Part of Advances in Neural Information Processing Systems 17 (NIPS 2004)
Liam Paninski
Log-concavity is an important property in the context of optimization, Laplace approximation, and sampling; Bayesian methods based on Gaus- sian process priors have become quite popular recently for classification, regression, density estimation, and point process intensity estimation. Here we prove that the predictive densities corresponding to each of these applications are log-concave, given any observed data. We also prove that the likelihood is log-concave in the hyperparameters controlling the mean function of the Gaussian prior in the density and point process in- tensity estimation cases, and the mean, covariance, and observation noise parameters in the classification and regression cases; this result leads to a useful parameterization of these hyperparameters, indicating a suitably large class of priors for which the corresponding maximum a posteriori problem is log-concave.