# State Space Expectation Propagation: Efficient Inference Schemes for Temporal Gaussian Processes

arXiv: https://arxiv.org/abs/2007.05994 · HTML: https://mlsanity.com/paper/2007.05994/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 12 Jul 2020.

## Built on by (most cited first, 10 of 10)

- [Spatio-Temporal Variational Gaussian Processes](https://mlsanity.com/paper/2111.01732/) (2021)
- [Sampling from Gaussian Process Posteriors using Stochastic Gradient Descent](https://mlsanity.com/paper/2306.11589/) (2023)
- [Markovian Gaussian Process Variational Autoencoders](https://mlsanity.com/paper/2207.05543/) (2022)
- [Bayes-Newton Methods for Approximate Bayesian Inference with PSD Guarantees](https://mlsanity.com/paper/2111.01721/) (2021)
- [Sparse Algorithms for Markovian Gaussian Processes](https://mlsanity.com/paper/2103.10710/) (2021)
- [Combining Pseudo-Point and State Space Approximations for Sum-Separable Gaussian Processes](https://mlsanity.com/paper/2106.10210/) (2021)
- [Linear Time GPs for Inferring Latent Trajectories from Neural Spike Trains](https://mlsanity.com/paper/2306.01802/) (2023)
- [A Look at Improving Robustness in Visual-inertial SLAM by Moment Matching](https://mlsanity.com/paper/2205.13821/) (2022)
- [Learning Coupled Subspaces for Multi-Condition Spike Data](https://mlsanity.com/paper/2410.19153/) (2024)
- [Recursive Entropic Variational Inference for Nonlinear State-Space Models](https://mlsanity.com/paper/2511.15409/) (2025)

## Abstract (arXiv)

We formulate approximate Bayesian inference in non-conjugate temporal and spatio-temporal Gaussian process models as a simple parameter update rule applied during Kalman smoothing. This viewpoint encompasses most inference schemes, including expectation propagation (EP), the classical (Extended, Unscented, etc.) Kalman smoothers, and variational inference. We provide a unifying perspective on these algorithms, showing how replacing the power EP moment matching step with linearisation recovers the classical smoothers. EP provides some benefits over the traditional methods via introduction of the so-called cavity distribution, and we combine these benefits with the computational efficiency of linearisation, providing extensive empirical analysis demonstrating the efficacy of various algorithms under this unifying framework. We provide a fast implementation of all methods in JAX.