Two classic flavours together, Gaussian Processes and state filters. There are other nonparametric state filters, e.g. Variational filters and particle filters.

This is a kind of a dual to using a state filter to calculate a Gaussian process regression as a computational shorthand.

Here we use Gaussian processes to define the filter, in particular to learn
nonparametric transition, observation or state densities for a generalized
Kalman filter.
This is what
[Turner, Deisenroth, and Rasmussen (2010); Frigola, Chen, and Rasmussen (2014); Frigola et al. (2013);EleftheriadisIdentification2017] do.
Also possible the same, *recurrent Gaussian Processes*?
đźŹ—
(Mattos et al. 2016, 2017; FĂ¶ll et al. 2017).

Eleftheriadis, Stefanos, Tom Nicholson, Marc Deisenroth, and James Hensman. 2017. â€śIdentification of Gaussian Process State Space Models.â€ť In *Advances in Neural Information Processing Systems 30*, edited by I. Guyon, U. V. Luxburg, S. Bengio, H. Wallach, R. Fergus, S. Vishwanathan, and R. Garnett, 5309â€“19. Curran Associates, Inc. http://papers.nips.cc/paper/7115-identification-of-gaussian-process-state-space-models.pdf.

FĂ¶ll, Roman, Bernard Haasdonk, Markus Hanselmann, and Holger Ulmer. 2017. â€śDeep Recurrent Gaussian Process with Variational Sparse Spectrum Approximation,â€ť November. http://arxiv.org/abs/1711.00799.

Frigola, Roger, Yutian Chen, and Carl Edward Rasmussen. 2014. â€śVariational Gaussian Process State-Space Models.â€ť In *Advances in Neural Information Processing Systems 27*, edited by Z. Ghahramani, M. Welling, C. Cortes, N. D. Lawrence, and K. Q. Weinberger, 3680â€“8. Curran Associates, Inc. http://papers.nips.cc/paper/5375-variational-gaussian-process-state-space-models.pdf.

Frigola, Roger, Fredrik Lindsten, Thomas B SchĂ¶n, and Carl Edward Rasmussen. 2013. â€śBayesian Inference and Learning in Gaussian Process State-Space Models with Particle MCMC.â€ť In *Advances in Neural Information Processing Systems 26*, edited by C. J. C. Burges, L. Bottou, M. Welling, Z. Ghahramani, and K. Q. Weinberger, 3156â€“64. Curran Associates, Inc. http://papers.nips.cc/paper/5085-bayesian-inference-and-learning-in-gaussian-process-state-space-models-with-particle-mcmc.pdf.

Huber, Marco F. 2014. â€śRecursive Gaussian Process: On-Line Regression and Learning.â€ť *Pattern Recognition Letters* 45 (August): 85â€“91. https://doi.org/10.1016/j.patrec.2014.03.004.

Mattos, CĂ©sar Lincoln C., Zhenwen Dai, Andreas Damianou, Guilherme A. Barreto, and Neil D. Lawrence. 2017. â€śDeep Recurrent Gaussian Processes for Outlier-Robust System Identification.â€ť *Journal of Process Control*, DYCOPS-CAB 2016, 60 (December): 82â€“94. https://doi.org/10.1016/j.jprocont.2017.06.010.

Mattos, CĂ©sar Lincoln C., Zhenwen Dai, Andreas Damianou, Jeremy Forth, Guilherme A. Barreto, and Neil D. Lawrence. 2016. â€śRecurrent Gaussian Processes.â€ť In *Proceedings of ICLR*. http://arxiv.org/abs/1511.06644.

Nickisch, Hannes, Arno Solin, and Alexander Grigorevskiy. 2018. â€śState Space Gaussian Processes with Non-Gaussian Likelihood.â€ť In *International Conference on Machine Learning*, 3789â€“98. http://proceedings.mlr.press/v80/nickisch18a.html.

Turner, Ryan, Marc Deisenroth, and Carl Rasmussen. 2010. â€śState-Space Inference and Learning with Gaussian Processes.â€ť In *Proceedings of the Thirteenth International Conference on Artificial Intelligence and Statistics*, 868â€“75. http://proceedings.mlr.press/v9/turner10a.html.