# MMD-Bayes: Robust Bayesian Estimation via Maximum Mean Discrepancy

arXiv: https://arxiv.org/abs/1909.13339 · HTML: https://mlsanity.com/paper/1909.13339/

## Key facts

- Cited by 35 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 29 Sep 2019.

## Built on by (most cited first, 30 of 35)

- [Robust Generalised Bayesian Inference for Intractable Likelihoods](https://mlsanity.com/paper/2104.07359/) (2021)
- [Learning Robust Statistics for Simulation-based Inference under Model Misspecification](https://mlsanity.com/paper/2305.15871/) (2023)
- [Robust Neural Posterior Estimation and Statistical Model Criticism](https://mlsanity.com/paper/2210.06564/) (2022)
- [Robust Bayesian Inference for Simulator-based Models via the MMD Posterior Bootstrap](https://mlsanity.com/paper/2202.04744/) (2022)
- [Probabilistic Forecasting with Generative Networks via Scoring Rule Minimization](https://mlsanity.com/paper/2112.08217/) (2021)
- [Addressing Misspecification in Simulation-based Inference through Data-driven Calibration](https://mlsanity.com/paper/2405.08719/) (2024)
- [Generalized Bayesian Inference for Scientific Simulators via Amortized Cost Estimation](https://mlsanity.com/paper/2305.15208/) (2023)
- [Composite Goodness-of-fit Tests with Kernels](https://mlsanity.com/paper/2111.10275/) (2021)
- [Robust and Scalable Bayesian Online Changepoint Detection](https://mlsanity.com/paper/2302.04759/) (2023)
- [Likelihood-Free Inference with Generative Neural Networks via Scoring Rule Minimization](https://mlsanity.com/paper/2205.15784/) (2022)
- [Targeted Separation and Convergence with Kernel Discrepancies](https://mlsanity.com/paper/2209.12835/) (2022)
- [Generalised Bayesian Inference for Discrete Intractable Likelihood](https://mlsanity.com/paper/2206.08420/) (2022)
- [Robust and Conjugate Gaussian Process Regression](https://mlsanity.com/paper/2311.00463/) (2023)
- [Optimally-Weighted Estimators of the Maximum Mean Discrepancy for Likelihood-Free Inference](https://mlsanity.com/paper/2301.11674/) (2023)
- [Integral Imprecise Probability Metrics](https://mlsanity.com/paper/2505.16156/) (2025)
- [Robust Bayesian Learning for Reliable Wireless AI: Framework and Applications](https://mlsanity.com/paper/2207.00300/) (2022)
- [Simulation-based Bayesian inference under model misspecification](https://mlsanity.com/paper/2503.12315/) (2025)
- [A Dictionary of Closed-Form Kernel Mean Embeddings](https://mlsanity.com/paper/2504.18830/) (2025)
- [Misspecification-robust Sequential Neural Likelihood for Simulation-based Inference](https://mlsanity.com/paper/2301.13368/) (2023)
- [Robust Kernel Hypothesis Testing under Data Corruption](https://mlsanity.com/paper/2405.19912/) (2024)
- [On the Robustness of Kernel Goodness-of-Fit Tests](https://mlsanity.com/paper/2408.05854/) (2024)
- [Robust PACm: Training Ensemble Models Under Misspecification and Outliers](https://mlsanity.com/paper/2203.01859/) (2022)
- [Deceptive Fairness Attacks on Graphs via Meta Learning](https://mlsanity.com/paper/2310.15653/) (2023)
- [Continual learning with the neural tangent ensemble](https://mlsanity.com/paper/2408.17394/) (2024)
- [A Semi-Bayesian Nonparametric Estimator of the Maximum Mean Discrepancy Measure: Applications in Goodness-of-Fit Testing and Generative Adversarial Networks](https://mlsanity.com/paper/2303.02637/) (2023)
- [A Primer on Variational Inference for Physics-Informed Deep Generative Modelling](https://mlsanity.com/paper/2409.06560/) (2024)
- [Signature Maximum Mean Discrepancy Two-Sample Statistical Tests](https://mlsanity.com/paper/2506.01718/) (2025)
- [Minimum Distance Summaries for Robust Neural Posterior Estimation](https://mlsanity.com/paper/2602.09161/) (2026)
- [Some challenges of calibrating differentiable agent-based models](https://mlsanity.com/paper/2307.01085/) (2023)
- [Decision Making under Model Misspecification: DRO with Robust Bayesian Ambiguity Sets](https://mlsanity.com/paper/2505.03585/) (2025)

## Abstract (arXiv)

In some misspecified settings, the posterior distribution in Bayesian statistics may lead to inconsistent estimates. To fix this issue, it has been suggested to replace the likelihood by a pseudo-likelihood, that is the exponential of a loss function enjoying suitable robustness properties. In this paper, we build a pseudo-likelihood based on the Maximum Mean Discrepancy, defined via an embedding of probability distributions into a reproducing kernel Hilbert space. We show that this MMD-Bayes posterior is consistent and robust to model misspecification. As the posterior obtained in this way might be intractable, we also prove that reasonable variational approximations of this posterior enjoy the same properties. We provide details on a stochastic gradient algorithm to compute these variational approximations. Numerical simulations indeed suggest that our estimator is more robust to misspecification than the ones based on the likelihood.