Pith. sign in

REVIEW 4 major objections 6 minor 58 references

Confidence intervals for functionals in constrained inverse problems via data-adaptive sampling-based calibration

T0 review · 4 major / 6 minor · reviewed 2026-08-09 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read By shrinking the parameter search to a data-dependent Berger–Boos set and splitting the coverage budget, this paper constructs four confidence intervals that keep the stated coverage and shorten intervals in high-dimensional inverse…

desk verdict Correct Berger-Boos coverage lemma with a useful HEP unfolding demonstration, but the abstract overclaims practical coverage and one appendix lemma is misstated. read the letter →

arxiv 2502.02674 v2 pith:WFNL5A5U submitted 2025-02-04 stat.ME stat.CO

Michael Stanley , Pau Batlle , Pratik Patil , Houman Owhadi , Mikael Kuusela This is my paper · ORCID
classification stat.MEstat.CO MSC 62F2562G0865J22
keywords inverseproblemsconfidenceintervalstestinversionquantileregressionBerger–Boossetill-posedunfoldinguncertaintyquantification
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

This paper tries to make the test-inversion confidence intervals of Batlle et al. (2023) practical and less conservative for functionals in ill-posed inverse problems. The key move is to replace the full constraint set by a data-dependent Berger–Boos set, a compact subset that contains the true parameter with probability at least $1-\eta$, and then maximize the $1-\gamma$ quantile of the log-likelihood-ratio statistic only over this subset, with $\gamma$ chosen so that the total error budget $\alpha$ is split as $\gamma + \eta$. The paper proves that this budget split preserves $1-\alpha$ coverage and that four sampling-based interval constructions converge in probability to the oracle intervals. In an 80-dimensional particle-unfolding simulation, the Sliced variants reach nominal coverage and are 11 to 19 percent shorter on average than the standard one-at-a-time strict bounds interval. The payoff is a recipe for constraint-aware intervals that remain valid when the forward model is rank-deficient and the parameter space is high-dimensional.

What carries the argument

The load-bearing object is the Berger–Boos set $B_\eta = \{x \in X : \|y - f(x)\|_2^2 \le \chi^2_{n,\eta}\}$, a $1-\eta$ confidence set for $x^*$ obtained as the pre-image of a chi-squared ball under the forward model. The argument splits the coverage budget as $1-\alpha = (1-\eta)(1-\gamma)$, so maximizing the larger $1-\gamma$ quantile over $B_\eta$ costs the same total error as maximizing the $1-\alpha$ quantile over $X$. The LLR statistic $\lambda(\mu, y) = \inf_{x \in \Phi_\mu \cap X} \|y - f(x)\|^2 - \inf_{x \in X} \|y - f(x)\|^2$ defines the tests being inverted; sampling design points inside $B_\eta$, estimating the quantile surface by Monte Carlo percentiles or quantile regression, and then either inverting the test or using the estimated cutoff in an endpoint optimization yields the four interval constructions.

What would settle it

For a fixed observation in any of the paper's low-dimensional examples, compute the true maximum quantile $q_{\gamma,\eta}$ by dense grid optimization over $B_\eta$, then compare it with the sampled or regressed estimate from Algorithms 1 and 2; finding any observation where the estimate lies below the true maximum, or where a coverage simulation using an exhaustive sampler exceeds one using the Polytope sampler, would refute the practical claim.

Watch

Extended reading notes

Core claim

The central claim is that valid confidence intervals for a one-dimensional functional can be built by inverting the constrained log-likelihood-ratio test over a shrunk, data-adaptive set instead of the entire constraint set. Specifically, Lemma 3.1 states that for any $\eta \in (0, \alpha)$, the set $C_{\mathrm{sl}}^{\alpha}(y; B_\eta) = \{\mu : \lambda(\mu, y) \le q_{\gamma,\eta}(\mu)\}$ is a $1-\alpha$ confidence set for $\mu^* = \varphi(x^*)$ whenever $\gamma \le \alpha - \eta$, where $q_{\gamma,\eta}(\mu) = \sup_{x \in B_\eta \cap \Phi_\mu} Q_x(1-\gamma)$; Corollary 3.2 gives the same guarantee for the global version. The paper further claims that the oracle quantiles can be approximated consistently: Theorem 4.1 shows that both inversion-based and optimization-based versions converge in probability to the oracle interval as the number of design points and LLR samples grows, assuming a consistent quantile regressor. Empirically, all four constructions reach nominal coverage in the tested settings, while the OSB interval fails in the 3D constrained-Gaussian case and in an adversarial 80D unfolding setting; the Sliced intervals reduce average length relative to OSB by 11 to 19 percent where OSB is valid.

