Posterior error bounds for prior-driven balancing in linear Gaussian inverse problems
Josie König, Han Cheng Lie
TL;DR
This work addresses how to quantify posterior approximation error when forward-model reductions are used in linear Gaussian inverse problems, with a focus on smoothing for high-dimensional linear time-invariant (LTI) systems. It develops a perturbation-based framework that bounds the difference in the approximate and exact posteriors, leveraging a local Lipschitz stability result for Gaussian posteriors and revealing how the impulse response of the prior-driven system corresponds to rows of the prior-preconditioned Hessian. The main contributions are the first a priori bounds for posterior mean and covariance under prior-driven model order reduction (PD-BT and PD-TLBT), with bounds expressed through trailing Hankel singular values of the prior-driven system, and the accommodation of singular priors. Numerical experiments on the ISS1R benchmark validate the theory, show how truncation rank and prior strength affect the posterior, and illustrate the practical viability of PD-TLBT; the work further discusses potential extensions to interpolatory MOR and other MOR frameworks, outlining future research directions toward broadening applicability and tightening bounds.
Abstract
In large-scale Bayesian inverse problems, it is often necessary to apply approximate forward models to reduce the cost of forward model evaluations, while controlling approximation quality. In the context of Bayesian inverse problems with linear forward models, Gaussian priors, and Gaussian noise, we use perturbation theory for inverses to bound the error in the approximate posterior mean and posterior covariance resulting from a linear approximate forward model. We then focus on the smoothing problem of inferring the initial condition of linear time-invariant dynamical systems, using finitely many partial state observations. For such problems, and for a specific model order reduction method based on balanced truncation, we show that the impulse response of a certain prior-driven system is closely related to the prior-preconditioned Hessian of the inverse problem. This reveals a novel connection between systems theory and inverse problems. We exploit this connection to prove the first a priori error bounds for system-theoretic model order reduction methods applied to smoothing problems. The bounds control the approximation error of the posterior mean and covariance in terms of the truncated Hankel singular values of the underlying system.