  We propose a novel Bayesian methodology for inference in functional linear
and logistic regression models based on the theory of reproducing kernel
Hilbert spaces (RKHS's). We introduce general models that build upon the RKHS
generated by the covariance function of the underlying stochastic process, and
whose formulation includes as particular cases all finite-dimensional models
based on linear combinations of marginals of the process, which can
collectively be seen as a dense subspace made of simple approximations. By
imposing a suitable prior distribution on this dense functional space we can
perform data-driven inference via standard Bayes methodology, estimating the
posterior distribution through reversible jump Markov chain Monte Carlo
methods. In this context, our contribution is two-fold. First, we derive a
theoretical result that guarantees posterior consistency, based on an
application of a classic theorem of Doob to our RKHS setting. Second, we show
that several prediction strategies stemming from our Bayesian procedure are
competitive against other usual alternatives in both simulations and real data
sets, including a Bayesian-motivated variable selection method.