Load-bearing premise

The implemented intervals keep their coverage guarantee only if the sampler and the quantile regressor do not underestimate the maximum $1-\gamma$ quantile over the Berger–Boos set; missing a high-quantile region makes the cutoff too small and the interval under-cover.

Editorial extensions

If this is right

  • If Lemma 3.1 and Corollary 3.2 hold, any $1-\eta$ confidence set for $x^*$ can be plugged into the construction, so the framework extends beyond Gaussian pre-image ellipsoids to other confidence sets with the same budget split.
  • The convergence result in Theorem 4.1 means that, with enough samples and a consistent quantile estimator, the inverted and optimized intervals coincide with the oracle Berger–Boos interval, so users can choose either form without changing the asymptotic guarantee.
  • In the smooth 80-dimensional unfolding setting, the Sliced constructions cut expected interval length by 11.1 to 18.7 percent relative to OSB while keeping at least nominal coverage, implying that the added computational cost buys shorter intervals in realistic high-energy-physics problems.
  • In the adversarial unfolding setting, all four constructions maintain nominal coverage where OSB under-covers, so the method promises valid uncertainty quantification when the true parameter is concentrated near constraint boundaries.

Reading between the lines

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

  • The same $\gamma + \eta$ budget argument could be applied to any test statistic with a computable quantile, not only the LLR, as long as a $1-\eta$ confidence set for $x^*$ is available; the paper notes the generalization in principle but does not pursue it.
  • The divergent behaviors of Sliced Inverted and Sliced Optimized in the simulations suggest the Optimized variant is more robust to local quantile-regression error, since it smooths the max-quantile curve; a single poorly predicted quantile can collapse the Inverted interval to a point.
  • One testable extension is to replace the Polytope sampler with a sampler biased toward high-quantile regions, as the appendix's importance-like sampler does for the 3D example; such bias could improve coverage robustness in higher dimensions without changing the budget argument.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 6 minor

Summary. The paper proposes four confidence-interval constructions for a scalar functional φ(x*) in constrained, possibly rank-deficient inverse problems with additive Gaussian noise and known parameter constraints. The construction restricts the test-inversion intervals of Batlle et al. (2023) to a data-adaptive Berger–Boos set Bη, defines sliced and global maximum quantiles q_{γ,η}(µ) and q_{γ,η}, and proves an oracle coverage guarantee in Lemma 3.1 and Corollary 3.2. To implement the oracle cutoffs, the paper proposes two samplers (the VGS sampler and a Vaidya-walk Polytope sampler) and a quantile-regression estimator based on gradient boosting, and states a consistency theorem (Theorem 4.1) for the resulting Global/Sliced × Inverted/Optimized intervals. Numerical experiments in two- and three-dimensional constrained Gaussian settings and in an 80-dimensional wide-bin unfolding problem compare the intervals with the OSB interval.

Significance. If the implementation matched the oracle construction, the paper would be a useful step: it gives a principled way to reduce the conservatism of OSB intervals while retaining a frequentist coverage guarantee, and it demonstrates the approach on a realistic high-energy-physics unfolding problem. The oracle coverage lemma is a clean union-bound argument and the Berger–Boos budget is handled correctly in that lemma. However, the practical interval constructions rely on unproven and partly unmet conditions: uniform consistency of the quantile regressor, positive-mass sampling of Bη, and a correct stability lemma for the Sliced Optimized endpoint. The numerical evidence is encouraging but does not by itself establish the theorem-level claim for the implemented intervals, and no code is provided. The contribution is real, but the gap between the oracle theory and the implemented procedure needs to be closed or explicitly acknowledged.

