Reversible jump Markov chain Monte Carlo for deconvolution.

To solve the problem of estimating an unknown input function to a linear time invariant system we propose an adaptive non-parametric method based on reversible jump Markov chain Monte Carlo (RJMCMC). We use piecewise polynomial functions (splines) to represent the input function. The RJMCMC algorithm allows the exploration of a large space of competing models, in our case the collection of splines corresponding to alternative positions of breakpoints, and it is based on the specification of transition probabilities between the models. RJMCMC determines: the number and the position of the breakpoints, and the coefficients determining the shape of the spline, as well as the corresponding posterior distribution of breakpoints, number of breakpoints, coefficients and arbitrary statistics of interest associated with the estimation problem. Simulation studies show that the RJMCMC method can obtain accurate reconstructions of complex input functions, and obtains better results compared with standard non-parametric deconvolution methods. Applications to real data are also reported.