A Unified Bayesian Framework for Data-Driven Smoothing, Prediction, and Control

Andrea Iannelli; Matthias A. M\"uller; Mingzhou Yin; Seyed Ali Nazari

arxiv: 2512.01475 · v3 · submitted 2025-12-01 · 📡 eess.SY · cs.SY

A Unified Bayesian Framework for Data-Driven Smoothing, Prediction, and Control

Mingzhou Yin , Andrea Iannelli , Seyed Ali Nazari , Matthias A. M\"uller This is my paper

Pith reviewed 2026-05-17 03:17 UTC · model grok-4.3

classification 📡 eess.SY cs.SY

keywords data-driven controlBayesian estimationstochastic systemssmoothingpredictiontrajectory estimationmaximum a posteriorilinear systems

0 comments

The pith

A unified Bayesian estimation method solves data-driven smoothing, prediction, and control tasks together for linear systems.

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

The paper establishes that stochastic data-driven tasks for linear systems can be addressed through one Bayesian estimation procedure instead of separate empirical fixes. It formulates a single trajectory estimation problem that incorporates different types of trajectory knowledge depending on whether the goal is smoothing, prediction, or control. The maximum a posteriori solution then blends this knowledge with a data-driven characterization of correlated input-output uncertainties that follow elliptical distributions. A sympathetic reader would care because the same structure recovers existing data-driven algorithms as special cases and applies across the three tasks without custom derivations for each.

Core claim

The central claim is that for linear systems whose input-output uncertainties are correlated and follow elliptical distributions, a Bayesian problem can be solved via maximum a posteriori estimation to find the trajectory that optimally combines task-specific trajectory knowledge with a data-driven characterization obtained from offline data. This unified trajectory estimation problem provides a systematic solution for smoothing, prediction, and control, and reduces to prior data-driven methods under suitable conditions on the uncertainties.

What carries the argument

Maximum a posteriori estimation of a unified trajectory estimation problem that merges specified trajectory knowledge with a data-driven model of correlated elliptical input-output uncertainties.

If this is right

Smoothing, prediction, and control are obtained from the identical estimation procedure by altering only the form of trajectory knowledge supplied to the problem.
Existing data-driven prediction and control algorithms appear as special cases when the uncertainty model satisfies particular conditions.
Numerical comparisons on benchmark examples show competitive or improved results relative to system identification and other data-driven baselines for all three tasks.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same formulation could serve as a template for deriving data-driven versions of additional tasks by defining new trajectory knowledge terms.
The framework may naturally support online settings if the offline data characterization is updated recursively as new measurements arrive.
If the elliptical distribution assumption is replaced by a more general one, the method could be tested on systems with heavier-tailed or multimodal noise to map its practical range.

Load-bearing premise

The underlying system must be linear and the input-output uncertainties must be correlated while following an elliptical distribution.

What would settle it

Apply the estimator to a linear system driven by correlated Gaussian noise and compare its accuracy to existing specialized predictors, or repeat the test after replacing the noise with a non-elliptical distribution such as uniform noise and check for degraded performance.

Figures

Figures reproduced from arXiv: 2512.01475 by Andrea Iannelli, Matthias A. M\"uller, Mingzhou Yin, Seyed Ali Nazari.

read the original abstract

Extending data-driven algorithms based on Willems' fundamental lemma to stochastic data often requires empirical and customized workarounds. This work presents a unified Bayesian framework for linear systems that provides a systematic and general method for handling stochastic data-driven tasks, including smoothing, prediction, and control, via maximum a posteriori estimation. This framework formulates a unified trajectory estimation problem for the three tasks by specifying different types of trajectory knowledge. Then, a Bayesian problem is solved that optimally combines trajectory knowledge with a data-driven characterization of the trajectory from offline data for correlated input-output uncertainties with elliptical distributions. Under specific conditions, this problem is shown to generalize existing data-driven prediction and control algorithms. Numerical examples demonstrate the performance of the unified approach for all three tasks against other data-driven and system identification approaches.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper unifies data-driven smoothing, prediction, and control in a Bayesian MAP framework for stochastic linear systems.

read the letter

The paper's main contribution is a Bayesian framework that unifies smoothing, prediction, and control as different instances of a single maximum a posteriori trajectory estimation problem for linear systems using stochastic offline data. It is new because it provides a systematic way to handle correlated input-output uncertainties with elliptical distributions by optimally combining trajectory knowledge with a data-driven characterization. This generalizes prior data-driven prediction and control algorithms under specific conditions, moving beyond the empirical workarounds often needed for stochastic extensions of Willems' fundamental lemma. The numerical examples test the approach on all three tasks and compare it to other data-driven and system identification methods, which helps show its practical value. The paper does well in laying out a coherent structure that ties the tasks together through one estimation problem. The assumptions about linear systems and the uncertainty model allow for tractable solutions, and the claim of generalization is presented clearly. On the soft spots, the abstract and available details do not include the full derivations or solver specifics, so it is difficult to verify if the mathematics fully supports the performance claims without any hidden assumptions or circular use of data. The numerical results are mentioned but lack details on the scale of improvements or the experimental setup, which makes it hard to judge how convincing they are. These are moderate concerns that a referee could address. This paper is for control theorists and engineers working on data-driven methods for uncertain linear systems. A reader interested in extending behavioral approaches to stochastic settings would find it useful. It shows clear thinking and honest engagement with the existing literature, so it deserves a serious referee. I recommend putting it through peer review.

