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.

翻译：标准贝叶斯方法在线性时不变（LTI）系统辨识中受限于参数不可辨识性；由此产生的复杂多峰后验分布使得推断效率低下且不切实际。我们通过将LTI系统的规范形式嵌入贝叶斯框架来解决此问题。我们严格证明，在这些最小参数化中进行推断能完整捕捉所有不变系统动态（例如传递函数、特征值、系统输出的预测分布），同时解决可辨识性问题。该方法使得有意义、结构感知的先验（例如通过特征值强制稳定性）得以应用，并确保了伯恩斯坦-冯·米塞斯定理的成立条件——该定理连接了贝叶斯与频率学派的大样本渐近理论，在标准参数化中此联系已被破坏。采用现代MCMC方法的大量仿真实验突显了规范形式相对于标准参数化的优势：规范形式实现了更高的计算效率，生成可解释且性质良好的后验分布，并能提供稳健的不确定性估计，尤其在数据有限的情况下。