  We introduce a new shrinkage prior on function spaces, called the functional
horseshoe prior (fHS), that encourages shrinkage towards parametric classes of
functions. Unlike other shrinkage priors for parametric models, the fHS
shrinkage acts on the shape of the function rather than inducing sparsity on
model parameters. We study the efficacy of the proposed approach by showing an
adaptive posterior concentration property on the function. We also demonstrate
consistency of the model selection procedure that thresholds the shrinkage
parameter of the functional horseshoe prior. We apply the fHS prior to
nonparametric additive models and compare its performance with procedures based
on the standard horseshoe prior and several penalized likelihood approaches. We
find that the new procedure achieves smaller estimation error and more accurate
model selection than other procedures in several simulated and real examples.
The supplementary material for this article, which contains additional
simulated and real data examples, MCMC diagnostics, and proofs of the
theoretical results, is available online.
