  If the probability distribution model aims to approximate the hidden mother
distribution, it is imperative to establish a useful criterion for the
resemblance between the mother and the model distributions.
  This study proposes a criterion that measures the Hellinger distance between
discretized (quantized) samples from both distributions. Unlike information
criteria such as AIC, this criterion does not require the probability density
function of the model distribution, which cannot be explicitly obtained for a
complicated model such as a deep learning machine. Second, it can draw a
positive conclusion (i.e., both distributions are sufficiently close) under a
given threshold, whereas a statistical hypothesis test, such as the
Kolmogorov-Smirnov test, cannot genuinely lead to a positive conclusion when
the hypothesis is accepted.
  In this study, we establish a reasonable threshold for the criterion deduced
from the Bayes error rate and also present the asymptotic bias of the estimator
of the criterion. From these results, a reasonable and easy-to-use criterion is
established that can be directly calculated from the two sets of samples from
both distributions.
