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We identify a new variational inference scheme for dynamical systems whosetransition function is modelled by a Gaussian process. Inference in thissetting has either employed computationally intensive MCMC methods, or reliedon factorisations of the variational posterior. As we demonstrate in ourexperiments, the factorisation between latent system states and transitionfunction can lead to a miscalibrated posterior and to learning unnecessarilylarge noise terms. We eliminate this factorisation by explicitly modelling thedependence between state trajectories and the Gaussian process posterior.Samples of the latent states can then be tractably generated by conditioning onthis representation. The method we obtain (VCDT: variationally coupled dynamicsand trajectories) gives better predictive performance and more calibratedestimates of the transition function, yet maintains the same time and spacecomplexities as mean-field methods. Code is available at:github.com/ialong/GPt.

Overcoming Mean-Field Approximations in Recurrent Gaussian Process Models