  We investigate the tightness and optimality of
thermodynamic-uncertainty-relation (TUR)-type inequalities from two aspects,
the choice of the Fisher information and the class of possible observables. We
show that there exists the best choice of the Fisher information, given by the
pseudo entropy production, and all other TUR-type inequalities in a certain
class can be reproduced by this tightest inequality. We also demonstrate that
if we observe not only generalized currents but generalized empirical measures,
the TUR-type inequality becomes optimal in the sense that it achieves its
equality in general nonequilibrium stationary systems. Combining these results,
we can draw a hierarchical structure of TUR-type inequalities.
