  We tackle the problem of multiscale regression for predictors that are
spatially or temporally indexed, or with a pre-specified multiscale structure,
with a Bayesian modular approach. The regression function at the finest scale
is expressed as an additive expansion of coarse to fine step functions. Our
Modular and Multiscale (M&M) methodology provides multiscale decomposition of
high-dimensional data arising from very fine measurements. Unlike more complex
methods for functional predictors, our approach provides easy interpretation of
the results. Additionally, it provides a quantification of uncertainty on the
data resolution, solving a common problem researchers encounter with simple
models on down-sampled data. We show that our modular and multiscale posterior
has an empirical Bayes interpretation, with a simple limiting distribution in
large samples. An efficient sampling algorithm is developed for posterior
computation, and the methods are illustrated through simulation studies and an
application to brain image classification. Source code is available as an R
package at https://github.com/mkln/bmms.