major comments (4)
  1. [Appendix B.4 (Lemma B.2)] The condition stated in Lemma B.2, namely that for all ε>0 there exists δ′>0 such that |f(µ)|>δ′ if and only if |µ−µ*|<ε, cannot hold for the functions to which the lemma is applied. At µ=µ* one typically has f(µ*)=0, so the left-hand side is false while the right-hand side is true; conversely, points outside the ε-neighborhood can have large |f|. The intended separation condition is presumably something like “|µ−µ*|≥ε implies |f(µ)|≥δ′”, and the proof of convergence of inf_{f_k≥0} to inf_{f≥0} requires that condition together with uniform convergence. As written, Lemma B.2 does not prove Statement 4 of Theorem 4.1.
  2. [Section 4, Theorem 4.1] The theorem assumes that the quantile regression in Algorithm 2 is consistent for all x, i.e., P(|bq_γ(x)−Q_x(1−γ)|>ε)→0 for every fixed x, but the implemented gradient-boosting estimator of Section 5.2 is not shown to satisfy this, and no regularity conditions or rates are given. In the proof, this assumption is used to control the first term of Eq. (S.7) for the empirical maximum over design points; pointwise consistency is not enough to control a maximum over M points as M grows, so a uniform consistency statement would be needed. Since the numerical experiments use Algorithm 2 with the gradient-boosting quantile regressor, the theoretical guarantee for the implemented intervals is not established.
  3. [Section 5.1.2 vs. Lemma B.1 / Theorem 4.1] Theorem 4.1 and Lemma B.1 require the design-point distribution to have positive mass on every Lebesgue-positive subset of Bη. The Polytope sampler of Algorithm 4 generates MCMC draws over a bounding polytope P_d ⊇ Bη, not i.i.d. draws from such a measure on Bη, and Section 5.1.2 states that in practice it “often has some difficulty reaching the corners of the generated polytope.” Those corners are exactly the regions where the LLR quantile of a constrained problem can be largest, and the paper’s own Section 6.2 needed a separate importance-like sampler (Algorithm 5) to reach the high-quantile boundary. If the sampler misses high-quantile regions, bq (or bm_γ) is biased downward, the LLR acceptance cutoff is too small, and coverage is lost. Thus the convergence theorem does not apply to the implemented sampler, and the abstract’s “both theoretically and in practice” claim is not supported for the implemented intervals.
  4. [Abstract and Section 4] The paper states that all four intervals “achieve nominal coverage … both theoretically and in practice.” The theoretical coverage in Lemma 3.1 concerns oracle cutoffs q_{γ,η}(µ) and q_{γ,η}; Theorem 4.1 provides only convergence in probability of the sampled endpoints to those oracle endpoints. That combination gives at best asymptotic coverage under the theorem’s assumptions, not a finite-sample guarantee, and the practical statement is based on selected simulations with 10^3 replications and no code. The wording should be softened to distinguish oracle coverage, asymptotic consistency of the implemented endpoints, and empirical performance.
minor comments (6)
  1. [Section 3.1, Remark 1] The remark says the Berger–Boos construction is “equivalent to a data-dependant reduction of the constraint set, replacing X for Bη both in the test statistic and the quantile optimization problems,” but the intervals in (23)–(24) use λ(µ,y;X) as defined in (11), with X, not Bη. Please clarify or correct the statement.
  2. [Appendix A.1, Lemma 3.1 proof] In the definition of A2, Q_x(1−γ) should be Q_{x*}(1−γ) with µ*=φ(x*); as written, x is undefined.
  3. [Algorithm 1, line 6] The notation λ({(1−γ)N}) is unclear; it should be written as the order statistic λ_{(⌈(1−γ)N⌉)} or with an equivalent explicit expression.
  4. [Section 4, Theorem 4.1] The condition that there exists a point µ ∈ φ(Bη) satisfying λ(µ,y;X) < q_{γ,η} is stated to be equivalent to the interval being non-empty for linear models, but no proof of this equivalence is provided.
  5. [Section 6.3] The paper reports that in the smooth setting the Global intervals and OSB dramatically over-cover, and in the adversarial setting the Sliced Optimized interval over-covers; these results are presented as strengths, but they also show that not all four constructions achieve close-to-nominal coverage in practice. The text should discuss why over-coverage is consistent with the stated goal that all four intervals achieve nominal coverage.
  6. [Section 5.2 and Section 6] No code or detailed hyperparameter settings for the gradient-boosting quantile regressor are provided beyond a reference to scikit-learn; given the tuning described in Section 5.2, this makes replication difficult.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the coverage guarantee is a Berger-Boos union bound; the sampling and quantile-regression steps are stated approximations rather than fitted predictions.

