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Gaussian process-based machine learning is a powerful Bayesian paradigm for nonparametric nonlinear regression and classification. In this article, we discuss connections of Gaussian process regression with Kalman filtering and present methods for converting spatiotemporal Gaussian process regression problems into infinite-dimensional state-space models. This formulation allows for use of computationally efficient infinite-dimensional Kalman filtering and smoothing methods, or more general Bayesian filtering and smoothing methods, which reduces the problematic cubic complexity of Gaussian process regression in the number of time steps into linear time complexity. The implication of this is that the use of machine-learning models in signal processing becomes computationally feasible, and it opens the possibility to combine machine-learning techniques with signal processing methods.