  We consider the problem of finding an optimal statistical model for a given
binary string. Following Kolmogorov, we use structure functions. In order to
get concrete results, we replace Turing machines by finite automata and
Kolmogorov complexity by Shallit and Wang's automatic complexity.
  The $p$-value of a model for given data $x$ is the probability that there
exists a model with as few states, accepting as few words, fitting uniformly
randomly selected data $y$.
  Deterministic and nondeterministic automata can give different optimal
models. For $x=011\, 110\, 110\, 11$, the best deterministic model has
$p$-value $0.3$, whereas the best nondeterministic model has $p$-value $0.04$.
  In the nondeterministic case, counting paths and counting words can give
different optimal models. For $x=01100\, 01000$, the best path-counting model
has $p$-value $0.79$, whereas the best word-counting model has $p$-value
$0.60$.
