Uncertainty Quantification for Markov Processes via Variational Principles and Functional Inequalities

Abstract

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 non-parametric model-form uncertainty. In this work, we combine such variational formulas with functional inequalities (Poincaré, log-Sobolev, Liapunov functions) to derive explicit uncertainty quantification bounds applicable to both discrete and continuoustime Markov processes. These bounds are well-behaved in the infinite-time limit and apply to steady-states.Keywords uncertainty quantification · Markov process · relative entropy · Poincaré inequality · log-Sobolev inequality · Liapunov function · Bernstein inequality Mathematics Subject Classification (2010) 47D07 · 39B72 · 60F10 · 60J25 1 Introduction Information-theory based variational principles have proven effective at providing uncertainty quantification (i.e. robustness) bounds for quantities-ofinterest in the presence of non-parametric model-form uncertainty [1,2,3,4,5,6,7,8,9,10]. In the present work, we combine these tools with functional

Quantification of model uncertainty on path-spaceviagoal-oriented relative entropy

“…is ℱ -measurable and so, if ∈ 1 ( ) ∩ 1 (̃︀), one can apply Corollary 4.3 tô︀ . Reparameterizing the infumum → , one finds that the bound is the same as equation (2.5), which was previously derived in [20] and used in [10]. As we will demonstrate in the example in Section 5.2 below, one can often use Theorem 3.6 to obtain tighter and more general UQ bounds on ergodic averages than those obtained from equation (2.5).…”

“…In [10,20,35], equation (2.4) was used to derive UQ bounds for ergodic averages on path-space, both in discrete and continuous time. The goal of the present work is to extend these methods to apply to more general path-space QoIs.…”

“…Bounds on CGFs are often used in the derivation of concentration inequalities; see [12] for an overview of this much-studied problem. For prior uses of CGF bounds in UQ, see [10,30,44]. With these points in mind, our focus will primarily be on the second problem; our examples in Section 5 will largely involve baseline models for which the CGF is relatively straightforward to compute.…”

“…In this example we show how Theorem 3.6 can be combined with the functional inequality methods from [10] to obtain improved UQ bounds for invariant measures of non-reversible SDEs. Specifically, consider the following class of SDEs on R :…”

“…Methods have been developed using relative entropy [15,17,27,30,39] and Rényi divergences [6,21]. For stochastic processes, the relative entropy rate has been used to derive UQ bounds on ergodic averages [10,20,29,35,46]. Ergodic averages constitute an important class of QoIs on path space, but many other key QoIs are not covered by these previous methods, such as the expectation of hitting times and exponentially discounted observables.…”