Probabilistic graphical models over continuous index sets

Placeholder for my notes on probabilistic graphical models over a continuum, i.e. with possibly-uncountably many nodes in the graph; or put another way, where the random field has an uncountable index set (but some kind of structure – a metric space, say.)

There is much formalising to be done here, which I do not propose to attempt right now. Here’s a concrete example. Consider Gaussian process whose covariance kerne \(K\) is continuous and of bounded support. Let it be over index space \(\mathcal{T}:=\mathbb{R}^n\) for the sake of argument. It implicitly defines an undirected graphical model where for any given observation index \(t_0\in\mathcal{T}\), the value \(x_0\) is influenced by the values of the field at \(\operatorname{supp}\{K(\cdot, t_0)\}\); (or a really a continuum of different strengths of influence depending on the magnitude of the kernel).

Does this kind of factoring buy us anything? Does the standard finite dimensional distribution argument get us anywhere in this setting if we can introduce some conditional independence?

I suspect that (Lauritzen 1996) is sufficiently general to cover this, but TBH I haven’t read it for long enough that I can’t remember. (Eichler, Dahlhaus, and Dueck 2016) is probably an example of what I mean; they construct a continuous index directed graphical model for point process fields, based on limiting cases of a discrete field, which seems like the obvious method of attack.

Bishop, Christopher M. 2006. Pattern Recognition and Machine Learning. Information Science and Statistics. New York: Springer.

Eichler, Michael, Rainer Dahlhaus, and Johannes Dueck. 2016. “Graphical Modeling for Multivariate Hawkes Processes with Nonparametric Link Functions.” Journal of Time Series Analysis, January, n/a–n/a.

Lauritzen, Steffen L. 1996. Graphical Models. Clarendon Press.