  We develop a closed form asymptotic formula to compute the marginal
likelihood of data given a naive Bayesian network model with two hidden states
and binary features. This formula deviates from the standard BIC score. Our
work provides a concrete example that the BIC score is generally not valid for
statistical models that belong to a stratified exponential family. This stands
in contrast to linear and curved exponential families, where the BIC score has
been proven to provide a correct approximation for the marginal likelihood.
