# Overcoming Mean-Field Approximations in Recurrent Gaussian Process Models

arXiv: https://arxiv.org/abs/1906.05828 · HTML: https://mlsanity.com/paper/1906.05828/

## Key facts

- Cited by 21 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 13 Jun 2019.

## Built on by (most cited first, 21 of 21)

- [Variational multiple shooting for Bayesian ODEs with Gaussian processes](https://mlsanity.com/paper/2106.10905/) (2021)
- [Structure-preserving Gaussian Process Dynamics](https://mlsanity.com/paper/2102.01606/) (2021)
- [Inferring the Structure of Ordinary Differential Equations](https://mlsanity.com/paper/2107.07345/) (2021)
- [Free-Form Variational Inference for Gaussian Process State-Space Models](https://mlsanity.com/paper/2302.09921/) (2023)
- [Output-Dependent Gaussian Process State-Space Model](https://mlsanity.com/paper/2212.07608/) (2022)
- [Recurrent Neural Networks and Universal Approximation of Bayesian Filters](https://mlsanity.com/paper/2211.00335/) (2022)
- [Towards Flexibility and Interpretability of Gaussian Process State-Space Model](https://mlsanity.com/paper/2301.08843/) (2023)
- [Ensemble Kalman Filtering Meets Gaussian Process SSM for Non-Mean-Field and Online Inference](https://mlsanity.com/paper/2312.05910/) (2023)
- [Towards Efficient Modeling and Inference in Multi-Dimensional Gaussian Process State-Space Models](https://mlsanity.com/paper/2309.01074/) (2023)
- [Out of Distribution Detection via Domain-Informed Gaussian Process State Space Models](https://mlsanity.com/paper/2309.06655/) (2023)
- [The Past Does Matter: Correlation of Subsequent States in Trajectory Predictions of Gaussian Process Models](https://mlsanity.com/paper/2211.11103/) (2022)
- [Sampling-Free Probabilistic Deep State-Space Models](https://mlsanity.com/paper/2309.08256/) (2023)
- [Efficient Transformed Gaussian Process State-Space Models for Non-Stationary High-Dimensional Dynamical Systems](https://mlsanity.com/paper/2503.18309/) (2025)
- [Deep Probabilistic Time Series Forecasting using Augmented Recurrent Input for Dynamic Systems](https://mlsanity.com/paper/2106.05848/) (2021)
- [Likelihood-Free Inference in State-Space Models with Unknown Dynamics](https://mlsanity.com/paper/2111.01555/) (2021)
- [Continual Learning of Multi-modal Dynamics with External Memory](https://mlsanity.com/paper/2203.00936/) (2022)
- [Towards AI-controlled FES-restoration of arm movements: Controlling for progressive muscular fatigue with Gaussian state-space models](https://mlsanity.com/paper/2301.04005/) (2023)
- [Traversing Time with Multi-Resolution Gaussian Process State-Space Models](https://mlsanity.com/paper/2112.03230/) (2021)
- [Recursive Gaussian Process State Space Model](https://mlsanity.com/paper/2411.14679/) (2024)
- [Simulation-Free Learning of GP-SDEs from Irregular Observations](https://mlsanity.com/paper/2609.33112/) (2026)
- [Amortized Structured Stochastic Variational Inference for Gaussian Process Latent Variable Models](https://mlsanity.com/paper/2610.03647/) (2026)

## Abstract (arXiv)

We identify a new variational inference scheme for dynamical systems whose transition function is modelled by a Gaussian process. Inference in this setting has either employed computationally intensive MCMC methods, or relied on factorisations of the variational posterior. As we demonstrate in our experiments, the factorisation between latent system states and transition function can lead to a miscalibrated posterior and to learning unnecessarily large noise terms. We eliminate this factorisation by explicitly modelling the dependence between state trajectories and the Gaussian process posterior. Samples of the latent states can then be tractably generated by conditioning on this representation. The method we obtain (VCDT: variationally coupled dynamics and trajectories) gives better predictive performance and more calibrated estimates of the transition function, yet maintains the same time and space complexities as mean-field methods. Code is available at: github.com/ialong/GPt.