  Information-theory based variational principles have proven effective at
providing scalable uncertainty quantification (i.e. robustness) bounds for
quantities of interest in the presence of nonparametric model-form uncertainty.
In this work, we combine such variational formulas with functional inequalities
(Poincar{\'e}, $\log$-Sobolev, Liapunov functions) to derive explicit
uncertainty quantification bounds for time-averaged observables, comparing a
Markov process to a second (not necessarily Markov) process. These bounds are
well-behaved in the infinite-time limit and apply to steady-states of both
discrete and continuous-time Markov processes.
