  Approximate Bayesian inference methods that scale to very large datasets are
crucial in leveraging probabilistic models for real-world time series. Sparse
Markovian Gaussian processes combine the use of inducing variables with
efficient Kalman filter-like recursions, resulting in algorithms whose
computational and memory requirements scale linearly in the number of inducing
points, whilst also enabling parallel parameter updates and stochastic
optimisation. Under this paradigm, we derive a general site-based approach to
approximate inference, whereby we approximate the non-Gaussian likelihood with
local Gaussian terms, called sites. Our approach results in a suite of novel
sparse extensions to algorithms from both the machine learning and signal
processing literature, including variational inference, expectation
propagation, and the classical nonlinear Kalman smoothers. The derived methods
are suited to large time series, and we also demonstrate their applicability to
spatio-temporal data, where the model has separate inducing points in both time
and space.
