  In data science and machine learning, hierarchical parametric models, such as
mixture models, are often used. They contain two kinds of variables: observable
variables, which represent the parts of the data that can be directly measured,
and latent variables, which represent the underlying processes that generate
the data. Although there has been an increase in research on the estimation
accuracy for observable variables, the theoretical analysis of estimating
latent variables has not been thoroughly investigated. In a previous study, we
determined the accuracy of a Bayes estimation for the joint probability of the
latent variables in a dataset, and we proved that the Bayes method is
asymptotically more accurate than the maximum-likelihood method. However, the
accuracy of the Bayes estimation for a single latent variable remains unknown.
In the present paper, we derive the asymptotic expansions of the error
functions, which are defined by the Kullback-Leibler divergence, for two types
of single-variable estimations when the statistical regularity is satisfied.
Our results indicate that the accuracies of the Bayes and maximum-likelihood
methods are asymptotically equivalent and clarify that the Bayes method is only
advantageous for multivariable estimations.
