  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 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 (by a factor between
$10^2$ and $10^3$). Furthermore, the simple structure of our models has
distinctive mathematical advantages, which translate into an improved
applicability in sequence generation and evaluation. Within these models, we
can easily estimate both the probability of a given sequence, and, using the
model's entropy, the size of the functional sequence space related to a
specific protein family. In the example of response regulators, we find a huge
number of ca. $10^{68}$ possible 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.
