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Canonical Bayesian Linear System Identification

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Pith's one-line read The paper claims that canonical minimal parameterizations of LTI systems make Bayesian system identification identifiable, preserve all invariant dynamics through equivalent posterior pushforwards, and restore Bernstein–von Mises…

desk verdict A promising but as-yet-unproven framework: canonical forms clearly help practical Bayesian LTI inference, but the main equivalence theorem omits covariance transformation and needs repair before the headline guarantees can be trusted. read the letter →

arxiv 2507.11535 v2 pith:IAS2PQ2C submitted 2025-07-15 stat.ML cs.LGcs.SYeess.SYstat.CO

Andrey Bryutkin , Matthew E. Levine , Iñigo Urteaga , Youssef Marzouk This is my paper
classification stat.MLcs.LGcs.SYeess.SYstat.CO MSC 93E1262F1593B1062F1265C4062M10
keywords Bayesiansystemidentificationlineartime-invariantsystemscanonicalformsidentifiabilityBernstein-vonMisestheoremFisherinformationmatrixstate-spacemodelsuncertaintyquantification
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper claims that standard Bayesian inference for linear time-invariant (LTI) systems is degraded by a gauge symmetry: infinitely many state-space parameterizations produce identical input-output behavior, so posteriors over the system matrices are multimodal and unidentifiable. It proposes doing inference on a canonical minimal parameterization, such as the SISO controller canonical form, which selects one representative per equivalence class. The central claims are that posterior distributions of every similarity-invariant quantity—transfer functions, eigenvalues, Hankel matrices, predictive output distributions—are identical under canonical and standard parameterizations, and that only the canonical posterior satisfies a Bernstein–von Mises theorem, becoming asymptotically normal with covariance given by the inverse Fisher information. If true, Bayesian system identification becomes computationally cheaper, supports priors defined directly on eigenvalues such as stability constraints, and produces interpretable, unimodal posteriors, especially with limited data.

What carries the argument

The load-bearing object is the SISO controller canonical form, a companion-matrix representation in which the dynamics matrix A_c is determined by the coefficients (a_0,...,a_{d_x-1}) of its characteristic polynomial, B_c is the unit vector, and C_c collects the numerator coefficients; this reduces the dynamical parameters from $d_x^{2}$ to 2d_x+1 and makes the parameterization identifiable. Around it, the argument runs on three mechanisms: the statistical-isomorphism characterization of Theorem 3.3, which identifies likelihood equivalence of minimal systems with the orbit of a similarity transformation; the pushforward-posterior equivalence of Theorem 3.8, which transfers any invariant quantity of interest from the standard to the canonical posterior; and the Fisher-information and local asymptotic normality analysis that yields the Bernstein–von Mises theorem. For priors, Vieta's formulas map eigenvalue specifications to the characteristic-polynomial coefficients of the canonical matrix, with a Vandermonde Jacobian, so a stability-enforcing prior on eigenvalues induces a well-defined prior on the canonical parameters.

What would settle it

Generate trajectories from a SISO system whose minimal state dimension is d0_x and run the canonical posterior with dx equal to d0_x, d0_x+1, and d0_x-1 using persistently exciting inputs. The theorem predicts BvM-valid Gaussian concentration only for dx = d0_x: for dx > d0_x the Fisher information should be singular, and for dx < d0_x the estimates should become inconsistent. If a posterior with the wrong dx still concentrates Gaussian and covers the true transfer function at nominal rates, the central claim would be falsified.

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Extended reading notes

Core claim

The discovery is that parameter non-identifiability in LTI systems is not a defect of the data but of the coordinate choice, and moving to a canonical form removes it without losing inferential content. Theorem 3.3 shows two minimal systems are statistically isomorphic—they induce identical output distributions for every input—exactly when they are related by an invertible similarity transformation. Theorem 3.8 then proves that for any similarity-invariant quantity of interest, the posterior pushforward under the canonical parameterization equals the pushforward under the standard parameterization, provided the priors are consistent. On the canonical parameter space the Fisher information matrix is non-singular under persistent excitation, so Theorem 5.3 establishes a Bernstein–von Mises theorem: the posterior concentrates around a square-root-T-consistent estimator and converges in total variation to a Gaussian with covariance $T^{-1}$ times the inverse asymptotic Fisher information. Proposition 5.4 shows the standard parameterization cannot satisfy BvM because its Fisher information is singular along the manifold of similarity-transformed parameters.

