Regression and classification using Gaussian process priors

Gaussian processes are a natural way of specifying prior distributions over functions of one or more input variables. When such a function defines the mean response in a regression model with Gaussian errors, inference can be done using matrix computations, which are feasible for datasets of up to about a thousand cases. The covariance function of the Gaussian process can be given a hierarchical prior, which allows the model to discover high-level properties of the data, such as which inputs are relevant to predicting the response. Inference for these covariance hyperparameters can be done using Markov chain sampling. Classification models can be defined using Gaussian processes for underlying latent values, which can also be sampled within the Markov chain. Gaussian processes are in my view the simplest and most obvious way of defining flexible Bayesian regression and classification models, but despite some past usage, they appear to have been rather neglected as a general-purpose technique. This may be partly due to a confusion between the properties of the function being modeled and the properties of the best predictor for this unknown function.