I am interested here in the trick which makes certain Gaussian process regression problems soluble by making them local, i.e. Markov, with respect to some assumed hidden state, in the same way Kalman filtering does Wiener filtering. This means you get to solve a GP as an SDE. This trick is explained in an intro article in S. Särkkä, Solin, and Hartikainen (2013), based on previous work (Reece and Roberts 2010; Lindgren, Rue, and Lindström 2011; Särkkä and Hartikainen 2012; Hartikainen and Särkkä 2010; Solin 2016). The state of the art seems to be @ Recent extensions include (Karvonen and Särkkä 2016; Nickisch, Solin, and Grigorevskiy 2018). The idea is that if your covariance kernel is, or can be well approximated by, say, a rational function then it is possible to factorise it into a state space model tractably, which makes it cheap due to the favourable properties of such models. That sounds simple enough conceptually; I wonder about the practice. Possibly related, but I have not yet actually read: (Huber 2014).
This complements, perhaps, the trick of fast Gaussian process calculations on lattices.
To learn: Is this a classic graphical model-style decomposition into message passing via factor graph decompositions? Publications like (Cox, van de Laar, and de Vries 2019) are suggestive that it is, but I need to take a better look. So is there anything special going on here? It seems like there is something special here, in that standard factor graph decompositions are based on discrete nodes in a graph, whereas Gaussian processes give us a function over the entire input space; as such, this particular trick gives us an angle of attack for continuous graphical models which are of general interest.
There is another concept which is kind of a dual to filtering of a causal Gaussian process, which uses Gaussian processes to define the process dynamics or observation distribution. I have no use for that at the moment, but it pops up in the same keyword searches.
miscellaneous notes towards implementations
- TemporalGPs.jl, introduced by Will Tebbutt, is a julia implementation of this.
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