  The functional generalized additive model (FGAM) was recently proposed in
McLean et al. (2013) as a more flexible alternative to the common functional
linear model (FLM) for regressing a scalar on functional covariates. In this
paper, we develop a Bayesian version of FGAM for the case of Gaussian errors
with identity link function. Our approach allows the functional covariates to
be sparsely observed and measured with error, whereas the estimation procedure
of McLean et al. (2013) required that they be noiselessly observed on a regular
grid. We consider both Monte Carlo and variational Bayes methods for fitting
the FGAM with sparsely observed covariates. Due to the complicated form of the
model posterior distribution and full conditional distributions, standard Monte
Carlo and variational Bayes algorithms cannot be used. The strategies we use to
handle the updating of parameters without closed-form full conditionals should
be of independent interest to applied Bayesian statisticians working with
nonconjugate models. Our numerical studies demonstrate the benefits of our
algorithms over a two-step approach of first recovering the complete
trajectories using standard techniques and then fitting a functional regression
model. In a real data analysis, our methods are applied to forecasting closing
price for items up for auction on the online auction website eBay.
