Path-Space Information Bounds for Uncertainty Quantification and Sensitivity Analysis of Stochastic Dynamics

Abstract

Uncertainty quantification is a primary challenge for reliable modeling and simulation of complex stochastic dynamics. Such problems are typically plagued with incomplete information that may enter as uncertainty in the model parameters, or even in the model itself. Furthermore, due to their dynamic nature, we need to assess the impact of these uncertainties on the transient and long-time behavior of the stochastic models and derive corresponding uncertainty bounds for observables of interest. A special class of such challenges is parametric uncertainties in the model and in particular sensitivity analysis along with the corresponding sensitivity bounds for stochastic dynamics. Moreover, sensitivity analysis can be further complicated in models with a high number of parameters that render straightforward approaches, such as gradient methods, impractical. In this paper, we derive uncertainty and sensitivity bounds for path-space observables of stochastic dynamics in terms of new goal-oriented divergences; the latter incorporate both observables and information theory objects such as the relative entropy rate. These bounds are tight, depend on the variance of the particular observable and are computable through Monte Carlo simulation. In the case of sensitivity analysis, the derived sensitivity bounds rely on the path Fisher Information Matrix, hence they depend only on local dynamics and are gradient-free. These features allow for computationally efficient implementation in systems with a high number of parameters, e.g., complex reaction networks and molecular simulations.Version: July 15, 2015 1 2. Uncertainty quantification information inequalities and sensitivity bounds.2.1. Distances and divergences of probability measures. Bounds of the type (1.1) are based on characterizing a distance or divergence between the measures, Q, P , under which the averages are evaluated. While our primary goal is to characterize the bounds based on relative entropy, other divergences can be c>0

Stochastic averaging and sensitivity analysis for two scale reaction networks

“…It can be seen that Var {W θi (T )} = O(T ), so one must balance choosing a terminal time T large enough to ensure sufficient decay of the bias E pT (θ) {f (X(T ))} − E π(θ) {f (X)}, yet as small as possible to contain the growth of the Var {W θi (T )}. While centering as in (15) helps to reduce the variance of the estimator, the variance is usually much larger than comparable finite difference of pathwise derivative methods [16][17][18] . Instead of using the terminal distribution f (X(T )) as an approximation of the steady-state distribution, one could instead use the ergodic-average (timeaverage) 1/T T 0 f (X(s))ds.…”

“…Statistics are taken at a termination time of t = 0.5s. Species averages are calculated as arithmetic averages over the independent trajectories while sensitivities are computed with the CLR method shown in (15). The error due to statistical averaging is estimated using t-test statistics for averages and a bootstrapping method for sensitivities.…”

“…The sensitivities give important insight into the system, indicating directions for gradient-descent type optimization as well as determining bounds for quantifying the uncertainty 15 . Current techniques for estimating the sensitivities have large variance, requiring many more samples than those for estimating E θ {f (X)} alone [16][17][18][19] .…”

From Atomistic to Systematic Coarse-Grained Models for Molecular Systems

“…Methods such as inverse Monte Carlo [1], inverse Boltzmann [2], force matching [3], relative entropy [4], provide parameterizations of coarse-grained effective potentials at equilibrium by minimizing a fitting functional over a parameter space. Then, we further extend these studies using path-space methods (relative entropy rate) for coarse-graining and uncertainty quantification for non-equilibrium processes, [5,6,7,8].…”

Uncertainty Quantification for Markov Processes via Variational Principles and Functional Inequalities

“…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…”