# On the well-posedness of Bayesian inverse problems

arXiv: https://arxiv.org/abs/1902.10257 · HTML: https://mlsanity.com/paper/1902.10257/

## Key facts

- Cited by 20 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 26 Feb 2019.

## Built on by (most cited first, 20 of 20)

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- [Conditional Generative Models are Provably Robust: Pointwise Guarantees for Bayesian Inverse Problems](https://mlsanity.com/paper/2303.15845/) (2023)
- [Bayesian Inversion for Nonlinear Imaging Models using Deep Generative Priors](https://mlsanity.com/paper/2203.10078/) (2022)
- [Learning from small data sets: Patch-based regularizers in inverse problems for image reconstruction](https://mlsanity.com/paper/2312.16611/) (2023)
- [Bayesian Posterior Perturbation Analysis with Integral Probability Metrics](https://mlsanity.com/paper/2303.01512/) (2023)
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- [Deep Microlocal Reconstruction for Limited-Angle Tomography](https://mlsanity.com/paper/2108.05732/) (2021)
- [NF-ULA: Langevin Monte Carlo with Normalizing Flow Prior for Imaging Inverse Problems](https://mlsanity.com/paper/2304.08342/) (2023)
- [Gaussian Measures Conditioned on Nonlinear Observations: Consistency, MAP Estimators, and Simulation](https://mlsanity.com/paper/2405.13149/) (2024)
- [Error Analysis of Bayesian Inverse Problems with Generative Priors](https://mlsanity.com/paper/2601.17374/) (2026)
- [InVAErt networks for amortized inference and identifiability analysis of lumped parameter hemodynamic models](https://mlsanity.com/paper/2408.08264/) (2024)
- [Further analysis of multilevel Stein variational gradient descent with an application to the Bayesian inference of glacier ice models](https://mlsanity.com/paper/2212.03366/) (2022)
- [Applications of ML-Based Surrogates in Bayesian Approaches to Inverse Problems](https://mlsanity.com/paper/2310.12046/) (2023)
- [Solving Bayesian inverse problems with diffusion priors and off-policy RL](https://mlsanity.com/paper/2503.09746/) (2025)
- [Prediction Markets as Bayesian Inverse Problems: Uncertainty Quantification, Identifiability, and Information Gain from Price-Volume Histories under Latent Types](https://mlsanity.com/paper/2601.18815/) (2026)
- [Neural network surrogates with uncertainty quantification for inverse problems in partial differential equations](https://mlsanity.com/paper/2606.20417/) (2026)

## Abstract (arXiv)

The subject of this article is the introduction of a new concept of well-posedness of Bayesian inverse problems. The conventional concept of (Lipschitz, Hellinger) well-posedness in [Stuart 2010, Acta Numerica 19, pp. 451-559] is difficult to verify in practice and may be inappropriate in some contexts. Our concept simply replaces the Lipschitz continuity of the posterior measure in the Hellinger distance by continuity in an appropriate distance between probability measures. Aside from the Hellinger distance, we investigate well-posedness with respect to weak convergence, the total variation distance, the Wasserstein distance, and also the Kullback--Leibler divergence. We demonstrate that the weakening to continuity is tolerable and that the generalisation to other distances is important. The main results of this article are proofs of well-posedness with respect to some of the aforementioned distances for large classes of Bayesian inverse problems. Here, little or no information about the underlying model is necessary; making these results particularly interesting for practitioners using black-box models. We illustrate our findings with numerical examples motivated from machine learning and image processing.