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We consider the problem of finding an optimal statistical model for a givenbinary string. Following Kolmogorov, we use structure functions. In order toget concrete results, we replace Turing machines by finite automata andKolmogorov complexity by Shallit and Wang's automatic complexity.  The $p$-value of a model for given data $x$ is the probability that thereexists a model with as few states, accepting as few words, fitting uniformlyrandomly selected data $y$.  Deterministic and nondeterministic automata can give different optimalmodels. 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 givedifferent optimal models. For $x=01100\, 01000$, the best path-counting modelhas $p$-value $0.79$, whereas the best word-counting model has $p$-value$0.60$.

Few paths, fewer words: model selection with automatic structure functions