Noise parameter `Sigma` can't be recovered from `PosteriorGPs`

[Crown421]

I noticed that once you have a PosteriorGP, it is no longer possible to recover the noise level that was given in the construction of the FiniteGP.

My quick fix for this would be to just add Sigma_y to the data field in the PosteriorGP. I have this currently implemented in a local copy, and am happy to make a quick PR.

[willtebbutt]

Hmm. I see where you're coming from, but I'm reluctant to add redundant fields into the PosteriorGP object. Is there a reason that it's hard to just keep the noise variance around in whatever your particular context is / a reason that it must go into the posterior object?

[Crown421]

For this became an issue when using decoupled sampling/ getting a realization from the GP. I think it is much nicer to have all information needed to get a realization from the GP itself.

Carrying it around separately seems unnecessarily error-prone.

[willtebbutt]

Ah, so you're trying to sample from the posterior at some points, and want samples + whatever noise the observations were made under? Could you provide a toy example of what you're trying to do?

[Crown421]

Not quite. The decoupled sampling/ matheron's rule works of generating samples from the noise distribution, so Normal(0, Sigma_y).
At least that is the case with the standard implementation

(from https://arxiv.org/pdf/2002.09309.pdf)

I can't think of a way to do this without knowing the observation noise.

[willtebbutt]

Ahhh, I see what you mean. I agree that you need access to the noise model when doing the pathwise sampling stuff. I think I also agree that the right time to introduce the approximations involved is when you've got hold of a PosteriorGP object (since the pathwise sampling stuff is really introducing an approximate posterior).

I'd definitely be happy to review this PR.