Load-bearing premise

The true minimal state dimension of the data-generating system is known in advance and equals the model dimension, so the canonical model is neither too small to capture the dynamics nor too large to have a nonsingular Fisher information matrix.

Editorial extensions

If this is right

  • Bayesian inference on the canonical parameter space is equivalent to inference on the standard space for every similarity-invariant quantity of interest, so transfer functions, poles, and predictive output distributions can be recovered without sampling the redundant parameters.
  • The canonical posterior is asymptotically Gaussian with covariance given by the inverse Fisher information, so in the large-sample limit posterior credible sets can also be read as frequentist confidence sets.
  • Standard parameterizations fail Bernstein–von Mises: their Fisher information matrix is singular along the equivalence-class manifold, so posterior uncertainty does not concentrate in a coordinate-free way.
  • Structure-aware priors, such as priors forcing all eigenvalues inside the unit disk, become computationally tractable because they are specified on eigenvalues and pushed forward through Vieta's formulas to the canonical coefficients.
  • MCMC sampling on the canonical space is substantially cheaper and mixes better than on the standard space, with gains in effective sample size per second that grow with trajectory length and state dimension.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A direct extension the paper leaves open is the MIMO setting: the same quotient-by-similarity logic should hold once a structurally identifiable MIMO canonical form and its Kronecker indices are fixed, shifting the practical bottleneck to selecting those indices rather than to posterior geometry.
  • One testable prediction of the identifiability argument is that plug-in estimators based on the posterior mean in the standard parameterization are systematically misleading because they average over modes, whereas the canonical posterior mean is a meaningful system estimate; this suggests posterior summaries under non-identifiable parameterizations should be avoided except for invariant quantitie
  • The BvM guarantee suggests a practical workflow for real data: first select the state dimension by an outer-loop model comparison, then run canonical inference with eigenvalue-based priors, and use the Fisher-information ellipsoid as a cheap Laplace-approximation diagnostic in the large-data regime.
  • Because the canonical Fisher information matrix is sparse and recursively computable, its automatic-differentiation evaluation could be used directly as an approximate posterior covariance during sampling, connecting the paper's asymptotics to finite-sample MCMC diagnostics.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. The paper proposes performing Bayesian inference for LTI system identification on minimal canonical state-space parameterizations instead of the standard, non-identifiable parameterization. It claims that canonical inference is statistically equivalent to standard inference for all invariant quantities of interest (Theorem 3.8), enables principled stability-preserving priors, and satisfies a Bernstein-von Mises theorem (Theorem 5.3), whereas the standard parameterization does not. The empirical sections compare MCMC performance and posterior geometry between canonical and standard forms, reporting substantial computational and statistical advantages for canonical inference.

Significance. If the theoretical guarantees were fully justified, the paper would make an important contribution: it would connect classical canonical-form system identification with modern Bayesian asymptotics, offering a principled way to impose structure-aware priors and to obtain efficient, well-calibrated posterior inference. The empirical study is careful and extensive, and the practical claims about computational efficiency and posterior geometry are credible and well supported. However, the load-bearing theoretical results currently contain serious gaps: the equivalence theorem omits the required transformation of noise covariances, and the proof of the central pushforward-posterior statement relies on an ill-defined quotient measure and an incorrect stabilizer computation. These issues must be resolved before the headline claims can be accepted.

