Relative entropy is directly related to the Fisher information metric. This can be made explicit as follows. Assume that the probability distributions  and  are both parameterized by some (possibly multi-dimensional) parameter . Consider then two close by values of  and  so that the parameter  differs by only a small amount from the parameter value . Specifically, up to first order one has (using the Einstein summation convention)
with  a small change of  in the  direction, and  the corresponding rate of change in the probability distribution. Since relative entropy has an absolute minimum 0 for , i.e. , it changes only to second order in the small parameters . More formally, as for any minimum, the first derivatives of the divergence vanish
and by the Taylor expansion one has up to second order
where the Hessian matrix of the divergence
must be positive semidefinite. Letting  vary (and dropping the subindex 0) the Hessian  defines a (possibly degenerate) Riemannian metric on the  parameter space, called the Fisher information metric.