Referee Report

1 major / 2 minor

Summary. The manuscript proposes a unified Bayesian framework for linear systems that addresses stochastic data-driven smoothing, prediction, and control tasks via maximum a posteriori (MAP) estimation. It formulates a single trajectory estimation problem by specifying different types of trajectory knowledge, then solves a Bayesian problem that combines this knowledge with a data-driven characterization of correlated input-output uncertainties drawn from offline data under elliptical distributions. The framework is claimed to generalize existing data-driven prediction and control algorithms under specific conditions, with numerical examples demonstrating performance relative to other data-driven and system-identification methods.

Significance. If the central derivations are correct, the work offers a systematic way to extend deterministic data-driven approaches such as Willems' fundamental lemma to stochastic settings without ad-hoc workarounds. Unifying smoothing, prediction, and control under one MAP formulation that explicitly handles correlated elliptical uncertainties is a potentially useful generalization. The numerical comparisons against baseline methods provide concrete evidence of practical utility.

major comments (1)

[Section 3] The abstract states that the Bayesian problem generalizes existing algorithms under specific conditions, but the manuscript must explicitly derive the reduction (e.g., showing that the MAP estimator recovers the data-driven predictor or controller when the prior is non-informative) to confirm the generalization is not tautological.

minor comments (2)

[Section 2] Clarify the precise definition of 'trajectory knowledge' for each task (smoothing, prediction, control) and how it enters the MAP objective; a short table summarizing the three cases would improve readability.
[Section 5] The numerical examples should report the exact parameter values used for the elliptical distribution (e.g., covariance or shape matrix) so that readers can reproduce the stochastic data generation.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful review, positive assessment of the work's significance, and recommendation for minor revision. We address the single major comment below.

read point-by-point responses

Referee: [Section 3] The abstract states that the Bayesian problem generalizes existing algorithms under specific conditions, but the manuscript must explicitly derive the reduction (e.g., showing that the MAP estimator recovers the data-driven predictor or controller when the prior is non-informative) to confirm the generalization is not tautological.

Authors: We agree that an explicit derivation of the reduction would strengthen the manuscript and remove any ambiguity regarding the generalization claim. In the revised version, we will add a new paragraph (or short subsection) in Section 3 that derives the reduction step by step. Specifically, we will show that when the prior on the trajectory is taken to be non-informative (i.e., the prior covariance tends to infinity), the MAP objective reduces exactly to the least-squares data-driven predictor and controller formulations that appear in the literature. This derivation will be presented both algebraically and via limiting arguments to confirm that the equivalence is substantive rather than tautological. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper formulates a unified Bayesian MAP trajectory estimation problem that combines specified trajectory knowledge with a data-driven characterization of correlated elliptical uncertainties from offline data. The abstract states that this generalizes existing data-driven prediction and control algorithms under specific conditions, but the provided text contains no equations demonstrating that any derived prediction or result reduces by construction to quantities already fitted from the same data. No self-definitional loops, fitted inputs renamed as predictions, or load-bearing self-citations are quoted or evident. The derivation chain appears self-contained against external benchmarks for the three tasks, with the central claim resting on the well-posedness of the Bayesian formulation rather than circular reduction to inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Based on the abstract, the central claim rests on domain assumptions about linearity and uncertainty distributions rather than new free parameters or invented entities. No explicit free parameters are named.

axioms (2)

domain assumption The underlying system is linear
Required for the data-driven trajectory characterization via Willems' fundamental lemma.
domain assumption Input-output uncertainties are correlated and follow elliptical distributions
Enables the Bayesian formulation that optimally combines trajectory knowledge with offline data.

pith-pipeline@v0.9.0 · 5441 in / 1341 out tokens · 47223 ms · 2026-05-17T03:17:00.041834+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

unified Bayesian framework ... via maximum a posteriori estimation ... elliptical distributions ... Willems’ fundamental lemma
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

MAP estimator ... Σg(g) ... hyperparameter estimation ... SQP

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

22 extracted references · 22 canonical work pages

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B˚ ankestad, M.M., Sj¨ olund, J., Taghia, J., and Sch¨ on, T.B. (2023). Variational elliptical processes.Transactions on Machine Learn- ing Research

work page 2023

[2] [2]

Berberich, J., K¨ ohler, J., M¨ uller, M.A., and Allg¨ ower, F. (2021). Data-driven model predictive control with stability and robustness guarantees.IEEE Transactions on Automatic Control, 66(4), 1702–1717

