Bayesian curve fitting for discontinuous functions using an overcomplete system with multiple kernels

We propose a fully Bayesian methodology for estimation of functions that have jump discontinuities. The proposed model is an extension of the LARK model, which enables functions to be represented by the small number of elements from an overcomplete system. In the proposed model, multiple kernels are used as the elements of an overcomplete system. Since these elements are composed of different types of functions such as Haar, Laplacian, and Gaussian kernel, the proposed model can estimate discontinuous as well as smooth functions without model selection. The location of jumps, the number of basis functions, and even the smoothness of the target function are automatically determined by the Levy random measure. A simulation study and a real data analysis illustrate that the proposed model performs better than the standard nonparametric methods for the estimation of discontinuous functions. Finally, we prove prior positivity of the model and show that the prior has sufficiently large support including discontinuous functions with a finite number of jumps.