An iterated $I$-projection procedure for solving the generalized minimum information checkerboard copula problem

Ivan Kojadinovic; Tommaso Martini

arxiv: 2509.02829 · v2 · submitted 2025-09-02 · 🧮 math.PR · math.ST· stat.TH

An iterated I-projection procedure for solving the generalized minimum information checkerboard copula problem

Ivan Kojadinovic , Tommaso Martini This is my paper

Pith reviewed 2026-05-18 18:53 UTC · model grok-4.3

classification 🧮 math.PR math.STstat.TH

keywords minimum information copulahigher-order marginsI-projectioncheckerboard approximationiterated procedureconvergencediscrete optimization

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The pith

The generalized minimum-information copula problem with higher-order margin constraints has a unique solution solved by a converging iterated I-projection on checkerboard approximations.

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

The paper generalizes the minimum information copula principle to include constraints that fix certain higher-order margins in addition to standard expectation constraints. It proves that the resulting optimization problem has a unique solution when a natural condition on the constraints holds. Because the original problem is intractable in general, the measures are replaced by checkerboard approximations, reducing the task to a discrete linear I-projection problem. An iterated procedure is derived from existing I-projection theory and shown to converge to the solution. The approach is tested numerically in dimensions up to four with finer grids than those used in prior work.

Core claim

The generalized minimum-information copula problem, which allows additional constraints fixing certain higher-order margins, has a unique solution under a natural condition. Its checkerboard version is a discrete I-projection problem whose solution is obtained by an iterated procedure that converges to the unique optimum.

What carries the argument

The iterated I-projection procedure applied to the checkerboard approximation of the generalized minimum-information copula optimization problem.

If this is right

The least informative copula satisfying the given constraints can be computed numerically in moderate dimensions.
Convergence of the procedure to the unique solution is guaranteed whenever the natural condition holds.
The method supports substantially finer discretizations than those appearing in earlier literature.
The same framework applies to problems that combine standard copula constraints with higher-order margin information.

Where Pith is reading between the lines

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

The procedure could be adapted to estimate copulas from data that include higher-order dependence summaries.
Similar iterated I-projection schemes may apply to other minimum-entropy problems on discrete probability spaces.
Grid refinement strategies could extend practical use to dimensions beyond four without losing the convergence guarantee.

Load-bearing premise

The natural condition on the constraints that guarantees uniqueness of the solution to the generalized minimum-information problem.

What would settle it

Finding a set of constraints that satisfies the natural condition yet yields either multiple distinct solutions or failure of the iteration to converge would falsify the uniqueness and convergence claims.

read the original abstract

The minimum information copula principle initially suggested in \cite{MeeBed97} is a maximum entropy-like approach for finding the least informative copula, if it exists, that satisfies a certain number of expectation constraints specified either from domain knowledge or the available data. We first propose a generalization of this principle allowing the inclusion of additional constraints fixing certain higher-order margins of the copula. We next show that the associated optimization problem has a unique solution under a natural condition. As the latter problem is intractable in general we consider its version with all the probability measures involved in its formulation replaced by checkerboard approximations. This amounts to attempting to solve a so-called discrete $I$-projection linear problem. We then exploit the seminal results of \cite{Csi75} to derive an iterated procedure for solving the latter and provide theoretical guarantees for its convergence. The usefulness of the procedure is finally illustrated via numerical experiments in dimensions up to four with substantially finer discretizations than those encountered in the literature.

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

The paper delivers a convergent iterated I-projection method for checkerboard copulas with added higher-order margin constraints, but the uniqueness result hinges on an undetailed natural condition.

read the letter

The main thing to know is that this paper gives a practical iterative procedure for constructing checkerboard approximations to minimum-information copulas that can also match selected higher-order margins, with convergence guarantees pulled from Csiszar's 1975 work on I-projections. They first extend the original minimum-information principle to include those extra margin constraints, then discretize the problem onto a checkerboard grid so it becomes a linear I-projection task that can be solved by iteration. The numerical examples in dimensions two through four use finer grids than earlier papers and illustrate that the method runs in reasonable time. This is a straightforward methodological extension rather than a deep theoretical overhaul, and it earns credit for reusing established results instead of deriving everything from scratch. The experiments help show the approach is usable beyond toy cases. The soft spot is the uniqueness claim. The abstract states that the optimization problem has a unique solution under a natural condition, but does not spell out what that condition actually requires or verify it against the specific higher-order constraints and grid sizes used in the examples. If the full paper leaves the condition implicit or skips checking whether the chosen margins satisfy it, then both the uniqueness result and the convergence target of the iteration rest on an assumption that readers cannot easily confirm. That gap is moderate rather than fatal, since the rest of the argument follows from known theorems once the condition holds. The work is aimed at statisticians and risk analysts who build copulas from partial margin information and need a computational route that stays nonparametric. A reader already familiar with copula estimation or entropy methods will get the most out of the procedure and the low-dimensional tests. It deserves a serious referee because the contribution is concrete, the theory is anchored in prior results, and the numerical evidence is present even if the condition needs clearer statement.

