  The thermodynamic uncertainty relation is an inequality stating that it is
impossible to attain higher precision than the bound defined by entropy
production. In statistical inference theory, information inequalities assert
that it is infeasible for any estimator to achieve an error smaller than the
prescribed bound. Inspired by the similarity between the thermodynamic
uncertainty relation and the information inequalities, we apply the latter to
systems described by Langevin equations and derive the bound for the
fluctuation of thermodynamic quantities. When applying the Cram\'er-Rao
inequality, the obtained inequality reduces to the fluctuation-response
inequality. We find that the thermodynamic uncertainty relation is a particular
case of the Cram\'er-Rao inequality, in which the Fisher information is the
total entropy production. Using the equality condition of the Cram\'er-Rao
inequality, we find that the stochastic total entropy production is the only
quantity which can attain equality in the thermodynamic uncertainty relation.
Furthermore, we apply the Chapman-Robbins inequality and obtain a relation for
the lower bound of the ratio between the variance and the sensitivity of
systems in response to arbitrary perturbations.
