# Deep Stochastic Processes via Functional Markov Transition Operators

arXiv: https://arxiv.org/abs/2305.15574 · HTML: https://mlsanity.com/paper/2305.15574/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 30 of its 57 references that are in the map.
- First posted on arXiv 24 May 2023.

## Builds on (most cited first, 30 of 30)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domains](https://mlsanity.com/paper/2006.10739/) (2020)
- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [Deep Sets](https://mlsanity.com/paper/1703.06114/) (2017)
- [Normalizing Flows for Probabilistic Modeling and Inference](https://mlsanity.com/paper/1912.02762/) (2019)
- [A Tutorial on Bayesian Optimization](https://mlsanity.com/paper/1807.02811/) (2018)
- [Neural Spline Flows](https://mlsanity.com/paper/1906.04032/) (2019)
- [Importance Weighted Autoencoders](https://mlsanity.com/paper/1509.00519/) (2015)
- [Conditional Neural Processes](https://mlsanity.com/paper/1807.01613/) (2018)
- [Attentive Neural Processes](https://mlsanity.com/paper/1901.05761/) (2019)
- [Monte Carlo Gradient Estimation in Machine Learning](https://mlsanity.com/paper/1906.10652/) (2019)
- [Scalable Gradients for Stochastic Differential Equations](https://mlsanity.com/paper/2001.01328/) (2020)
- [Deep Bayesian Bandits Showdown: An Empirical Comparison of Bayesian Deep Networks for Thompson Sampling](https://mlsanity.com/paper/1802.09127/) (2018)
- [Convolutional Conditional Neural Processes](https://mlsanity.com/paper/1910.13556/) (2019)
- [Modern Bayesian Experimental Design](https://mlsanity.com/paper/2302.14545/) (2023)
- [Generative Models as Distributions of Functions](https://mlsanity.com/paper/2102.04776/) (2021)
- [Meta-Learning Stationary Stochastic Process Prediction with Convolutional Neural Processes](https://mlsanity.com/paper/2007.01332/) (2020)
- [Prediction-Oriented Bayesian Active Learning](https://mlsanity.com/paper/2304.08151/) (2023)
- [Neural Diffusion Processes](https://mlsanity.com/paper/2206.03992/) (2022)
- [Student-t Processes as Alternatives to Gaussian Processes](https://mlsanity.com/paper/1402.4306/) (2014)
- [MetaFun: Meta-Learning with Iterative Functional Updates](https://mlsanity.com/paper/1912.02738/) (2019)
- [The Gaussian Neural Process](https://mlsanity.com/paper/2101.03606/) (2021)
- [Transforming Gaussian Processes With Normalizing Flows](https://mlsanity.com/paper/2011.01596/) (2020)
- [Parametric generation of conditional geological realizations using generative neural networks](https://mlsanity.com/paper/1807.05207/) (2018)
- [Group Equivariant Conditional Neural Processes](https://mlsanity.com/paper/2102.08759/) (2021)
- [Generating Realistic Geology Conditioned on Physical Measurements with Generative Adversarial Networks](https://mlsanity.com/paper/1802.03065/) (2018)
- [On Contrastive Representations of Stochastic Processes](https://mlsanity.com/paper/2106.10052/) (2021)
- [BRUNO: A Deep Recurrent Model for Exchangeable Data](https://mlsanity.com/paper/1802.07535/) (2018)

## Built on by (most cited first, 7 of 7)

- [Flow Matching Neural Processes](https://mlsanity.com/paper/2512.23853/) (2025)
- [Learning multi-modal generative models with permutation-invariant encoders and tighter variational objectives](https://mlsanity.com/paper/2309.00380/) (2023)
- [Spectral Convolutional Conditional Neural Processes](https://mlsanity.com/paper/2404.13182/) (2024)
- [Exploring Pseudo-Token Approaches in Transformer Neural Processes](https://mlsanity.com/paper/2504.14416/) (2025)
- [Incremental Transformer Neural Processes](https://mlsanity.com/paper/2602.18955/) (2026)
- [Continual learning via probabilistic exchangeable sequence modelling](https://mlsanity.com/paper/2503.20725/) (2025)
- [Revisiting Neural Processes via Fourier Transform and Volterra Series](https://mlsanity.com/paper/2606.01172/) (2026)

## Abstract (arXiv)

We introduce Markov Neural Processes (MNPs), a new class of Stochastic Processes (SPs) which are constructed by stacking sequences of neural parameterised Markov transition operators in function space. We prove that these Markov transition operators can preserve the exchangeability and consistency of SPs. Therefore, the proposed iterative construction adds substantial flexibility and expressivity to the original framework of Neural Processes (NPs) without compromising consistency or adding restrictions. Our experiments demonstrate clear advantages of MNPs over baseline models on a variety of tasks.