  Nonstationary thermodynamic quantities depend on the full details of
nonstationary probability distributions, making them difficult to measure
directly in experiments and numerics. We propose a method to infer
thermodynamic quantities in relaxation processes by measuring only a few
observables, using additional information obtained from measurements in tilted
equilibrium, i.e., equilibrium with external fields applied. Our method is
applicable to arbitrary classical stochastic systems, possibly underdamped,
that relax to equilibrium. The method allows us to compute the exact value of
the minimum entropy production (EP) compatible with the nonstationary
observations, giving a tight lower bound on the true EP. Under a certain
additional condition, it also allows the inference of the EP rate,
thermodynamic forces, and a constraint on relaxation paths. Our method uses a
Legendre transform of EP at the level of probability distributions, which we
develop based on a similar Legendre transform in information geometry.
