I want to get a GP which includes a region with “continuous” observation. For example, S=\{ (x,x), x\in[-4,4] \}, observation y(s)=-1, \forall s \in S. I can use finite discrete observations to approximate it like:

But I wonder if there is a more efficient way to encode the condition? If so, can it apply to impose such condition on a region given by polygon like:

In fact, it is a region of the Solomon Sea and polygons denote some islands. I want to represent that function values take a special value (such as -1) in the “land” region, while works as usual in the sea region.

I have no idea whether continued observation will work as I expected. For example, will the “continuous” observation region remove any uncertainty in the outer region? (though I don’t observe it in the gif example.)

I don’t think the event {I observed u(x) for all x \in [a,b]} is measurable. If you construct the GP through a Kolmogorov extension (ie through finite dimensional distributions which is the usual way to do it), the sigma algebra associated with it only contains events that touch the GP at most a countable number of times. It’s why proof of continuity etc are a) so long and weird, and b) talk about their being a continuous version rather than the field itself being continuous.

All derivatives of the latent GP are continuous, I guess. If so, you cannot impose exact equality of the latent process to a value, say -1, within polygons, for then the latent process would be constant everywhere.

You can however, at least in principle, construct evidence of the latent process being close to -1 on the islands. That evidence would add an extra term to the log-likelihood. Coefficient of the term, for the strength of the evidence, would then appear as a new degree of freedom (the appearance of this extra coefficient is related to the non-measurability of the “event”). Maybe someone can give a generative interpretation for such evidence as a prior, I can’t.

And I don’t know how you would implement the model with Stan. An implementation would be possible in a more stochastic environment where you could effectively integrate over the area of the islands by sampling.