full rationale

The central coverage claim, Lemma 3.1, is derived independently in Appendix A.1 from the definition of the Berger-Boos set B_eta as a 1-eta confidence set for x* and the definition of q_{gamma,eta}(mu) as the supremum of the LLR quantile over B_eta intersect Phi_mu. The proof is a standard union bound: P(miss) <= P(x* not in B_eta) + P(lambda > Q_{x*}(1-gamma)) <= eta + gamma, with gamma = alpha - eta. No fitted parameter, estimated quantile, or data-derived value is used to establish this oracle coverage; the oracle intervals are defined before any estimation. The computational algorithms (Algorithms 1 and 2) are presented as approximations, and Theorem 4.1 explicitly assumes consistency of the quantile regressor rather than deriving coverage from it. The self-citation to Batlle et al. (2023) supplies the background test-inversion framework and OSB comparison results, but Lemma 3.1 is proven in the appendix and does not reduce to that citation. The acknowledged practical limitations of the Polytope sampler (Section 5.1.2: it 'often has some difficulty reaching the corners') and the heuristic importance-like sampler of Appendix C.1 (explicitly lacking a theoretical guarantee) are finite-sample implementation risks, not circular reductions. The misstatement in Lemma B.2 is a correctness issue in one auxiliary convergence argument, not a case of the derivation being equivalent to its inputs. Therefore no claimed prediction is forced by construction.

Assumptions & free parameters 5 free parameters · 5 assumptions · 0 invented entities

The central coverage claim rests on the Gaussian noise assumption and the exactness of the quantile Q_x; the practical convergence rests on compactness, continuity, and sampler/regressor consistency assumptions that are plausible but unverified for the specific implementations.

free parameters (5)
  • eta (Berger-Boos uncertainty budget) = 0.01 in experiments
    User-chosen probability that the true parameter lies outside the bounding set; added to the uncertainty budget. Small value recommended but not optimized.
  • Rolling window size T = not specified
    Controls the rolling maximum estimate of the sliced max-quantile; affects Sliced interval endpoints.
  • Vaidya walk proposal radius = 0.5
    Hand-tuned to give acceptance probability about 33.3%; affects sampler mixing.
  • Importance-like sampler inverse length scale gamma_p and norm order q = hand-tuned
    Used only for the 3D example to oversample near the constraint boundary; no theoretical guarantee.
  • Gradient Boosting quantile regressor hyperparameters = chosen by 10-fold CV in pilot study
    Affect quantile surface accuracy and hence coverage of Algorithm 2-based intervals.
assumptions (5)
  • domain assumption Observation noise is Gaussian with known covariance, standardized to identity
    Used to define B_eta = {x : ||y - f(x)||^2 <= chi^2_{n,eta}} as a 1-eta confidence set for x* (Section 3.1, Eq. 20). Coverage guarantee fails under misspecified noise.
  • domain assumption Quantile function Q_x(1-gamma) is continuous in x
    Assumed in Theorem 4.1 for convergence of the sampling-based intervals (Section 4).
  • domain assumption B_eta is compact without isolated points
    Assumed in Theorem 4.1 so that sampling can approach every point; may fail when the forward model has null directions unbounded in the constraint set (Section 5.1.2).
  • ad hoc to paper Quantile regression in Algorithm 2 is consistent for all x
    Assumed in Theorem 4.1; the specific Gradient Boosting implementation has no consistency proof.
  • standard math For linear-Gaussian case the LLR chi-squared analysis of Batlle et al. (2023) holds
    Used to identify OSB and SSB benchmarks and to prove OSB validity in the 2D example (Section 6.1).

how reviews work

0 comments
Cite this review

Pith. "Pith review of Confidence intervals for functionals in constrained inverse problems via data-adaptive sampling-based calibration." pith.science (2026). https://pith.science/paper/WFNL5A5U

