Reconciliation of overlapping Gaussian processes
Combining Gaussian processes on the same domain; consistency; coherence; generalized phase retrieval
August 12, 2024 — August 12, 2024
Suppose I have two random functions
I am sure that this must be well-studied, but it is one of those things that is rather hard to google for and ends up being easier to work out by hand, which is what I do here.
1 Overlapping GPs
Suppose I have two GP priors
How do we reconcile these two GPs into a single GP
The standard answer for Gaussian processes is to find a new one whose density is the product of the density of the two components.
2 Enveloping overlapping GPs
3 Connection to classical methods
This idea bears a resemblance to Griffin-Lim iteration phase recovery, where we have two overlapping signals and we want to combine them into a single signal that is consistent with both. That case is somewhat different because it assumes
- a point estimate will do; we do not talk about random functions, and
- The covariance kernels are implicitly stationary, which we do not assume here (and as such, there is not necessarily a “phase” to “recover”).