# Overcoming Mean-Field Approximations in Recurrent Gaussian Process Models

## TL;DR

This paper introduces VCDT, a novel variational inference method for recurrent Gaussian process models that explicitly models dependencies between states and transition functions, improving accuracy and calibration without increasing computational complexity.

## Contribution

It eliminates the mean-field factorisation in variational inference for Gaussian process dynamical systems, enabling better posterior calibration and predictive performance.

## Key findings

- VCDT outperforms mean-field methods in predictive accuracy.
- VCDT provides more calibrated estimates of transition functions.
- The method maintains computational efficiency comparable to existing approaches.

## Abstract

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.