Bayesian adaptive nonparametric M-regression

Nonparametric regression has been popularly used in curve fitting, signal denosing, and image processing. In such applications, the underlying functions (or signals) may vary irregularly, and it is very common that data are contaminated with outliers. Adaptive and robust techniques are needed to extract clean and accurate information. In this paper, we develop adaptive nonparametric M-regression with a Bayesian approach. This general approach fits M-regression using piecewise polynomial functions with an unknown number of knots at unknown locations, all treated as parameters to be inferred through Reversible Jump Markov Chain Monte Carlo (RJMCMC) of Green (1995, [9]). The Bayesian solution presented in this paper with computational details can be considered as an approximation to the general optimal solution for M-regression with free knots as described in Stone (2005, [22]). Numerical results show that the Bayesian approach performs well in various cases, especially with discontinuous underlying functions.