# Fast Variational Learning in State-Space Gaussian Process Models

arXiv: https://arxiv.org/abs/2007.04731 · HTML: https://mlsanity.com/paper/2007.04731/

## Key facts

- Cited by 20 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 9 Jul 2020.

## Built on by (most cited first, 20 of 20)

- [The Bayesian Learning Rule](https://mlsanity.com/paper/2107.04562/) (2021)
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- [Sparse Algorithms for Markovian Gaussian Processes](https://mlsanity.com/paper/2103.10710/) (2021)
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- [Fantasizing with Dual GPs in Bayesian Optimization and Active Learning](https://mlsanity.com/paper/2211.01053/) (2022)
- [Physics-Informed Variational State-Space Gaussian Processes](https://mlsanity.com/paper/2409.13876/) (2024)
- [Linear Time GPs for Inferring Latent Trajectories from Neural Spike Trains](https://mlsanity.com/paper/2306.01802/) (2023)
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- [Improving Hyperparameter Learning under Approximate Inference in Gaussian Process Models](https://mlsanity.com/paper/2306.04201/) (2023)
- [Neighbour-Driven Gaussian Process Variational Autoencoders for Scalable Structured Latent Modelling](https://mlsanity.com/paper/2505.16481/) (2025)
- [Short-term Prediction and Filtering of Solar Power Using State-Space Gaussian Processes](https://mlsanity.com/paper/2302.00388/) (2023)
- [A Look at Improving Robustness in Visual-inertial SLAM by Moment Matching](https://mlsanity.com/paper/2205.13821/) (2022)
- [Deep Gaussian Markov Random Fields for Graph-Structured Dynamical Systems](https://mlsanity.com/paper/2306.08445/) (2023)
- [Diffusion-aware Censored Gaussian Processes for Demand Modelling](https://mlsanity.com/paper/2501.12354/) (2025)
- [Recursive Entropic Variational Inference for Nonlinear State-Space Models](https://mlsanity.com/paper/2511.15409/) (2025)
- [Designing an Optimal Sensor Network via Minimizing Information Loss](https://mlsanity.com/paper/2512.05940/) (2025)

## Abstract (arXiv)

Gaussian process (GP) regression with 1D inputs can often be performed in linear time via a stochastic differential equation formulation. However, for non-Gaussian likelihoods, this requires application of approximate inference methods which can make the implementation difficult, e.g., expectation propagation can be numerically unstable and variational inference can be computationally inefficient. In this paper, we propose a new method that removes such difficulties. Building upon an existing method called conjugate-computation variational inference, our approach enables linear-time inference via Kalman recursions while avoiding numerical instabilities and convergence issues. We provide an efficient JAX implementation which exploits just-in-time compilation and allows for fast automatic differentiation through large for-loops. Overall, our approach leads to fast and stable variational inference in state-space GP models that can be scaled to time series with millions of data points.