Hida-Mat\'ern Kernel

We present the class of Hida-Mat\'ern kernels, which is the canonical family of covariance functions over the entire space of stationary Gauss-Markov Processes. It extends upon Mat\'ern kernels, by allowing for flexible construction of priors over processes with oscillatory components. Any stationary kernel, including the widely used squared-exponential and spectral mixture kernels, are either directly within this class or are appropriate asymptotic limits, demonstrating the generality of this class. Taking advantage of its Markovian nature we show how to represent such processes as state space models using only the kernel and its derivatives. In turn this allows us to perform Gaussian Process inference more efficiently and side step the usual computational burdens. We also show how exploiting special properties of the state space representation enables improved numerical stability in addition to further reductions of computational complexity.