A New Algorithm for Totally Positive Approximations

Lutz Duembgen; Philip Stange

A New Algorithm for Totally Positive Approximations

T0 review · 1 major / 0 minor · reviewed 2026-06-26 · grok-4.3

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Pith's one-line read A new algorithm computes the maximum-likelihood TP2 approximation to any bivariate distribution with finite support.

desk verdict The paper claims a new algorithm for max-likelihood TP2 approximation of bivariate distributions but the abstract supplies no method, analysis, or results to evaluate it. read the letter →

arxiv 2606.22622 v1 pith:KOV2BAGR submitted 2026-06-21 stat.CO

Lutz Duembgen , Philip Stange This is my paper · ORCID

classification stat.CO

keywords totallypositiveTP2maximumlikelihoodapproximationbivariatedistributionalgorithmcontingencytablestatisticalcomputing

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 revisits the task of replacing a given bivariate discrete distribution with the closest TP2 distribution, where closeness is measured by maximum likelihood. TP2 distributions encode a strong form of positive dependence through the inequality p(i,j)p(k,l) >= p(i,l)p(k,j) for i < k and j < l. The authors supply an algorithm intended to solve this constrained optimization problem directly. A reader would care because the resulting approximation preserves monotone dependence while remaining computationally tractable for contingency-table data.

What carries the argument

An iterative procedure that projects onto the TP2 cone while ascending the likelihood surface.

What would settle it

Execute the algorithm on a 3-by-3 contingency table whose global TP2 maximizer can be enumerated by hand or by exhaustive search over the cone and check whether the output matches that known optimum.

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Extended reading notes

Core claim

We introduce a new algorithm that finds the maximum-likelihood estimator inside the cone of totally positive distributions of order two with finite support.

Load-bearing premise

The non-convex maximum-likelihood problem over the TP2 cone admits an efficient, globally convergent algorithm that needs no post-processing or escape from poor local solutions.

Editorial extensions

If this is right

The fitted distribution satisfies all TP2 inequalities by construction.
The procedure yields a regularized estimate that automatically enforces positive quadrant dependence.
It applies directly to observed frequency tables without requiring external smoothing parameters.

Reading between the lines

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

The same iteration might be adapted to other cones defined by pairwise inequalities, such as those arising in shape-constrained estimation.
If the algorithm scales linearly with table size, it would make TP2 modeling routine for moderately large categorical datasets.

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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.

Desk Editor's Note

read the letter

The one thing to know is that this paper claims a new algorithm for the max-likelihood approximation of bivariate distributions by TP2 ones, but the abstract alone does not let us check if the algorithm is real or reliable.

They do a decent job of stating the problem in clear terms. The task is to find a TP2 distribution that maximizes the likelihood for approximating a given bivariate one with finite support. That framing is straightforward and matches what the reader highlighted as the strongest claim.

What is new is the algorithm itself, according to the title and abstract. The paper positions it as a revisit with a new method. If the full text delivers a working procedure with some guarantee, that could be useful for people doing bivariate modeling where TP2 structure is desired.

The soft spots are obvious and central. There is no description of the algorithm, no mention of how it handles the likely non-convex optimization, no convergence analysis, and no empirical results. The stress-test note correctly flags the missing global optimality guarantee. For a non-convex problem over the TP2 cone, a local method could easily get stuck, and without evidence against that, the claim is unverified. The soundness score of 2.0 from the reader matches what the abstract shows.

This is for a narrow group of statisticians working on totally positive distributions or shape-constrained estimation. A reader outside that niche gets no value. It does not look like it deserves peer review yet because the central claim cannot be evaluated. I would not bring it to a reading group or cite it based on this.

Recommendation: desk reject or ask for a revision with the actual method and tests before sending to referees.

Referee Report

1 major / 0 minor

Summary. The manuscript revisits the maximum-likelihood approximation of a bivariate finite-support distribution by a totally positive of order two (TP2) distribution and presents a new algorithm for solving this optimization problem over the TP2 cone.

