsanity.com/paper/2306.16378/) (2023)
- [Accelerated training of deep learning surrogate models for surface displacement and flow, with application to MCMC-based history matching of CO2 storage operations](https://mlsanity.com/paper/2408.10717/) (2024)
- [Bayesian neural network priors for edge-preserving inversion](https://mlsanity.com/paper/2112.10663/) (2021)
- [Data assimilation for subsurface flow using latent diffusion model parameterization: performance of ensemble-Kalman and Monte Carlo techniques](https://mlsanity.com/paper/2606.11140/) (2026)
- [Multimodal Auto-regressive Transformer Surrogate for Modeling Variable Operations and Quantifying Uncertainty in Geological Carbon Storage](https://mlsanity.com/paper/2608.02629/) (2026)

## Abstract (arXiv)

The methodology developed in this article is motivated by a wide range of prediction and uncertainty quantification problems that arise in Statistics, Machine Learning and Applied Mathematics, such as non-parametric regression, multi-class classification and inversion of partial differential equations. One popular formulation of such problems is as Bayesian inverse problems, where a prior distribution is used to regularize inference on a high-dimensional latent state, typically a function or a field. It is common that such priors are non-Gaussian, for example piecewise-constant or heavy-tailed, and/or hierarchical, in the sense of involving a further set of low-dimensional parameters, which, for example, control the scale or smoothness of the latent state. In this formulation prediction and uncertainty quantification relies on efficient exploration of the posterior distribution of latent states and parameters. This article introduces a framework for efficient MCMC sampling in Bayesian inverse problems that capitalizes upon two fundamental ideas in MCMC, non-centred parameterisations of hierarchical models and dimension-robust samplers for latent Gaussian processes. Using a range of diverse applications we showcase that the proposed framework is dimension-robust, that is, the efficiency of the MCMC sampling does not deteriorate as the dimension of the latent state gets higher. We showcase the full potential of the machinery we develop in the article in semi-supervised multi-class classification, where our sampling algorithm is used within an active learning framework to guide the selection of input data to manually label in order to achieve high predictive accuracy with a minimal number of labelled data.