@misc{pith2026250202674,
  author       = {Pith},
  title        = {Pith review of: Confidence intervals for functionals in constrained inverse problems via data-adaptive sampling-based calibration},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WFNL5A5U}},
  note         = {Machine review of arXiv:2502.02674}
}
read the original abstract

We address functional uncertainty quantification for ill-posed inverse problems where it is possible to evaluate a possibly rank-deficient forward model, the observation noise distribution is known, and there are known parameter constraints. We present four constraint-aware confidence intervals extending the work of Batlle et al. (2023) by making the intervals both computationally feasible and less conservative. Our approach first shrinks the potentially unbounded constraint set compact in a data-adaptive way, obtains samples of the relevant test statistic inside this set to estimate a quantile function, and then uses these computed quantities to produce the intervals. Our data-adaptive bounding approach is based on the approach by Berger and Boos (1994), and involves defining a subset of the constraint set where the true parameter exists with high probability. This probabilistic guarantee is then incorporated into the final coverage guarantee in the form of an uncertainty budget. We then propose custom sampling algorithms to efficiently sample from this subset, even when the parameter space is high-dimensional. Optimization-based interval methods formulate confidence interval computation as two endpoint optimizations, where the optimization constraints can be set to achieve different types of interval calibration while seamlessly incorporating parameter constraints. However, choosing valid optimization constraints has been elusive. We show that all four proposed intervals achieve nominal coverage for a particular functional both theoretically and in practice, with numerical examples demonstrating superior performance of our intervals over the OSB interval in terms of both coverage and expected length. In particular, we show the superior performance in a realistic unfolding simulation from high-energy physics that is severely ill-posed and involves a rank-deficient forward model.

Figures

Figures reproduced from arXiv: 2502.02674 by the authors.

Figure 1.1
Figure 1.1. (Left) A particular quantile surface, Qx(1 − α), where x ≥ 0 and α = 0.32. This surface was obtained via Monte Carlo sampling the LLR test statistic over a grid of x’s defined by the two-dimensional constrained-Gaussian scenario similar to that in Section 6.1, but with h := 1 1⊤ . (Right-Top) An illustration of the Berger–Boos set and other, a 1−η confidence set for x ∗ , which prevents having to contend with an un… view at source ↗
Figure 5
Figure 5. in [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
Figure 5.1
Figure 5.1. Numerical illustrations of the VGS Sampler’s infeasibility in high dimensional regimes. The [PITH_FULL_IMAGE:figures/full_fig_p018_5_1.png] view at source ↗
Figures from the paper (8 more)
Figure 5.2
Figure 5.2. Figure 5.2: Polytope sampler output for a realization of the 80-dimensional ill-posed inverse problem studied [PITH_FULL_IMAGE:figures/full_fig_p021_5_2.png]
Figure 6.1
Figure 6.1. Figure 6.1: Estimated coverages and expected lengths across all four interval constructions and OSB for [PITH_FULL_IMAGE:figures/full_fig_p025_6_1.png]
Figure 6.2
Figure 6.2. Figure 6.2: (Left) Four realizations of the data-generating process where the observations are shown in red. For each realization, the blue points are uniformly distributed samples from its Berger–Boos set, sampled using the VGS sampler. (Center) For a realization of the data-ge…
Figure 6.3
Figure 6.3. Figure 6.3: Estimated coverage and expected length across all four interval constructions and OSB for com [PITH_FULL_IMAGE:figures/full_fig_p026_6_3.png]
Figure 6.4
Figure 6.4. Figure 6.4: Confidence interval lengths in the Berger–Boos setting, averaged over values of [PITH_FULL_IMAGE:figures/full_fig_p027_6_4.png]
Figure 6.5
Figure 6.5. Figure 6.5: Parameter values for the smooth and ad￾versarial settings for x ∗ used to illustrate our inter￾val construction versus the OSB interval. The ad￾versarial setting is made more difficult by the sharp jumps in parameter values. 6.3.1 Smooth setting Using the smooth x ∗ …
Figure 6.6
Figure 6.6. Figure 6.6: Estimated coverage and expected length across all four interval constructions and OSB at the [PITH_FULL_IMAGE:figures/full_fig_p029_6_6.png]
Figure 6.7
Figure 6.7. Figure 6.7: Estimated coverage and expected length across all four interval constructions and OSB at the [PITH_FULL_IMAGE:figures/full_fig_p030_6_7.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