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Breschi, V., Chiuso, A., and Formentin, S. (2023). Data-driven predictive control in a stochastic setting: a unified framework. Automatica, 152, 110961

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Coulson, J., Lygeros, J., and D¨ orfler, F. (2019). Data-enabled predictive control: In the shallows of the DeePC. InEuropean Control Conference (ECC), 307–312

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Coulson, J., Lygeros, J., and Dorfler, F. (2022). Distributionally robust chance constrained data-enabled predictive control.IEEE Transactions on Automatic Control, 67(7), 3289–3304

work page 2022

[6] [6]

Damen, A., Van den Hof, P., and Hajdasinski, A. (1982). Approxi- mate realization based upon an alternative to the Hankel matrix: the Page matrix.Systems & Control Letters, 2(4), 202–208. D¨ orfler, F., Coulson, J., and Markovsky, I. (2023). Bridging direct & indirect data-driven control formulations via regularizations and relaxations.IEEE Transactions o...

work page 1982

[7] [7]

and Lucia, S

Fiedler, F. and Lucia, S. (2021). On the relationship between data- enabled predictive control and subspace predictive control. In European Control Conference (ECC), 222–229

work page 2021

[8] [8]

and Lindskog, F

Hult, H. and Lindskog, F. (2002). Multivariate extremes, aggregation and dependence in elliptical distributions.Advances in Applied Probability, 34(3), 587–608

work page 2002

[9] [9]

Iannelli, A., Yin, M., and Smith, R.S. (2021). Design of input for data-driven simulation with Hankel and Page matrices. InIEEE Conference on Decision and Control (CDC), 139–145

work page 2021

[10] [10]

Lian, Y., Shi, J., Koch, M., and Jones, C.N. (2023). Adaptive robust data-driven building control via bilevel reformulation: An experimental result.IEEE Transactions on Control Systems Technology, 31(6), 2420–2436

work page 2023

[11] [11]

and D¨ orfler, F

Markovsky, I. and D¨ orfler, F. (2021). Behavioral systems theory in data-driven analysis, signal processing, and control.Annual Reviews in Control, 52, 42–64

work page 2021

[12] [12]

and D¨ orfler, F

Markovsky, I. and D¨ orfler, F. (2022). Data-driven dynamic interpo- lation and approximation.Automatica, 135, 110008

work page 2022

[13] [13]

and D¨ orfler, F

Markovsky, I. and D¨ orfler, F. (2023). Identifiability in the behav- ioral setting.IEEE Transactions on Automatic Control, 68(3), 1667–1677

work page 2023

[14] [14]

and Polyak, B.T

Nesterov, Y. and Polyak, B.T. (2006). Cubic regularization of Newton method and its global performance.Mathematical Pro- gramming, 108(1), 177–205

work page 2006

[15] [15]

Pan, G., Ou, R., and Faulwasser, T. (2023). On a stochastic fundamental lemma and its use for data-driven optimal control. IEEE Transactions on Automatic Control, 68(10), 5922–5937

work page 2023

[16] [16]

Smith, R.S., Abdalmoaty, M., and Yin, M. (2024). Data-driven formulation of the Kalman filter and its application to predictive control. InIEEE Conference on Decision and Control (CDC), 2633–2639. Van Overschee, P. and De Moor, B. (2012).Subspace identifica- tion for linear systems: Theory, implementation, applications

work page 2024

[17] [17]

van Waarde, H.J., De Persis, C., Camlibel, M.K., and Tesi, P

Springer, New York, NY. van Waarde, H.J., De Persis, C., Camlibel, M.K., and Tesi, P. (2020). Willems’ fundamental lemma for state-space systems and its extension to multiple datasets.IEEE Control Systems Letters, 4(3), 602–607

work page 2020

[18] [18]

Willems, J.C., Rapisarda, P., Markovsky, I., and De Moor, B.L.M. (2005). A note on persistency of excitation.Systems & Control Letters, 54(4), 325–329

work page 2005

[19] [19]

Yin, M., Iannelli, A., and Smith, R.S. (2021). Maximum likelihood signal matrix model for data-driven predictive control. InPro- ceedings of the 3rd Conference on Learning for Dynamics and Control, 1004–1014

work page 2021

[20] [20]

Yin, M., Iannelli, A., and Smith, R.S. (2022). Data-driven prediction with stochastic data: Confidence regions and minimum mean- squared error estimates. InEuropean Control Conference (ECC), 853–858

work page 2022

[21] [21]

Yin, M., Iannelli, A., and Smith, R.S. (2023). Maximum likelihood estimation in data-driven modeling and control.IEEE Transac- tions on Automatic Control, 68(1), 317–328

work page 2023

[22] [22]

Yin, M., Iannelli, A., and Smith, R.S. (2024). Stochastic data- driven predictive control: Regularization, estimation, and con- straint tightening.IFAC-PapersOnLine, 58(15), 79–84

work page 2024