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REVIEW 3 minor 6 references

Uncertainty Quantification in Data-Driven Inverse Optimization via Bayesian Inference

T0 review · 0 major / 3 minor · reviewed 2026-06-29 · grok-4.3

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Pith's one-line read A hierarchical Bayesian framework quantifies parameter uncertainty in data-driven inverse optimization via MCMC and establishes posterior consistency.

desk verdict This paper supplies a hierarchical Bayesian model and two MCMC algorithms to get credible regions around parameters inferred from observed decisions in optimization. read the letter →

arxiv 2605.25288 v1 pith:KDCCHNJ3 submitted 2026-05-24 math.OC

Timothy C.Y. Chan , Nathan Sandholtz , Nasrin Yousefi This is my paper · ORCID
classification math.OC
keywords inverseoptimizationBayesianinferenceuncertaintyquantificationMarkovchainMonteCarloposteriorconsistencycredibleregionsdata-driven
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 develops a Bayesian approach to move beyond point estimates in inverse optimization by treating the unknown parameters as random variables with a posterior distribution. It introduces two MCMC algorithms tailored to two different data-generating processes and proves that the posterior converges to the true parameter under standard identifiability assumptions. Credible regions built from the posterior samples are shown numerically to attain coverage close to the nominal level and to contract as more observed decisions become available.

What carries the argument

Hierarchical Bayesian model whose posterior over the unknown parameter vector is sampled by two Markov chain Monte Carlo algorithms, one for each data-generating process considered.

What would settle it

Generate data from the assumed model, run the MCMC procedure, and check whether the constructed credible regions contain the true parameter at a frequency that deviates systematically from the nominal level.

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Extended reading notes

Core claim

The central claim is that a hierarchical Bayesian model, estimated by two specialized Markov chain Monte Carlo algorithms, yields a consistent posterior for the parameter vector of an inverse optimization problem; the resulting credible regions exhibit near-nominal empirical coverage that improves with the number of observed decisions.

Load-bearing premise

The observed decisions are generated exactly by one of the two models in the paper and the inverse optimization problem meets the standard identifiability conditions needed for posterior consistency.

Editorial extensions

If this is right

  • Credible regions for the recovered parameters can be reported instead of single point estimates.
  • The posterior contracts and the regions shrink as the number of observed decisions grows.
  • Posterior consistency holds whenever the standard identifiability conditions are satisfied.
  • Numerical coverage of the credible regions stays close to the nominal probability in the tested cases.

Reading between the lines

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

  • Downstream decisions that use the recovered parameters can now incorporate the full posterior rather than a point estimate.
  • The same MCMC machinery could be adapted to inverse problems whose forward models differ from the two cases treated here.
  • The approach supplies a concrete route to attach uncertainty statements to any inverse optimization pipeline that already satisfies identifiability.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

0 major / 3 minor

Summary. The paper proposes a hierarchical Bayesian framework for uncertainty quantification in data-driven inverse optimization. For two data-generating processes, it develops MCMC algorithms to sample the posterior over the unknown parameter vector, constructs credible regions from the posterior, proves posterior consistency under standard identifiability conditions, and reports numerical experiments showing near-nominal empirical coverage that improves with more observed decisions.

Significance. If the derivations and experiments hold, the work fills a clear gap: most inverse optimization methods return only point estimates, while this supplies a principled posterior and credible sets with consistency guarantees. The explicit handling of two DGPs and the use of standard identifiability conditions for the consistency proof are strengths; reproducible code or machine-checked proofs would further strengthen the contribution.

minor comments (3)
  1. The abstract and introduction should explicitly state the precise form of the two data-generating processes (e.g., whether noise is additive on the objective or on the constraints) so that readers can immediately map the claimed consistency result to the model assumptions.
  2. Notation for the credible regions (e.g., how the highest-posterior-density or equal-tailed intervals are defined from the MCMC samples) should be introduced once in a dedicated subsection rather than scattered across the algorithm and experiment sections.
  3. The numerical experiments section would benefit from a table reporting the exact empirical coverage percentages, average region volumes, and wall-clock times for each MCMC sampler across the tested sample sizes, rather than only qualitative statements about “near-nominal” coverage.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive review, the recognition of the contribution in providing a principled posterior and credible sets with consistency guarantees, and the recommendation of minor revision. We appreciate the comments on the strengths of handling two DGPs and using standard identifiability conditions.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity identified

full rationale

The derivation relies on a hierarchical Bayesian model with MCMC sampling to obtain posteriors and credible regions, plus a consistency result stated under external standard identifiability conditions that are not constructed from the paper's fitted parameters or internal definitions. No equations or steps reduce a claimed prediction to a fitted input by construction, and no load-bearing uniqueness theorem or ansatz is imported via self-citation. The empirical coverage checks are independent validations rather than tautological. The central claims remain self-contained against the stated assumptions.

