Multi-task learning of GP meta-parameters

[etienne-thuillier]

Hi,

I have a question which might be ill-advised as I am relatively new at using Gaussian Processes.

I have a task in which the acquisition of any data point is costly such that we can only afford a verse sparse sampling. For this reason, I would like to leverage relatively densely sampled records that I have of past related tasks to learn the meta-parameters of the Gaussian Process. More specifically, I would like to obtain a single set of meta-parameter values from the past tasks taken as a whole (all tasks at once). My understanding is that I should minimise the total negative log marginal likelihood obtained by summing the individual negative log marginal likelihoods related to each task.

A practical issue that I have with this approach is that even though I have a single prior, it seems that I have to define several posteriors in GPJax:
posterior_1 = prior * likelihood_1
posterior_2 = prior * likelihood_2
...
where each likelihood is a Gaussian object specified for a distinct number of observed data points, e.g.
likelihood_1 =gpx.Gaussian(num_datapoints=dataset_1.n)
where dataset_1 is specific to task 1 and has a different data point count than dataset_2.

This is puzzling to me because I have now have several models (posterior_1, posterior_2, ...) to initialise even though I want the meta-parameters to be shared across all tasks...

Am I missing something here?

With thanks!