major comments (4)
  1. [§3.2, Theorem 3.8 and Eq. (D.9)] The likelihood equality in Eq. (3.1), which underpins Theorem 3.8, is only valid if the noise covariances transform as in Theorem 3.3, i.e., Σ' = T^{-1}ΣT^{-⊤} and P0' = T^{-1}P0T^{-⊤}. However, Theorem 3.8 restricts the parameter spaces ΘM_l to the dynamics matrices (A,B,C,D) only, and the proof's Eq. (D.9) invokes Theorem 3.3 while fixing the covariances. For a scalar system with parameters (a,b,c,Σ=σ_w^2), the controller-form image (a,1,bc,Σ=σ_w^2) has the same transfer function but a different process-noise contribution to the output distribution unless Σ is also transformed. Consequently, p(y[T]|LΘs,u[T]) ≠ p(y[T]|LΘc,u[T]) in general when covariances are held fixed, and the claim that canonical inference fully captures predictive distributions is not established by the stated theorems.
  2. [Appendix D.4, Eqs. (D.12)-(D.14)] The stabilizer computation used to define the quotient measure is incorrect for minimal realizations. For a controllable pair (A,B) in controller canonical form, the only similarity transformation fixing (A,B,C) is the identity: T = λI does not fix B_c = (0,...,0,1)^T unless λ=1, and for the full parameterization it also does not fix covariances. The similarity action on minimal realizations is therefore free, so μ(Stab(Θc)) = 0 and the quotient measure defined in Eq. (D.14) is ill-defined. This invalidates the fiber-integration argument in the proof of Theorem 3.8 and leaves the equivalence claim without a well-defined induced prior.
  3. [Appendix D.3, Lemma 3.7] The induced prior on the canonical parameterization is not well-defined by the Radon-Nikodym argument as written. The proof asserts that the pushforward measure μc is absolutely continuous with respect to Lebesgue measure on Θc, but the quotient space Θc does not come with a canonical Lebesgue measure, and the map τ : Θmin_s → Θc is many-to-one. Defining a density on Θc requires an additional choice of a fiber measure or a fundamental domain; without such a choice, the phrase 'corresponding induced prior' in Theorem 3.8 is ambiguous. The later quotient-measure construction in Appendix D.4 attempts to supply this choice but fails for the reason stated above.
  4. [Appendix C.1, Theorem 5.3] The proof of the Bernstein-von Mises theorem largely restates the desired conclusion as assumptions. In particular, the local asymptotic normality (LAN) condition (C.1) is assumed rather than derived from the state-space model, and the positive definiteness of the asymptotic Fisher information I(Θ0_c) is asserted with the sentence 'Following the argument above we conclude that the FIM is non-singular' without a rigorous argument. Given that the canonical likelihood with fixed covariances is not the true data-generating likelihood for systems obtained from a standard-form realization, the theorem does not currently connect the canonical posterior to the standard parameterization in the way the abstract claims. The proof needs to verify the LAN and nonsingularity conditions explicitly, including the treatment of noise covariances.
minor comments (5)
  1. [§3.2, Eq. (3.7)] In the definition of the maps Sl, the subscript 'q' appears in 'Sl : ΘM_q → Q'; this is likely a typo for 'l'.
  2. [§6.6, Figure 8 caption] The notation 'α0, C2' in the inset labels is not defined in the caption or the text; please specify which canonical parameters these correspond to.
  3. [§6.5, Figure 6 caption] The word 'produed' should be 'produced'.
  4. [Appendix A, last paragraph] The acknowledgment that the balancing transformation induces off-diagonal process noise covariances, while the inference assumes diagonal covariances, is important for interpreting all experiments with nonzero process noise; this caveat should be stated in Section 6 rather than only in Appendix A.
  5. [§3.2, Theorem 3.8] The claim that 'most quantities of interest' including filtering and smoothing distributions can be computed from the canonical posterior is imprecise, since filtering/smoothing distributions are not deterministic pushforwards of Θ; the statement should specify which quantities are deterministic maps and how distributional quantities are handled.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: canonical-equivalence and BvM results are derived from stated isomorphism and identifiability conditions, not from their own conclusions.

