# Bayesian Inference and Learning in Gaussian Process State-Space Models with Particle MCMC

arXiv: https://arxiv.org/abs/1306.2861 · HTML: https://mlsanity.com/paper/1306.2861/

## Key facts

- Cited by 24 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 12 Jun 2013.

## Built on by (most cited first, 24 of 24)

- [Deep networks for system identification: a Survey](https://mlsanity.com/paper/2301.12832/) (2023)
- [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)
- [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)
- [Learning-Based Optimal Control with Performance Guarantees for Unknown Systems with Latent States](https://mlsanity.com/paper/2303.17963/) (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)
- [Active Learning in Gaussian Process State Space Model](https://mlsanity.com/paper/2108.00819/) (2021)
- [Hybrid Gaussian Process Modeling Applied to Economic Stochastic Model Predictive Control of Batch Processes](https://mlsanity.com/paper/2108.06430/) (2021)
- [The Bayesian Context Trees State Space Model for time series modelling and forecasting](https://mlsanity.com/paper/2308.00913/) (2023)
- [Efficient Transformed Gaussian Process State-Space Models for Non-Stationary High-Dimensional Dynamical Systems](https://mlsanity.com/paper/2503.18309/) (2025)
- [LatentFlow: A General Framework for Conditioning Stochastic Processes](https://mlsanity.com/paper/2607.12922/) (2026)
- [PDE-constrained Gaussian process surrogate modeling with uncertain data locations](https://mlsanity.com/paper/2305.11586/) (2023)
- [Semi-Supervised Model-Free Bayesian State Estimation from Compressed Measurements](https://mlsanity.com/paper/2407.07368/) (2024)
- [Bayesian Inference and Learning in Nonlinear Dynamical Systems: A Framework for Incorporating Explicit and Implicit Prior Knowledge](https://mlsanity.com/paper/2508.15345/) (2025)
- [Adaptive Graph Convolutional Network Framework for Multidimensional Time Series Prediction](https://mlsanity.com/paper/2205.04885/) (2022)
- [A projected nonlinear state-space model for forecasting time series signals](https://mlsanity.com/paper/2311.13247/) (2023)
- [Learning the Dynamic Correlations and Mitigating Noise by Hierarchical Convolution for Long-term Sequence Forecasting](https://mlsanity.com/paper/2312.16790/) (2023)
- [Recursive Gaussian Process State Space Model](https://mlsanity.com/paper/2411.14679/) (2024)
- [pDANSE: Particle-based Data-driven Nonlinear State Estimation from Nonlinear Measurements](https://mlsanity.com/paper/2510.27503/) (2025)
- [Coordinate ascent neural Kalman-MLE for state estimation](https://mlsanity.com/paper/2511.01855/) (2025)
- [Fast Riemannian-manifold Hamiltonian Monte Carlo for hierarchical Gaussian-process models](https://mlsanity.com/paper/2511.06407/) (2025)
- [Simulation-Free Learning of GP-SDEs from Irregular Observations](https://mlsanity.com/paper/2609.33112/) (2026)

## Abstract (arXiv)

State-space models are successfully used in many areas of science, engineering and economics to model time series and dynamical systems. We present a fully Bayesian approach to inference and learning (i.e. state estimation and system identification) in nonlinear nonparametric state-space models. We place a Gaussian process prior over the state transition dynamics, resulting in a flexible model able to capture complex dynamical phenomena. To enable efficient inference, we marginalize over the transition dynamics function and infer directly the joint smoothing distribution using specially tailored Particle Markov Chain Monte Carlo samplers. Once a sample from the smoothing distribution is computed, the state transition predictive distribution can be formulated analytically. Our approach preserves the full nonparametric expressivity of the model and can make use of sparse Gaussian processes to greatly reduce computational complexity.