# Martingale Posterior Neural Processes

arXiv: https://arxiv.org/abs/2304.09431 · HTML: https://mlsanity.com/paper/2304.09431/

## Key facts

- Cited by 16 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 33 references that are in the map.
- First posted on arXiv 19 Apr 2023.

## Builds on (most cited first, 15 of 15)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Layer Normalization](https://mlsanity.com/paper/1607.06450/) (2016)
- [Deep Learning Face Attributes in the Wild](https://mlsanity.com/paper/1411.7766/) (2014)
- [Deep Sets](https://mlsanity.com/paper/1703.06114/) (2017)
- [Conditional Neural Processes](https://mlsanity.com/paper/1807.01613/) (2018)
- [Neural Processes](https://mlsanity.com/paper/1807.01622/) (2018)
- [Attentive Neural Processes](https://mlsanity.com/paper/1901.05761/) (2019)
- [Convolutional Conditional Neural Processes](https://mlsanity.com/paper/1910.13556/) (2019)
- [Learning in Implicit Generative Models](https://mlsanity.com/paper/1610.03483/) (2016)
- [Meta-Learning Stationary Stochastic Process Prediction with Convolutional Neural Processes](https://mlsanity.com/paper/2007.01332/) (2020)
- [Pre-trained Gaussian Processes for Bayesian Optimization](https://mlsanity.com/paper/2109.08215/) (2021)
- [Martingale posterior distributions](https://mlsanity.com/paper/2103.15671/) (2021)
- [The Functional Neural Process](https://mlsanity.com/paper/1906.08324/) (2019)
- [Bootstrapping Neural Processes](https://mlsanity.com/paper/2008.02956/) (2020)

## Built on by (most cited first, 16 of 16)

- [Estimating the Hallucination Rate of Generative AI](https://mlsanity.com/paper/2406.07457/) (2024)
- [Bayesian Optimization of Antibodies Informed by a Generative Model of Evolving Sequences](https://mlsanity.com/paper/2412.07763/) (2024)
- [Posterior Uncertainty Quantification in Neural Networks using Data Augmentation](https://mlsanity.com/paper/2403.12729/) (2024)
- [TabMGP: Martingale Posterior with TabPFN](https://mlsanity.com/paper/2510.25154/) (2025)
- [Quasi-Bayes meets Vines](https://mlsanity.com/paper/2406.12764/) (2024)
- [Dimension Agnostic Neural Processes](https://mlsanity.com/paper/2502.20661/) (2025)
- [Spectral Convolutional Conditional Neural Processes](https://mlsanity.com/paper/2404.13182/) (2024)
- [On Uncertainty Quantification for Near-Bayes Optimal Algorithms](https://mlsanity.com/paper/2403.19381/) (2024)
- [Reevaluating Meta-Learning Optimization Algorithms Through Contextual Self-Modulation](https://mlsanity.com/paper/2410.01655/) (2024)
- [Can Generative AI Solve Your In-Context Learning Problem? A Martingale Perspective](https://mlsanity.com/paper/2412.06033/) (2024)
- [R'enyi Neural Processes](https://mlsanity.com/paper/2405.15991/) (2024)
- [Concentration and Calibration in Predictive Bayesian Inference](https://mlsanity.com/paper/2605.00455/) (2026)
- [Self-Supervised Laplace Approximation for Bayesian Uncertainty Quantification](https://mlsanity.com/paper/2605.12208/) (2026)
- [Federated Martingale Posterior Samping](https://mlsanity.com/paper/2605.18554/) (2026)
- [Revisiting Neural Processes via Fourier Transform and Volterra Series](https://mlsanity.com/paper/2606.01172/) (2026)
- [Score-Based Martingale Posteriors for Deep Neural Networks](https://mlsanity.com/paper/2606.15725/) (2026)

## Abstract (arXiv)

A Neural Process (NP) estimates a stochastic process implicitly defined with neural networks given a stream of data, rather than pre-specifying priors already known, such as Gaussian processes. An ideal NP would learn everything from data without any inductive biases, but in practice, we often restrict the class of stochastic processes for the ease of estimation. One such restriction is the use of a finite-dimensional latent variable accounting for the uncertainty in the functions drawn from NPs. Some recent works show that this can be improved with more "data-driven" source of uncertainty such as bootstrapping. In this work, we take a different approach based on the martingale posterior, a recently developed alternative to Bayesian inference. For the martingale posterior, instead of specifying prior-likelihood pairs, a predictive distribution for future data is specified. Under specific conditions on the predictive distribution, it can be shown that the uncertainty in the generated future data actually corresponds to the uncertainty of the implicitly defined Bayesian posteriors. Based on this result, instead of assuming any form of the latent variables, we equip a NP with a predictive distribution implicitly defined with neural networks and use the corresponding martingale posteriors as the source of uncertainty. The resulting model, which we name as Martingale Posterior Neural Process (MPNP), is demonstrated to outperform baselines on various tasks.