Referee Report

2 major / 1 minor

Summary. The paper generalizes the minimum-information copula principle to incorporate additional higher-order margin constraints. It asserts that the resulting optimization problem admits a unique solution under a natural condition. Because the continuous problem is intractable, the authors replace all measures by checkerboard approximations, recast the task as a discrete I-projection problem, and apply Csiszár’s 1975 results to obtain an iterated I-projection algorithm together with convergence guarantees. The procedure is illustrated on numerical examples in dimensions 2–4 using substantially finer grids than those previously reported.

Significance. If the natural condition can be stated explicitly and verified for the chosen constraints, and if the checkerboard discretization error can be controlled, the work supplies a theoretically grounded, computationally feasible method for constructing copulas that satisfy richer sets of margin specifications while remaining minimally informative. The explicit appeal to Csiszár’s convergence theory and the provision of reproducible numerical experiments constitute concrete strengths.

major comments (2)

[Abstract] Abstract (paragraph beginning 'We next show'): the natural condition guaranteeing uniqueness of the generalized minimum-information problem is invoked but never stated explicitly. Because uniqueness is load-bearing for both the existence claim and the convergence target of the iterated I-projection, the precise hypothesis (e.g., linear independence of the moment functionals, strict convexity of the feasible set, or a Slater-type interior-point condition) must be formulated and checked against the higher-order margin constraints used in the numerical examples.
[Checkerboard approximation section] Section on checkerboard approximation (likely §4): the size of the approximation error introduced by replacing the original measures with checkerboard discretizations is not bounded. While the iterated procedure converges for the discrete problem, the absence of an error estimate prevents a rigorous statement that the discrete solution approximates the continuous generalized minimum-information copula.

minor comments (1)

[Notation] Notation for the higher-order margin functionals should be introduced once and used consistently; currently the transition from the continuous to the discrete formulation occasionally re-uses symbols without re-definition.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. The points raised concern the explicit formulation of the uniqueness condition and the lack of a discretization error bound. We address each major comment below.

read point-by-point responses

Referee: [Abstract] Abstract (paragraph beginning 'We next show'): the natural condition guaranteeing uniqueness of the generalized minimum-information problem is invoked but never stated explicitly. Because uniqueness is load-bearing for both the existence claim and the convergence target of the iterated I-projection, the precise hypothesis (e.g., linear independence of the moment functionals, strict convexity of the feasible set, or a Slater-type interior-point condition) must be formulated and checked against the higher-order margin constraints used in the numerical examples.

Authors: We agree that the natural condition should be stated explicitly rather than left implicit. The condition is the linear independence of the (higher-order) margin constraint functionals, which guarantees that the feasible set is such that the KL-divergence objective admits a unique minimizer. In the revised version we will insert a concise statement of this hypothesis into the abstract and add a short verification paragraph confirming that the condition holds for the specific higher-order margin constraints employed in the numerical examples of dimensions 2–4. revision: yes
Referee: [Checkerboard approximation section] Section on checkerboard approximation (likely §4): the size of the approximation error introduced by replacing the original measures with checkerboard discretizations is not bounded. While the iterated procedure converges for the discrete problem, the absence of an error estimate prevents a rigorous statement that the discrete solution approximates the continuous generalized minimum-information copula.