58 extracted references · 54 canonical work pages

  1. [1]

    Ashton, G. et al. (2022). Nested sampling for physical scientists. Nature Reviews Methods Primers , 2

    work page 2022

  2. [2]

    Batlle, P., Stanley, M., Patil, P., Kuusela, M., and Owhadi, H. (2023). Optimization-based frequentist confidence intervals for functionals in constrained inverse problems: Resolving the burrus conjecture. arXiv preprint arXiv:2310.02461

    work page Pith review arXiv 2023

  3. [3]

    Berger, R. L. and Boos, D. D. (1994). P values maximized over a confidence set for the nuisance parameter. Journal of the American Statistical Association , 89:1012--1016

    work page 1994

  4. [4]

    and Vandenberghe, L

    Boyd, S. and Vandenberghe, L. (2004). Convex Optimization . Cambridge University Press

    2004

  5. [5]

    Buchner, J. (2023). Nested sampling methods. arXiv preprint arXiv:2101.09675v4

    arXiv 2023

  6. [6]

    Burrus, W., Commission, U. A. E., and Laboratory, O. R. N. (1965). Utilization of a Priori Information by Means of Mathematical Programming in the Statistical Interpretation of Measured Distributions . Oak Ridge National Laboratory

    work page 1965

  7. [7]

    Carpenter, J. (1999). Test inversion bootstrap confidence intervals. Journal of the Royal Statistical Society: Series B (Statistical Methodology) , 61(1):159--172

    work page 1999

  8. [8]

    and Berger, R

    Casella, G. and Berger, R. L. (2002). Statistical Inference . Duxbery, 2 edition

    work page 2002

Show all 58 references
  1. [9]

    J., and Yu, B

    Chen, Y., Dwivedi, R., Wainwright, M. J., and Yu, B. (2018). Fast mcmc sampling algorithms on polytopes. Journal of Machine Learning Research , 19:1--86

    2018

  2. [10]

    Measurement of differential cross sections for H iggs boson production in the diphoton decay channel in pp collisions at s = 8 TeV

    CMS Collaboration (2016). Measurement of differential cross sections for H iggs boson production in the diphoton decay channel in pp collisions at s = 8 TeV . The European Physical Journal C , 76(13)

    2016

  3. [11]

    Measurement of inclusive and differential H iggs boson production cross sections in the diphoton decay channel in proton-proton collisions at s = 13 TeV

    CMS Collaboration (2019). Measurement of inclusive and differential H iggs boson production cross sections in the diphoton decay channel in proton-proton collisions at s = 13 TeV . Journal of High Energy Physics , 2019(183)

    2019

  4. [12]

    Dalmasso, N., Izbicki, R., and Lee, A. (2020). Confidence sets and hypothesis testing in a likelihood-free inference setting. In International Conference on Machine Learning , pages 2323--2334. PMLR

    2020

  5. [13]

    Dalmasso, N., Masserano, L., Zhao, D., Izbicki, R., and Lee, A. B. (2024). Likelihood-free frequentist inference: Bridging classical statistics and machine learning for reliable simulator-based inference. Electronic Journal of Statistics , pages 5045--5090

    2024

  6. [14]

    Fisher, E., Schweiger, R., and Rosset, S. (2020). Efficient construction of test inversion confidence intervals using quantile regression. Journal of Computational and Graphical Statistics , 29(1):140--148

    2020

  7. [15]

    Garthwaite, P. H. and Buckland, S. T. (1992). Generating monte carlo confidence intervals by the robbins-monro process. Journal of the Royal Statistical Society , 41(1):159--171

    1992

  8. [16]

    and Xie, L

    Geng, X. and Xie, L. (2019). Data-driven decision making in power systems with probabilistic guarantees: Theory and applications of chance-constrained optimization. Annual reviews in control , 47:341--363

    2019

  9. [17]

    Gouriéroux, C., Holly, A., and Monfort, A. (1982). Likelihood ratio test, wald test, and kuhn-tucker test in linear models with inequality constraints on the regression parameters. Econometrica , 50:63--80