Assumptions & free parameters 0 free parameters · 1 assumptions · 0 invented entities

Ledger constructed from abstract only; full paper may contain additional fitted parameters or assumptions not visible here.

assumptions (1)
  • domain assumption standard identifiability conditions
    Invoked to establish posterior consistency (abstract).

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Cite this review

Pith. "Pith review of Uncertainty Quantification in Data-Driven Inverse Optimization via Bayesian Inference." pith.science (2026). https://pith.science/paper/KDCCHNJ3

@misc{pith2026260525288,
  author       = {Pith},
  title        = {Pith review of: Uncertainty Quantification in Data-Driven Inverse Optimization via Bayesian Inference},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KDCCHNJ3}},
  note         = {Machine review of arXiv:2605.25288}
}
read the original abstract

Inverse optimization (IO) is used to estimate unknown parameters of an optimization model from observed decisions. In the data-driven context, the estimated parameters are inherently uncertain, yet quantifying this uncertainty has received limited attention in the literature, where existing methods return a point estimate. In this paper, we propose a hierarchical Bayesian framework for parameter uncertainty quantification in data-driven inverse optimization. Considering two data-generating processes, we develop two Markov chain Monte Carlo algorithms to estimate the posterior distribution of the unknown parameter vector, which is used to construct credible regions. We establish posterior consistency under standard identifiability conditions. Numerical experiments demonstrate near-nominal empirical coverage of the credible regions and show that the regions shrink as the number of observed decisions increases.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

6 extracted references · 1 canonical work pages

  1. [1]

    Chan, Sandholtz, and Yousefi:Uncertainty Quantification in IO 21 Ahuja RK, Orlin JB (2001) Inverse optimization.Operations research49(5):771–783

    Absil PA, Mahony R, Sepulchre R (2008)Optimization algorithms on matrix manifolds(Princeton University Press). Chan, Sandholtz, and Yousefi:Uncertainty Quantification in IO 21 Ahuja RK, Orlin JB (2001) Inverse optimization.Operations research49(5):771–783. Ajayi T, Lee T, Schaefer AJ (2022) Objective selection for cancer treatment: An inverse optimization...

    2008

  2. [2]

    Esfahani PM, Shafieezadeh-Abadeh S, Hanasusanto GA, Kuhn D (2018) Data-driven inverse optimization with imper- fect information.Mathematical Programming167(1):191–234

    Chow JY, Ritchie SG, Jeong K (2014) Nonlinear inverse optimization for parameter estimation of commodity-vehicle- decoupled freight assignment.Transportation Research Part E: Logistics and Transportation Review67:71–91. Esfahani PM, Shafieezadeh-Abadeh S, Hanasusanto GA, Kuhn D (2018) Data-driven inverse optimization with imper- fect information.Mathemati...

    2014

  3. [3]

    Gurobi Optimization, LLC (2026) Gurobi Optimizer Reference Manual

    Gupta R, Zhang Q (2022) Decomposition and adaptive sampling for data-driven inverse linear optimization.INFORMS Journal on Computing34(5):2720–2735. Gurobi Optimization, LLC (2026) Gurobi Optimizer Reference Manual. URLhttps://www.gurobi.com. Haario H, Saksman E, Tamminen J (2001) An adaptive Metropolis algorithm.Bernoulli7(2):223–242. Hastings WK (1970) ...

    2022

  4. [4]

    Keshavarz A, Wang Y, Boyd S (2011) Imputing a convex objective function.2011 IEEE international symposium on intelligent control, 613–619 (IEEE)

    Iyengar G, Kang W (2005) Inverse conic programming with applications.Operations Research Letters33(3):319–330. Keshavarz A, Wang Y, Boyd S (2011) Imputing a convex objective function.2011 IEEE international symposium on intelligent control, 613–619 (IEEE). Lin B, Delage E, Chan T (2024) Conformal inverse optimization.Advances in Neural Information Process...

    work page arXiv 2005

  5. [5]

    In our implementation, however, we adopt a symmetric approximation for numerical efficiency

    Because the Gaussian distribution has support onR 𝑛 rather thanS 𝑛−1, this approach induces a proposal Chan, Sandholtz, and Yousefi:Uncertainty Quantification in IO 27 density that is not exactly symmetric, so the acceptance probability should include the proposal ratio 𝑞(𝜽 𝑡−1 |𝜽 𝑐)/𝑞(𝜽 𝑐 |𝜽 𝑡−1). In our implementation, however, we adopt a symmetric appr...

    2001

  6. [6]

    In low dimensions (𝑛≤5), all three proposals yielded comparable posterior summaries (𝛼RMS and coverage) and similar sampling effi- ciency. In higher dimensions, the full-covariance Gaussian proposal substantially outperformed both alternatives in efficiency, requiring substantially fewer iterations (by an order of magnitude), to reach comparable PSRF valu...

    2021

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Reviewed June 29, 2026 · model on record in the stance chip above.