  In modern contexts, some types of data are observed in high-resolution,
essentially continuously in time. Such data units are best described as taking
values in a space of functions. Subject units carrying the observations may
have intrinsic relations among themselves, and are best described by the nodes
of a large graph. It is often sensible to think that the underlying signals in
these functional observations vary smoothly over the graph, in that neighboring
nodes have similar underlying signals. This qualitative information allows
borrowing of strength over neighboring nodes and consequently leads to more
accurate inference. In this paper, we consider a model with Gaussian functional
observations and adopt a Bayesian approach to smoothing over the nodes of the
graph. We characterize the minimax rate of estimation in terms of the
regularity of the signals and their variation across nodes quantified in terms
of the graph Laplacian. We show that an appropriate prior constructed from the
graph Laplacian can attain the minimax bound, while using a mixture prior, the
minimax rate up to a logarithmic factor can be attained simultaneously for all
possible values of functional and graphical smoothness. We also show that in
the fixed smoothness setting, an optimal sized credible region has arbitrarily
high frequentist coverage. A simulation experiment demonstrates that the method
performs better than potential competing methods like the random forest. The
method is also applied to a dataset on daily temperatures measured at several
weather stations in the US state of North Carolina.
