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The variational framework for learning inducing variables (Titsias, 2009a)has had a large impact on the Gaussian process literature. The framework may beinterpreted as minimizing a rigorously defined Kullback-Leibler divergencebetween the approximating and posterior processes. To our knowledge thisconnection has thus far gone unremarked in the literature. In this paper wegive a substantial generalization of the literature on this topic. We give anew proof of the result for infinite index sets which allows inducing pointsthat are not data points and likelihoods that depend on all function values. Wethen discuss augmented index sets and show that, contrary to previous works,marginal consistency of augmentation is not enough to guarantee consistency ofvariational inference with the original model. We then characterize an extracondition where such a guarantee is obtainable. Finally we show how ourframework sheds light on interdomain sparse approximations and sparseapproximations for Cox processes.

On Sparse variational methods and the Kullback-Leibler divergence between stochastic processes