In this paper, we introduce the notion of Gaussian processes indexed by probability density functions for extending the Matérn family of covariance functions. We use some tools from information geometry to improve the efficiency and the computational aspects of the Bayesian learning model. We particularly show how a Bayesian inference with a Gaussian process prior (covariance parameters estimation and prediction) can be put into action on the space of probability density functions. Our framework has the capacity of classifiying and infering on data observations that lie on nonlinear subspaces. Extensive experiments on multiple synthetic, semi-synthetic and real data demonstrate the effectiveness and the efficiency of the proposed methods in comparison with current state-of-the-art methods.