## Variational Multinomial Logit Gaussian Process

Kian Ming A. Chai; 13(56):1745−1808, 2012.

### Abstract

Gaussian process prior with an appropriate likelihood function is a flexible non-parametric model for a variety of learning tasks. One important and standard task is multi-class classification, which is the categorization of an item into one of several fixed classes. A usual likelihood function for this is the multinomial logistic likelihood function. However, exact inference with this model has proved to be difficult because high-dimensional integrations are required. In this paper, we propose a variational approximation to this model, and we describe the optimization of the variational parameters. Experiments have shown our approximation to be tight. In addition, we provide data-independent bounds on the marginal likelihood of the model, one of which is shown to be much tighter than the existing variational mean-field bound in the experiments. We also derive a proper lower bound on the predictive likelihood that involves the Kullback-Leibler divergence between the approximating and the true posterior. We combine our approach with a recently proposed sparse approximation to give a variational sparse approximation to the Gaussian process multi-class model. We also derive criteria which can be used to select the inducing set, and we show the effectiveness of these criteria over random selection in an experiment.

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