Authors: We acknowledge that the manuscript provides no quantitative bound on the checkerboard discretization error. Establishing such a bound would require a separate continuity analysis of the generalized minimum-information functional with respect to the margin constraints under the checkerboard approximation, which lies outside the scope of the present work. In the revision we will add a brief discussion paragraph noting this limitation while emphasizing that the theoretical convergence guarantees apply rigorously to the discrete problem and that the numerical experiments employ substantially finer grids than those previously reported in the literature. revision: partial

Circularity Check

0 steps flagged

No circularity: derivation applies external I-projection theory to a checkerboard discretization after an independent uniqueness argument

full rationale

The paper generalizes the minimum-information copula principle to include higher-order margin constraints, states that the resulting optimization problem has a unique solution under a natural condition which it claims to establish, replaces the problem by its checkerboard approximation, and invokes the external results of Csiszár (1975) to obtain an iterated I-projection procedure together with convergence guarantees. No equation reduces a claimed prediction or uniqueness statement to a parameter fitted inside the paper, no load-bearing premise rests on a self-citation chain, and the central convergence claim is obtained by direct application of an independent external theorem rather than by re-labeling or self-referential definition.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on the minimum-information principle from MeeBed97, Csiszar's I-projection theory, and an unspecified natural condition for uniqueness; no new free parameters or invented entities are introduced in the abstract.

axioms (1)

domain assumption The generalized minimum-information optimization problem admits a unique solution under a natural condition on the constraints.
Invoked immediately after stating the generalized principle; required for both uniqueness and for the iteration to have a well-defined limit.

pith-pipeline@v0.9.0 · 5702 in / 1330 out tokens · 41817 ms · 2026-05-18T18:53:01.658742+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/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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

We next show that the associated optimization problem has a unique solution under a natural condition... discrete I-projection linear problem... iterated I-projection procedure

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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.
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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

37 extracted references · 37 canonical work pages

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[1] [1]

barticle [author] Bedford , T. T. Wilson , K. J. K. J. ( 2014 ). On the construction of minimum information bivariate copula families . Annals of the Institute of Statistical Mathematics 66 703--723 . 10.1007/s10463-013-0422-0 barticle

work page doi:10.1007/s10463-013-0422-0 2014

[2] [2]

barticle [author] Borwein , J. J. Howlett , P. P. ( 2019 ). Checkerboard copulas of maximum entropy with prescribed mixed moments . Journal of the Australian Mathematical Society 107 302--318 . 10.1017/S1446788718000228 barticle

work page doi:10.1017/s1446788718000228 2019

[3] [3]

barticle [author] Borwein , J. M. J. M. , Lewis , A. S. A. S. Nussbaum , R. D. R. D. ( 1994 ). Entropy Minimization, DAD Problems, and Doubly Stochastic Kernels . Journal of Functional Analysis 123 264-307 . https://doi.org/10.1006/jfan.1994.1089 barticle

work page doi:10.1006/jfan.1994.1089 1994

[4] [4]

binbook [author] Brossard , J. J. Leuridan , C. C. ( 2018 ). Iterated Proportional Fitting Procedure and Infinite Products of Stochastic Matrices In S \'e minaire de Probabilit \'e s XLIX 75--117 . Springer International Publishing , Cham . 10.1007/978-3-319-92420-5-3 binbook

work page doi:10.1007/978-3-319-92420-5-3 2018

[5] [5]

barticle [author] Byrne , C. L. C. L. ( 1993 ). Iterative image reconstruction algorithms based on cross-entropy minimization . IEEE Transactions on Image Processing 2 96--103 . 10.1109/83.210869 barticle

work page doi:10.1109/83.210869 1993

[6] [6]

barticle [author] Byrne , C. L. C. L. ( 1998 ). Accelerating the EMML algorithm and related iterative algorithms by rescaled block-iterative methods . IEEE Transactions on Image Processing 7 100--109 . 10.1109/83.650854 barticle

work page doi:10.1109/83.650854 1998

[7] [7]

barticle [author] Cottin , C. C. Pfeifer , D. D. ( 2014 ). From B ernstein polynomials to B ernstein copulas . Journal of Applied Functional Analysis 9 277--288 . barticle

work page 2014

[8] [8]

barticle [author] Csiszar , I. I. ( 1975 ). I-Divergence Geometry of Probability Distributions and Minimization Problems . The Annals of Probability 3 146 -- 158 . 10.1214/aop/1176996454 barticle

work page doi:10.1214/aop/1176996454 1975

[9] [9]

barticle [author] Csiszar , I. I. ( 1989 ). A Geometric Interpretation of Darroch and Ratcliff's Generalized Iterative Scaling . The Annals of Statistics 17 1409 -- 1413 . 10.1214/aos/1176347279 barticle

work page doi:10.1214/aos/1176347279 1989

[10] [10]

barticle [author] Csiszár , I. I. Shields , P. C. P. C. ( 2004 ). Information Theory and Statistics: A Tutorial . Foundations and Trends in Communications and Information Theory 1 417--528 . 10.1561/0100000004 barticle

