Ensemble Kalman Filtering-Aided Variational Inference for Gaussian Process State-Space Models

Gaussian process state-space models (GPSSMs) provide a principled and flexible approach to model latent state dynamics observed through emission models. However, existing variational methods for learning GPSSMs face a substantial challenge in optimizing a large number of parameters, particularly with the introduction of amortized inference networks. To address this challenge, we leverage the ensemble Kalman filter (EnKF), a well-established model-based filtering technique, to approximate the posterior distribution of latent states within the variational inference framework. This approach eliminates the need for inference networks, significantly reducing the number of variational parameters. Moreover, we demonstrate that with the aid of EnKF, the straightforward evaluation of approximated evidence lower bound (ELBO) in the variational inference can be easily obtained through the summation of multiple terms with closed-form solutions. By leveraging automatic differentiation tools, we thus can maximize the ELBO and train the GPSSM efficiently. We also extend the proposed method to an online setting and provide comprehensive algorithm analyses and insights. Extensive testing on diverse real and simulated datasets demonstrates that our variational inference algorithms, integrated with EnKF, outperform existing methods in terms of learning and inference performance.