  We develop a fully Bayesian framework for function-on-scalars regression with
many predictors. The functional data response is modeled nonparametrically
using unknown basis functions, which produces a flexible and data-adaptive
functional basis. We incorporate shrinkage priors that effectively remove
unimportant scalar covariates from the model and reduce sensitivity to the
number of (unknown) basis functions. For variable selection in functional
regression, we propose a decision theoretic posterior summarization technique,
which identifies a subset of covariates that retains nearly the predictive
accuracy of the full model. Our approach is broadly applicable for Bayesian
functional regression models, and unlike existing methods provides joint rather
than marginal selection of important predictor variables. Computationally
scalable posterior inference is achieved using a Gibbs sampler with linear time
complexity in the number of predictors. The resulting algorithm is empirically
faster than existing frequentist and Bayesian techniques, and provides joint
estimation of model parameters, prediction and imputation of functional
trajectories, and uncertainty quantification via the posterior distribution. A
simulation study demonstrates improvements in estimation accuracy, uncertainty
quantification, and variable selection relative to existing alternatives. The
methodology is applied to actigraphy data to investigate the association
between intraday physical activity and responses to a sleep questionnaire.
