Incorporating individual measurement uncertainties into Gaussian Process

[Paeyus]

I have a set of observations, f_i=f(x_i), and I want to construct a probabilistic surrogate, f(x) ~ N[mu(x), sigma(x)], where N is a normal distribution. Each observed output, f_i, is associated with a measurement uncertainty, sigma_i. I would like to incorporate these measurement uncertainties into my surrogate, f_i, so that mu(x) predicts the observations, f_i(x_i), and that the predicted standard deviation, sigma(x_i), envelops the uncertainty in the observed output, epsilon_i.

[Balandat]

If you just want to predict the posterior over the latent f (rather than the observations y), this is straightforward to do by using a FixedNoiseGaussianLikelihood (which takes the tensor of noise levels) instead of a GaussianLikelihood (which infers the noise level). If you want to be able to compute the posterior of y at arbitrary (unobserved) points you need to do a bit more work and also model the noise function across the domain. You can take a look at BoTorch's HeteroskedasticSingleTaskGP model to see how this could work.