  Generative models emerge as promising candidates for novel sequence-data
driven approaches to protein design, and for the extraction of structural and
functional information about proteins deeply hidden in rapidly growing sequence
databases. Here we propose simple autoregressive models as highly accurate but
computationally extremely efficient generative sequence models. We show that
they perform similarly to existing approaches based on Boltzmann machines or
deep generative models, but at a substantially lower computational cost.
Furthermore, the simple structure of our models has distinctive mathematical
advantages, which translate into an improved applicability in sequence
generation and evaluation. Using these models, we can easily estimate both the
model probability of a given sequence, and the size of the functional sequence
space related to a specific protein family. In the case of response regulators,
we find a huge number of ca. $10^{68}$ sequences, which nevertheless constitute
only the astronomically small fraction $10^{-80}$ of all amino-acid sequences
of the same length. These findings illustrate the potential and the difficulty
in exploring sequence space via generative sequence models.
