  Gaussian latent tree models, or more generally, Gaussian latent forest models
have Fisher-information matrices that become singular along interesting
submodels, namely, models that correspond to subforests. For these
singularities, we compute the real log-canonical thresholds (also known as
stochastic complexities or learning coefficients) that quantify the
large-sample behavior of the marginal likelihood in Bayesian inference. This
provides the information needed for a recently introduced generalization of the
Bayesian information criterion. Our mathematical developments treat the general
setting of Laplace integrals whose phase functions are sums of squared
differences between monomials and constants. We clarify how in this case real
log-canonical thresholds can be computed using polyhedral geometry, and we show
how to apply the general theory to the Laplace integrals associated with
Gaussian latent tree and forest models. In simulations and a data example, we
demonstrate how the mathematical knowledge can be applied in model selection.
