  Bayesian inference is a widely used statistical method. The free energy and
generalization loss, which are used to estimate the accuracy of Bayesian
inference, are known to be small in singular models that do not have a unique
optimal parameter. However, their characteristics are not yet known when there
are multiple optimal probability distributions. In this paper, we theoretically
derive the asymptotic behaviors of the generalization loss and free energy in
the case that the optimal probability distributions are not unique and show
that they contain asymptotically different terms from those of the conventional
asymptotic analysis.
