Gaussian process training decomposes into inference of the (approximate) posterior and learning of the hyperparameters. For non-Gaussian (non-conjugate) likelihoods, two common choices for approximate inference are Expectation Propagation (EP) and Variational Inference (VI), which have complementary strengths and weaknesses. While VI's lower bound to the marginal likelihood is a suitable objective for inferring the approximate posterior, it does not automatically imply it is a good learning objective for hyperparameter optimization. We design a hybrid training procedure where the inference leverages conjugate-computation VI and the learning uses an EP-like marginal likelihood approximation. We empirically demonstrate on binary classification that this provides a good learning objective and generalizes better.
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