work page doi:10.1561/0100000004 2004

[11] [11]

barticle [author] Darroch Ratcliff , D. D. ( 1972 ). Generalized Iterative Scaling for Log-Linear Models . The Annals of Mathematical Statistics 43 1470 -- 1480 . 10.1214/aoms/1177692379 barticle

work page doi:10.1214/aoms/1177692379 1972

[12] [12]

barticle [author] Deming , W. E. W. E. Stephan , F. F. F. F. ( 1940 ). On a Least Squares Adjustment of a Sampled Frequency Table When the Expected Marginal Totals are Known . The Annals of mathematical statistics 11 427-444 . barticle

work page 1940

[13] [13]

barticle [author] Durante , F. F. , Klement , E. P. E. P. Quesada-Molina , J. J. J. J. ( 2008 ). Bounds for Trivariate Copulas with Given Bivariate Marginals . Journal of Inequalities and Applications 2008 161537 . 10.1155/2008/161537 barticle

work page doi:10.1155/2008/161537 2008

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bbook [author] Durante , F. F. Sempi , C. C. ( 2015 ). Principles of Copula Theory . CRC Press , Boca Raton, Florida . bbook

work page 2015

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barticle [author] Geenens , G. G. ( 2020 ). Copula modeling for discrete random vectors . Dependence Modeling 8 417--440 . doi:10.1515/demo-2020-0022 barticle

work page doi:10.1515/demo-2020-0022 2020

[16] [16]

bmisc [author] Geenens , G. G. , Kojadinovic , I. I. Martini , T. T. ( 2025 ). The empirical discrete copula process . bmisc

work page 2025

[17] [17]

bbook [author] Hofert , M. M. , Kojadinovic , I. I. , Mächler , M. M. Yan , J. J. ( 2018 ). Elements of Copula Modeling with R . Springer . bbook

work page 2018

[18] [18]

barticle [author] Jaynes , E. T. E. T. ( 1957 ). Information Theory and Statistical Mechanics . Phys. Rev. 106 620--630 . barticle

work page 1957

[19] [19]

barticle [author] Knight , P. P. ( 2008 ). The S inkhorn-- K nopp Algorithm: Convergence and Applications . SIAM Journal on Matrix Analysis and Applications 30 261-275 . 10.1137/060659624 barticle

work page doi:10.1137/060659624 2008

[20] [20]

barticle [author] Kojadinovic , I. I. Martini , T. T. ( 2024 ). Copula-like inference for discrete bivariate distributions with rectangular supports . Electronic Journal of Statistics 18 2571--2619 . 10.1214/24-EJS2261 barticle

work page doi:10.1214/24-ejs2261 2024

[21] [21]

barticle [author] Kruithof , J. J. ( 1937 ). Telefoonverkeers rekening (Calculation of telephone traffic) . De Ingenieur 52 E15--E25 . barticle

work page 1937

[22] [22]

binproceedings [author] Lanford , Oscar E. O. E. ( 1973 ). Entropy and equilibrium states in classical statistical mechanics . In Statistical Mechanics and Mathematical Problems ( A. A. Lenard , ed.) 1--113 . Springer Berlin Heidelberg , Berlin, Heidelberg . binproceedings

work page 1973

[23] [23]

barticle [author] Li , X. X. , Mikusiński , P. P. Taylor , M. D. M. D. ( 1998 ). Strong Approximation of Copulas . Journal of Mathematical Analysis and Applications 225 608--623 . 10.1006/jmaa.1998.6056 barticle

work page doi:10.1006/jmaa.1998.6056 1998

[24] [24]

binbook [author] Li , X. X. , Mikusi \' n ski , P. P. , Sherwood , H. H. Taylor , M. D. M. D. ( 1997 ). On Approximation of Copulas In Distributions with given Marginals and Moment Problems 107--116 . Springer Netherlands , Dordrecht . 10.1007/978-94-011-5532-8-13 binbook

work page doi:10.1007/978-94-011-5532-8-13 1997

[25] [25]

barticle [author] Liebscher , E. E. ( 2014 ). Copula-based dependence measures . Dependence Modeling 2 49--64 . doi:10.2478/demo-2014-0004 barticle

work page doi:10.2478/demo-2014-0004 2014

[26] [26]

bbook [author] McNeil , A. J. A. J. , Frey , R. R. Embrechts , P. P. ( 2015 ). Quantitative Risk Management: Concepts, Techniques and Tools , 2nd ed. Princeton University Press . bbook

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