  Hierarchical probabilistic models, such as Gaussian mixture models, are
widely used for unsupervised learning tasks. These models consist of observable
and latent variables, which represent the observable data and the underlying
data-generation process, respectively. Unsupervised learning tasks, such as
cluster analysis, are regarded as estimations of latent variables based on the
observable ones. The estimation of latent variables in semi-supervised
learning, where some labels are observed, will be more precise than that in
unsupervised, and one of the concerns is to clarify the effect of the labeled
data. However, there has not been sufficient theoretical analysis of the
accuracy of the estimation of latent variables. In a previous study, a
distribution-based error function was formulated, and its asymptotic form was
calculated for unsupervised learning with generative models. It has been shown
that, for the estimation of latent variables, the Bayes method is more accurate
than the maximum-likelihood method. The present paper reveals the asymptotic
forms of the error function in Bayesian semi-supervised learning for both
discriminative and generative models. The results show that the generative
model, which uses all of the given data, performs better when the model is well
specified.
