Canonical Bayesian Linear System Identification
This addresses a fundamental bottleneck in system identification for control and signal processing, offering a novel solution to improve Bayesian inference in this domain.
The paper tackles the problem of parameter non-identifiability in Bayesian linear time-invariant system identification, which leads to inefficient inference, by embedding canonical forms to resolve identifiability and enable meaningful priors, resulting in higher computational efficiency, interpretable posteriors, and robust uncertainty estimates in simulations.
Standard Bayesian approaches for linear time-invariant (LTI) system identification are hindered by parameter non-identifiability; the resulting complex, multi-modal posteriors make inference inefficient and impractical. We solve this problem by embedding canonical forms of LTI systems within the Bayesian framework. We rigorously establish that inference in these minimal parameterizations fully captures all invariant system dynamics (e.g., transfer functions, eigenvalues, predictive distributions of system outputs) while resolving identifiability. This approach unlocks the use of meaningful, structure-aware priors (e.g., enforcing stability via eigenvalues) and ensures conditions for a Bernstein--von Mises theorem -- a link between Bayesian and frequentist large-sample asymptotics that is broken in standard forms. Extensive simulations with modern MCMC methods highlight advantages over standard parameterizations: canonical forms achieve higher computational efficiency, generate interpretable and well-behaved posteriors, and provide robust uncertainty estimates, particularly from limited data.