    1982

  10. [18]

    and Kartvelishvili, V

    Höcker, A. and Kartvelishvili, V. (1996). Svd approach to data unfolding. Nuclear Instruments and Methods in Physics Research A , 372:469--481

    1996

  11. [19]

    and Narayanan, H

    Kannan, R. and Narayanan, H. (2012). Random walks on polytopes and an affine interior point method for linear programming. Mathematics of Operations Research , 37:1 -- 20

    2012

  12. [20]

    Kibzun, A. I. and Kan, Y. S. (1997). Stochastic programming problems with probability and quantile functions. Journal of the Operational Research Society , 48(8):849–849

    1997

  13. [21]

    Koenker, R. (2005). Quantile regression , volume 38. Cambridge university press

    2005

  14. [22]

    and Bassett Jr, G

    Koenker, R. and Bassett Jr, G. (1978). Regression quantiles. Econometrica: journal of the Econometric Society , pages 33--50

    1978

  15. [23]

    and Hallock, K

    Koenker, R. and Hallock, K. F. (2001). Quantile regression. Journal of Economic Perspectives , 15(4):143--156

    2001

  16. [24]

    Kuusela, M. (2016). Uncertainty quantification in unfolding elementary particle spectra at the Large Hadron Collider . PhD thesis, École Polytechnique Fédérale de Lausanne

    2016

  17. [25]

    and Panaretos, V

    Kuusela, M. and Panaretos, V. M. (2015). Statistical unfolding of elementary particle spectra: Empirical bayes estimation and bias-corrected uncertainty quantification. The Annals of Applied Statistics , 9(3):1671--1705

    2015

  18. [26]

    and Stark, P

    Kuusela, M. and Stark, P. B. (2017). Shape-constrained uncertainty quantification in unfolding steeply falling elementary particle spectra. The Annals of Applied Statistics , 11(3):1671--1710

    2017

  19. [27]

    Liu, J., Bowman, K., Meemong, L., et al. (2016). Carbon monitoring system flux estimation and attribution: impact of ACOS - GOSAT X _ CO_2 sampling on the inference of terrestrial biospheric sources and sinks. Tellus B: Chemical and Physical Meteorology , 66

    2016

  20. [28]

    Lovasz, L. (1999). Hit-and-run mixes fast. Mathematical Programming , 86:443 -- 461

    1999

  21. [29]

    and Vempala, S

    Lovasz, L. and Vempala, S. (2006). Hit-and-run from a corner. SIAM Journal on Computing , 35:985 -- 1005

    2006

  22. [30]

    Masserano, L., Dorigo, T., Izbicki, R., Kuusela, M., and Lee, A. (2023). Simulation-based inference with waldo: Confidence regions by leveraging prediction algorithms or posterior estimators for inverse problems. AISTATS

    2023

  23. [31]

    Masserano, L., Shen, A., Doro, M., Dorigo, T., Izbiki, R., and Lee, A. (2024). Classification under nuisance parameters and generalized label shift in likelihood-free inference. arXiv preprint arXiv:2402.05330

    2024 arXiv

  24. [32]

    Meinshausen, N. (2006). Quantile regression forests. Journal of Machine Learning Research , 7:983--999

    2006

  25. [33]

    and Verbeke, G

    Molenberghs, G. and Verbeke, G. (2007). Likelihood ratio, score, and wald tests in a constrained parameter space. The American Statistician , 61(1):22--27

    2007

  26. [34]

    A., Alvey, J., and Weniger, C

    Montel, N. A., Alvey, J., and Weniger, C. (2023). Scalable inference with autoregressive neural ratio estimation. arXiv preprint arXiv:2308:08597v1

    2023

  27. [35]

    Narayanan, H. (2016). Ranomized interior point methods for sampling and optimization. The Annals of Applied Probability , 26:597--641

    2016

  28. [36]

    Ok, E. A. (2007). Real Analysis with Economic Applications . Econometric Society Monographs. Princeton University Press, Princeton, NJ, 1st edition. Includes bibliographical references and index

    2007

  29. [37]

    and Paninski, L

    Pakman, A. and Paninski, L. (2014). Exact hamiltonian monte carlo for truncated multivariate gaussians. Journal of Computational and Graphical Statistics , 23:518 -- 542

    2014

  30. [38]

    Panaretos, V. M. (2016). Statistics for Mathematicians . Springer International Publishing

    2016

  31. [39]

    Patil, P., Kuusela, M., and Hobbs, J. (2022). Objective frequentist uncertainty quantification for atmospheric CO _2 retrievals. SIAM/ASA Journal on Uncertainty Quantification , 10

    2022

  32. [40]

    Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., Vanderplas, J., Passos, A., Cournapeau, D., Brucher, M., Perrot, M., and Duchesnay, E. (2011). Scikit-learn: Machine learning in P ython. Jou...