Significance. A reliable algorithm for this constrained approximation problem would be useful in statistical modeling of positive dependence structures. However, the absence of any derivation, convergence analysis, or empirical validation in the provided material prevents assessment of whether the claimed advance is substantive.

major comments (1)

The central claim is the existence of a new algorithm that solves the ML problem over the TP2 cone. Because the likelihood is typically non-convex in the probability masses while the constraint set is convex, any local solver risks returning stationary points that are not globally optimal; no section supplies a global-convergence argument, convex reformulation, or exhaustive-search guarantee.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their report and the opportunity to respond. We address the major comment below.

read point-by-point responses

Referee: The central claim is the existence of a new algorithm that solves the ML problem over the TP2 cone. Because the likelihood is typically non-convex in the probability masses while the constraint set is convex, any local solver risks returning stationary points that are not globally optimal; no section supplies a global-convergence argument, convex reformulation, or exhaustive-search guarantee.

Authors: We agree that the likelihood is non-convex while the TP2 constraint set is convex, so the problem is non-convex and our algorithm is a local solver without a global convergence guarantee, convex reformulation, or exhaustive-search property. The manuscript does not claim global optimality. Its contribution is an efficient procedure for handling the TP2 constraints within an iterative scheme that yields good practical approximations, supported by the derivation of the update steps. We will revise the text to explicitly acknowledge the local character of the method and to discuss the non-convexity limitation. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity detected; derivation self-contained

full rationale

The provided abstract and context describe a new algorithm for maximum-likelihood approximation of bivariate finite-support distributions onto the TP2 cone. No equations, fitted parameters, self-citations, or derivation steps are exhibited that reduce a claimed prediction or result to an input by construction. The central claim is the existence of an algorithm solving the stated optimization problem; absent any visible self-definitional, fitted-input, or self-citation load-bearing elements, the work does not exhibit circularity. This matches the default expectation for papers without such reductions.

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

Abstract supplies no explicit free parameters, axioms, or invented entities; ledger left empty pending full text.

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Cite this review

Pith. "Pith review of A New Algorithm for Totally Positive Approximations." pith.science (2026). https://pith.science/paper/KOV2BAGR

@misc{pith2026260622622,
  author       = {Pith},
  title        = {Pith review of: A New Algorithm for Totally Positive Approximations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KOV2BAGR}},
  note         = {Machine review of arXiv:2606.22622}
}

read the original abstract

We revisit the problem of approximating a bivariate distribution with finite support by another such distribution which is totally positive or order two (TP2). Approximation is meant in a maximum likelihood sense.

Figures

Figures reproduced from arXiv: 2606.22622 by the authors.

**Figure 2.** Candidate f after one and seven pre-iterations. 10 20 30 40 5 10 15 20 25 30 Iteration 1: L = 159.562 i j 10 20 30 40 5 10 15 20 25 30 Iteration 2: L = 155.415 i j 10 20 30 40 5 10 15 20 25 30 Iteration 3: L = 154.74 i j 10 20 30 40 5 10 15 20 25 30 Final fit: L =154.671 i j [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗

**Figure 3.** Candidate f after one, two, three and eight iterations. 8 [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

3 extracted references

[1]

o sching, A. and D\

M\" o sching, A. and D\" u mbgen, L. (2024). Estimation of a likelihood ratio ordered family of distributions. Stat. Comput. 34 Paper No. 58

2024
[2]

Owen, A. B. (2001). Empirical Likelihood. Chapman and Hall/CRC, New York

2001
[3]

, Wright, F

Robertson, T. , Wright, F. T. and Dykstra, R. L. (1988). Order restricted statistical inference. John Wiley & Sons, Ltd., Chichester

1988

Pith tools

Reviewed June 26, 2026 · model on record in the stance chip above.

[1] [1]

o sching, A. and D\

M\" o sching, A. and D\" u mbgen, L. (2024). Estimation of a likelihood ratio ordered family of distributions. Stat. Comput. 34 Paper No. 58

2024

[2] [2]

Owen, A. B. (2001). Empirical Likelihood. Chapman and Hall/CRC, New York

2001

[3] [3]

, Wright, F

Robertson, T. , Wright, F. T. and Dykstra, R. L. (1988). Order restricted statistical inference. John Wiley & Sons, Ltd., Chichester

1988