# Canonical Bayesian Linear System Identification

arXiv: https://arxiv.org/abs/2507.11535 · HTML: https://mlsanity.com/paper/2507.11535/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 48 references that are in the map.
- First posted on arXiv 15 Jul 2025.

## Builds on (most cited first, 9 of 9)

- [Gradient Descent Learns Linear Dynamical Systems](https://mlsanity.com/paper/1609.05191/) (2016)
- [Non-asymptotic Identification of LTI Systems from a Single Trajectory](https://mlsanity.com/paper/1806.05722/) (2018)
- [On Particle Methods for Parameter Estimation in State-Space Models](https://mlsanity.com/paper/1412.8695/) (2014)
- [A New Approach to Learning Linear Dynamical Systems](https://mlsanity.com/paper/2301.09519/) (2023)
- [Computationally Efficient Bayesian Learning of Gaussian Process State Space Models](https://mlsanity.com/paper/1506.02267/) (2015)
- [Bayesian System ID: Optimal management of parameter, model, and measurement uncertainty](https://mlsanity.com/paper/2003.02359/) (2020)
- [Nonparametric Finite Time LTI System Identification](https://mlsanity.com/paper/1902.01848/) (2019)
- [On the Sample Complexity of Stabilizing LTI Systems on a Single Trajectory](https://mlsanity.com/paper/2202.07187/) (2022)
- [Bayesian inference for a covariance matrix](https://mlsanity.com/paper/1408.4050/) (2014)

## Built on by (most cited first, 1 of 1)

- [When is a System Discoverable from Data? Discovery Requires Chaos](https://mlsanity.com/paper/2511.08860/) (2025)

## Abstract (arXiv)

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.