# Martingale posterior distributions

arXiv: https://arxiv.org/abs/2103.15671 · HTML: https://mlsanity.com/paper/2103.15671/

## Key facts

- Cited by 45 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 29 Mar 2021.

## Built on by (most cited first, 30 of 45)

- [Is In-Context Learning in Large Language Models Bayesian? A Martingale Perspective](https://mlsanity.com/paper/2406.00793/) (2024)
- [Rethinking Aleatoric and Epistemic Uncertainty](https://mlsanity.com/paper/2412.20892/) (2024)
- [BED-LLM: Intelligent Information Gathering with LLMs and Bayesian Experimental Design](https://mlsanity.com/paper/2508.21184/) (2025)
- [Outlier-robust Kalman Filtering through Generalised Bayes](https://mlsanity.com/paper/2405.05646/) (2024)
- [Estimating the Hallucination Rate of Generative AI](https://mlsanity.com/paper/2406.07457/) (2024)
- [Robust and Scalable Bayesian Online Changepoint Detection](https://mlsanity.com/paper/2302.04759/) (2023)
- [Martingale Posterior Neural Processes](https://mlsanity.com/paper/2304.09431/) (2023)
- [On Subjective Uncertainty Quantification and Calibration in Natural Language Generation](https://mlsanity.com/paper/2406.05213/) (2024)
- [Robust and Conjugate Gaussian Process Regression](https://mlsanity.com/paper/2311.00463/) (2023)
- [TabMGP: Martingale Posterior with TabPFN](https://mlsanity.com/paper/2510.25154/) (2025)
- [AI-Powered Bayesian Inference](https://mlsanity.com/paper/2502.19231/) (2025)
- [Adaptive Elicitation of Latent Information Using Natural Language](https://mlsanity.com/paper/2504.04204/) (2025)
- [Quasi-Bayes meets Vines](https://mlsanity.com/paper/2406.12764/) (2024)
- [Dimension Agnostic Neural Processes](https://mlsanity.com/paper/2502.20661/) (2025)
- [Uncertainty Quantification for Prior-Data Fitted Networks using Martingale Posteriors](https://mlsanity.com/paper/2505.11325/) (2025)
- [Martingale Posterior Neural Networks for Fast Sequential Decision Making](https://mlsanity.com/paper/2506.11898/) (2025)
- [Quasi-Bayesian Nonparametric Density Estimation via Autoregressive Predictive Updates](https://mlsanity.com/paper/2206.06462/) (2022)
- [On Uncertainty Quantification for Near-Bayes Optimal Algorithms](https://mlsanity.com/paper/2403.19381/) (2024)
- [Predictive variational inference: Learn the predictively optimal posterior distribution](https://mlsanity.com/paper/2410.14843/) (2024)
- [From Drift to Coherence: Stabilizing Beliefs in LLMs](https://mlsanity.com/paper/2606.17832/) (2026)
- [Robust Classification by Coupling Data Mollification with Label Smoothing](https://mlsanity.com/paper/2406.01494/) (2024)
- [Can Generative AI Solve Your In-Context Learning Problem? A Martingale Perspective](https://mlsanity.com/paper/2412.06033/) (2024)
- [Architectural and Inferential Inductive Biases For Exchangeable Sequence Modeling](https://mlsanity.com/paper/2503.01215/) (2025)
- [Deep Ensembles for Epistemic Uncertainty: A Frequentist Perspective](https://mlsanity.com/paper/2510.22063/) (2025)
- [Multi-Output Robust and Conjugate Gaussian Processes](https://mlsanity.com/paper/2510.26401/) (2025)
- [Uncertainty Quantification and Data Efficiency in AI: An Information-Theoretic Perspective](https://mlsanity.com/paper/2512.05267/) (2025)
- [Optimizing Data Augmentation through Bayesian Model Selection](https://mlsanity.com/paper/2505.21813/) (2025)
- [Scalable Posterior Uncertainty for Flexible Density-Based Clustering](https://mlsanity.com/paper/2603.03188/) (2026)
- [A Predictive View on Streaming Hidden Markov Models](https://mlsanity.com/paper/2604.09208/) (2026)
- [What does a Bayes-filtered transformer believe? A predictive Monte Carlo approach](https://mlsanity.com/paper/2607.17060/) (2026)

## Abstract (arXiv)

The prior distribution on parameters of a sampling distribution is the usual starting point for Bayesian uncertainty quantification. In this paper, we present a different perspective which focuses on missing observations as the source of statistical uncertainty, with the parameter of interest being known precisely given the entire population. We argue that the foundation of Bayesian inference is to assign a distribution on missing observations conditional on what has been observed. In the conditionally i.i.d. setting with an observed sample of size n, the Bayesian would thus assign a predictive distribution on the missing Yn+1:∞ conditional on Y1:n, which then induces a distribution on the parameter. Demonstrating an application of martingales, Doob shows that choosing the Bayesian predictive distribution returns the conventional posterior as the distribution of the parameter. Taking this as our cue, we relax the predictive machine, avoiding the need for the predictive to be derived solely from the usual prior to posterior to predictive density formula. We introduce the textitmartingale posterior distribution, which returns Bayesian uncertainty directly on any statistic of interest without the need for the likelihood and prior, and this distribution can be sampled through a computational scheme we name textitpredictive resampling. To that end, we introduce new predictive methodologies for multivariate density estimation, regression and classification that build upon recent work on bivariate copulas.