  The total entropy production quantifies the extent of irreversibility in
thermodynamic systems, which is nonnegative for any feasible dynamics. When
additional information such as the initial and final states or moments of an
observable is available, it is known that tighter lower bounds on the entropy
production exist according to the classical speed limits and the thermodynamic
uncertainty relations. Here, we obtain a universal lower bound on the total
entropy production in terms of probability distributions of an observable in
the time forward and backward processes. For a particular case, we show that
our universal relation reduces to a classical speed limit, imposing a
constraint on the speed of the system's evolution in terms of the Hatano--Sasa
entropy production. Notably, the newly obtained classical speed limit is
tighter than the previously reported bound by a constant factor. Moreover, we
demonstrate that a generalized thermodynamic uncertainty relation can be
derived from another particular case of the universal relation. Our new
uncertainty relation holds for systems with time-reversal symmetry breaking and
recovers several existing bounds. Our approach provides a unified perspective
on two closely related classes of inequality: classical speed limits and
thermodynamic uncertainty relations.
