Abstract
We provide a prior distribution for a functional parameter so thatits trajectories are smooth and vanish on a given subset. This dis-tribution can be interpreted as the distribution of an initial Gaussianprocess conditioned to be zero on a given subset. Precisely, we showthat the initial Gaussian process is the sum of the conditioned processand an independent process with probability one and that all the pro-cesses have the same almost sure regularity. This prior distribution isuse to provide an interpretable estimate of the coefficient function inthe linear scalar-on-function regression; by interpretable, we mean asmooth function that may possibly be zero on some intervals. We ap-ply our model in a simulation and real case studies with two differentpriors for the null region of the coefficient function. In one case, thenull region is known to be an unknown single interval. In the othercase, it can be any unknown unions of intervals.