  A fundamental aspect of biological information processing is the ubiquity of
sequence-function relationships -- functions that map the sequence of DNA, RNA,
or protein to a biochemically relevant activity. Most sequence-function
relationships in biology are quantitative, but only recently have experimental
techniques for effectively measuring these relationships been developed. The
advent of such "massively parallel" experiments presents an exciting
opportunity for the concepts and methods of statistical physics to inform the
study of biological systems. After reviewing these recent experimental
advances, we focus on the problem of how to infer parametric models of
sequence-function relationships from the data produced by these experiments.
Specifically, we retrace and extend recent theoretical work showing that
inference based on mutual information, not the standard likelihood-based
approach, is often necessary for accurately learning the parameters of these
models. Closely connected with this result is the emergence of "diffeomorphic
modes" -- directions in parameter space that are far less constrained by data
than likelihood-based inference would suggest. Analogous to Goldstone modes in
physics, diffeomorphic modes arise from an arbitrarily broken symmetry of the
inference problem. An analytically tractable model of a massively parallel
experiment is then described, providing an explicit demonstration of these
fundamental aspects of statistical inference. This paper concludes with an
outlook on the theoretical and computational challenges currently facing
studies of quantitative sequence-function relationships.
