  Evaluation of the marginal likelihood plays an important role in model
selection problems. The widely applicable Bayesian information criterion (WBIC)
and singular Bayesian information criterion (sBIC) give approximations to the
log marginal likelihood, which can be applied to both regular and singular
models. When the real log canonical thresholds are known, the performance of
sBIC is considered to be better than that of WBIC, but only few real log
canonical thresholds are known. In this paper, we propose a new estimator of
the real log canonical thresholds based on the variance of thermodynamic
integration with an inverse temperature. In addition, we propose an application
to make sBIC widely applicable. Finally, we investigate the performance of the
estimator and model selection by simulation studies and application to real
data.
