  Latent Dirichlet allocation (LDA) obtains essential information from data by
using Bayesian inference. It is applied to knowledge discovery via dimension
reducing and clustering in many fields. However, its generalization error had
not been yet clarified since it is a singular statistical model where there is
no one-to-one mapping from parameters to probability distributions. In this
paper, we give the exact asymptotic form of its generalization error and
marginal likelihood, by theoretical analysis of its learning coefficient using
algebraic geometry. The theoretical result shows that the Bayesian
generalization error in LDA is expressed in terms of that in matrix
factorization and a penalty from the simplex restriction of LDA's parameter
region. A numerical experiment is consistent to the theoretical result.