    2011

  33. [41]

    R., and Wachter, A

    Pena-Ordieres, A., Luedtke, J. R., and Wachter, A. (2020). Solving chance-constrained problems via a smooth sample-based nonlinear approximation. SIAM Journal on Optimization , 30(3):2221--2250

    2020

  34. [42]

    T., and Dykstra, R

    Robertson, T., Wright, F. T., and Dykstra, R. (1988). Order Restricted Statistical Inference . Wiley

    1988

  35. [43]

    Rodgers, C. D. (2000). Inverse Methods for Atmospheric Sounding . World Scientific Publishing

    2000

  36. [44]

    Rust, B. W. and Burrus, W. R. (1972). Mathematical Programming and the Numerical Solution of Linear Equations . American Elsevier

    1972

  37. [45]

    Rust, B. W. and O'Leary, D. P. (1994). Confidence intervals for discrete approximations to ill-posed problems. Journal of Computational and Graphical Statistics , 3(1):67--96

    1994

  38. [46]

    Schmitt, S. (2012). Tunfold, an algorithm for correcting migration effects in high energy physics. Journal of Instrumentation , 7:T10003

    2012

  39. [47]

    Shapiro, A. (1988). Towards a unified theory of inequality constrained testing in multivariate analysis. International Statistical Review/Revue Internationale de Statistique , pages 49--62

    1988

  40. [48]

    Skilling, J. (2004). Nested sampling. AIP Conference Proceedings , 735:295

    2004

  41. [49]

    Smith, R. L. (1984). Efficient monte carlo procedures for generating points uniformly distributed over bounded regions. Operations Research , 32:1296--1308

    1984

  42. [50]

    Stanley, M., Patil, P., and Kuusela, M. (2022). Uncertainty quantification for wide-bin unfolding: one-at-a-time strict bounds and prior-optimized confidence intervals. Journal of Instrumentation , 17

    2022

  43. [51]

    Stark, P. B. (1992a). Inference in infinite-dimensional inverse problems: discretization and duality. Journal of Geophysical Research: Solid Earth , 97(B10):14055--14082

    1992

  44. [52]

    Stark, P. B. (1992b). Inference in infinite-dimensional inverse problems: Discretization and duality. Journal of Geophysical Research , 97(B10):14055--14082

    1992

  45. [53]

    and Christmann, A

    Steinwart, I. and Christmann, A. (2011). Estimating conditional quantiles with the help of the pinball loss. Bernoulli , 117(1):211--225

    2011

  46. [54]

    V., Sears, T

    Takeuchi, I., Le, Q. V., Sears, T. D., and Smola, A. J. (2006). Nonparametric quantile estimation. Journal of Machine Learning , 7:1231--1264

    2006

  47. [55]

    Tenorio, L., Fleck, A., and Moses, K. (2007). Confidence intervals for linear discrete inverse problems with a non-negativity constraint. Inverse problems , 23(2):669

    2007

  48. [56]

    R., Gosmann, J., and Stewart, T

    Voelker, A. R., Gosmann, J., and Stewart, T. C. (2017). Simultaneous confidence intervals for linear estimates of linear functionals. Technical report, Centre for Theoretical Neuroscience

    2017

  49. [57]

    Wolak, F. A. (1987). An exact test for multiple inequality and equality constraints in the linear regression model. Journal of the American Statistical Association , 82:782--793

    1987

  50. [58]

    Wolak, F. A. (1989). Testing inequality constraints in linear economentric models. Journal of Econometrics , 41:205--235

    1989

Pith tools

Reviewed August 9, 2026 · model on record in the stance chip above.