full rationale

The paper's central claims do not reduce to their inputs. Theorem 3.8 is a mathematical consequence of (i) Theorem 3.3, which characterizes likelihood-preserving similarity transformations and explicitly requires covariances to transform as Σ' = T^{-1}ΣT^{-⊤}, P0' = T^{-1}P0T^{-⊤}, Γ' = Γ, and (ii) Lemma 3.7, which defines the canonical prior as the pushforward of the standard prior under the canonical projection τ. Given those assumptions, the equality of pushforward posteriors for similarity-invariant QoIs follows by Bayes' theorem rather than by definition or by fitting. The BvM theorem (Theorem 5.3) is conditional on Assumption 2 (known minimal state dimension), which the paper explicitly treats as an outer-loop problem; Proposition 5.4 derives the standard-form failure from a singular FIM. The covariance-transformation omission in the statement/proof of Theorem 3.8 is a correctness concern—with fixed covariances, a non-orthogonal T does not preserve the likelihood—but it is not circularity, because the claimed result is not built into the definition of the inputs or obtained by renaming a fitted quantity. No load-bearing self-citation or fitted-input-called-prediction step was found.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central derivations rely on standard Kalman filtering, control-theoretic canonical forms, and Bayesian asymptotic theory. Key non-standard assumptions are Gaussian noise with known input, known consistent state dimension, persistent excitation, and minimality of the true system. No free parameters are fitted in the theoretical argument; experiment-specific noise variances are data-generating settings rather than fitted constants.

assumptions (6)
  • domain assumption Gaussian process and measurement noise with known deterministic input (Assumption 1).
    Used throughout to obtain Gaussian likelihood via Kalman filter recursions (Theorem B.1).
  • domain assumption The data-generating system is minimal (controllable and observable) and the prior gives zero mass to non-minimal systems (Lemma 3.7).
    Needed for Theorem 3.3's similarity-transformation characterization and for the canonical projection to be well-defined.
  • domain assumption The minimal state dimension of the data-generating system is known and equals the model dimension (Assumption 2).
    Required for the BvM theorem; the paper explicitly defers estimation of dx to an outer-loop problem.
  • domain assumption The input sequence is persistently exciting of order dx (Definition 5.2).
    Used to argue the limiting FIM is positive definite and LAN holds (Appendix C.1).
  • standard math Standard BvM regularity conditions: smooth likelihood, local asymptotic normality, positive and continuous prior density near the true parameter (Theorem C.1).
    Invoked directly as the BvM framework; the paper sketches why these hold for the canonical form without a complete proof.
  • ad hoc to paper The asymptotic Fisher information matrix for the canonical parameterization is positive definite under the stated conditions.
    Asserted in Appendix C.1 on the basis of identifiability and persistent excitation, but not proven in detail; it is the key condition making BvM applicable.

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Pith. "Pith review of Canonical Bayesian Linear System Identification." pith.science (2026). https://pith.science/paper/IAS2PQ2C

@misc{pith2026250711535,
  author       = {Pith},
  title        = {Pith review of: Canonical Bayesian Linear System Identification},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IAS2PQ2C}},
  note         = {Machine review of arXiv:2507.11535}
}
read the original abstract

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.

Figures

Figures reproduced from arXiv: 2507.11535 by the authors.

Figure 1
Figure 1. Visualization of samples from different prior distributions over eigenvalues [PITH_FULL_IMAGE:figures/full_fig_p013_1.png] view at source ↗
Figure 2
Figure 2. Inference of an LTI system using the canonical form ( [PITH_FULL_IMAGE:figures/full_fig_p019_2.png] view at source ↗
Figure 3
Figure 3. Visualization of posterior geometry ( [PITH_FULL_IMAGE:figures/full_fig_p020_3.png] view at source ↗
Figures from the paper (6 more)
Figure 4
Figure 4. Figure 4: Effective sample size per second, in log-scale, for MCMC sampling; comparison [PITH_FULL_IMAGE:figures/full_fig_p021_4.png]
Figure 5
Figure 5. Figure 5: Comparing the computational efficiency and accuracy of inference for representative [PITH_FULL_IMAGE:figures/full_fig_p022_5.png]
Figure 6
Figure 6. Figure 6: Comparison of inferred trajectories using canonical (Θ [PITH_FULL_IMAGE:figures/full_fig_p023_6.png]
Figure 7
Figure 7. Figure 7: Prior sensitivity analysis: Median errors (over 30 systems)—of the Hankel matrix, [PITH_FULL_IMAGE:figures/full_fig_p024_7.png]
Figure 8
Figure 8. Figure 8: Consistency and BvM convergence analysis ( [PITH_FULL_IMAGE:figures/full_fig_p026_8.png]
Figure 9
Figure 9. Figure 9: Scalability analysis comparing standard vs. canonical forms for increasing system [PITH_FULL_IMAGE:figures/full_fig_p030_9.png]

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Cited by 1 Pith paper

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.