  In this paper, we derive a useful lower bound for the Kullback-Leibler
divergence (KL-divergence) based on the Hammersley-Chapman-Robbins bound
(HCRB). The HCRB states that the variance of an estimator is bounded from below
by the Chi-square divergence and the expectation value of the estimator. By
using the relation between the KL-divergence and the Chi-square divergence, we
show that the lower bound for the KL-divergence which only depends on the
expectation value and the variance of a function we choose. This lower bound
can also be derived from an information geometric approach. Furthermore, we
show that the equality holds for the Bernoulli distributions and show that the
inequality converges to the Cram\'{e}r-Rao bound when two distributions are
very close. We also describe application examples and examples of numerical
calculation.
