Gaussian Processes. Latent vs Marginal

The latent GP is typically used when you have a non-normal likelihood, or when you want to add additional components to the model before passing it to the likelihood. For example, if you are analyzing count data, you will likely assume a Poisson or negative binomial data generating process, so you would model the mean as a latent GP, and transform it before passing it to the likelihood. The interpretation is really no different. You should be able to use a periodic covariance with a latent GP, AFAIK.

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