  We interpret likelihood-based test functions from a geometric perspective
where the Kullback-Leibler (KL) divergence is adopted to quantify the distance
from a distribution to another. Such a test function can be seen as a
sub-Gaussian random variable, and we propose a principled way to calculate its
corresponding sub-Gaussian norm. Then an error bound for binary hypothesis
testing can be obtained in terms of the sub-Gaussian norm and the KL
divergence, which is more informative than Pinsker's bound when the
significance level is prescribed. For $M$-ary hypothesis testing, we also
derive an error bound which is complementary to Fano's inequality by being more
informative when the number of hypotheses or the sample size is not large.
