A prior near-ignorance Gaussian process model for nonparametric regression

This paper proposes a prior near-ignorance model for regression based on a set of Gaussian Processes (GP). GPs are natural prior distributions for Bayesian regression. They offer a great modeling flexibility and have found widespread application in many regression problems. However, a GP requires the prior elicitation of its mean function, which represents our prior belief about the shape of the regression function, and of the covariance between any two function values.In the absence of prior information, it may be difficult to fully specify these infinite dimensional parameters. In this work, by modeling the prior mean of the GP as a linear combination of a set of basis functions and assuming as prior for the combination coefficients a set of conjugate distributions obtained as limits of truncate exponential priors, we have been able to model prior ignorance about the mean of the GP. The resulting model satisfies translation invariance, learning and, under some constraints, convergence, which are desirable properties for a prior near-ignorance model. Moreover, it is shown in this paper how this model can be extended to allow for a weaker specification of the GP covariance between function values, by letting each basis function to vary in a set of functions.Application to hypothesis testing has shown how the use of this model induces the capability of automatically detecting when a reliable decision cannot be made based on the available data.