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LikelihoodGeometry: Macaulay2 Package

T0 review · 3 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read This paper presents a Macaulay2 package that constructs the likelihood ideal of a discrete statistical model and computes its ML-degree, with a toric shortcut that cut one example from over an hour to 0.06 seconds.

desk verdict A working Macaulay2 package for likelihood geometry with real toric speedups, but the saturation shortcut is only conditionally correct and MLdegree is underspecified, so referee time is warranted rather than unconditional acceptance. read the letter →

arxiv 2411.11165 v1 pith:WXN2FSHE submitted 2024-11-17 stat.CO math.AC

David Barnhill , John Cobb , Matthew Faust This is my paper · ORCID
classification stat.COmath.AC MSC 62R0114M2513P10
keywords LikelihoodGeometrycorrespondenceML-degreetoricmodelshierarchicallog-linearundirectedgraphicalMacaulay2algebraicstatistics
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 introduces a software package that builds the likelihood correspondence of a discrete algebraic statistical model: the variety whose points pair data vectors with the critical points of the likelihood function, so that every maximum-likelihood problem on the model is captured at once. The package computes the vanishing ideal of this correspondence, the likelihood ideal, and reads off the model's ML-degree. For toric models, it uses a saturation shortcut by the single hyperplane $\sum p_i$, which the paper reports running a rational-normal-scroll example in 0.06 seconds where the generic Lagrange-multiplier algorithm took more than 3600 seconds. It also supplies constructors for toric, hierarchical log-linear, and undirected graphical models, including explicit Gröbner bases for complete and joint independence models.

What carries the argument

The central object is the likelihood correspondence $L_M \subset \mathbb P^p \times \mathbb P^u$, the universal family of critical sets of the log-likelihood; its vanishing ideal, the likelihood ideal, encodes all MLE information. The mechanism that carries the argument is Algorithm 1 for toric models, which saturates the ideal generated by the toric equations and the $2\times2$ minors of $A\cdot M$ by the single hyperplane $\sum p_i$. For non-toric input the package uses the Lagrange-multiplier elimination of Algorithm 6 in [HKS04], and for complete or joint independence graphical models it returns the Gröbner basis from the companion paper [BCF23].

What would settle it

Run computeLC on a toric model for which the single-hyperplane saturation returns a non-prime ideal, then compare that output with the ideal obtained by saturating with the full product $(\sum p_i)(\prod p_i)$; a mismatch would show that the implemented shortcut can produce an incorrect likelihood ideal.

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Extended reading notes

Core claim

The central claim is that likelihood ideals of discrete algebraic statistical models can be constructed in one package by combining a general Lagrange-multiplier elimination for arbitrary ideals with a much faster toric algorithm. For a toric model given by a matrix $A$, Algorithm 1 forms the toric ideal $I_X$, appends the $2\times2$ minors of $A\cdot M$, and saturates the sum only by the probability-sum hyperplane $V(\sum p_i)$, rather than by all coordinate hyperplanes. This shortcut is reported to be orders of magnitude faster than the generic route, and for complete and joint independence graphical models the package returns explicit Gröbner bases for the likelihood ideal. The ML-degree is then obtained as the degree of the fiber of a generic map from the likelihood correspondence down to the model.

Load-bearing premise

The load-bearing assumption is that saturating the toric computation by the single hyperplane $\sum p_i$, instead of by the full product of coordinates, gives the true likelihood ideal; the paper guarantees this only when the saturated output ideal is prime and does not say what the package does when it is not.

Editorial extensions

If this is right

  • Toric models that would previously have been impractical for generic elimination become accessible, since the benchmark example finished in 0.06 seconds instead of over an hour.
  • A single command, computeLC, chooses the applicable algorithm automatically depending on whether the input is a general ideal, a toric variety, or a complete/joint-independence graphical model.
  • For complete and joint independence models, the likelihood ideal has explicit Gröbner-basis generators rather than being the output of a lengthy elimination.
  • The ML-degree computation gives an algebraic route to the number of complex critical points of the likelihood function, complementing numerical approaches.

Reading between the lines

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

  • The reported speed gap suggests that toric models with much larger state spaces than the benchmark are now within reach; the paper does not run such a scaling test.
  • Because correctness of the single-hyperplane saturation is only guaranteed when the saturated ideal is prime, a practical safeguard would be to check primality and, on failure, fall back to saturating by the full product of coordinates.
  • The ML-degree is described only as the fiber of a generic map; checking it against known ML-degrees for rational normal curves would close that specification gap.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper presents LikelihoodGeometry, a Macaulay2 package for constructing the likelihood correspondence and computing the ML-degree of discrete algebraic statistical models. It implements algorithms from the authors' earlier work (BCF23) for toric models, the general Lagrange-multiplier algorithm of Hosten-Khetan-Sturmfels (HKS04), and specialized Gröbner-basis methods for complete and joint independence graphical models. The package is demonstrated on examples including the Hardy-Weinberg curve, rational normal scrolls, hierarchical log-linear models, and undirected graphical models, with a reported dramatic speedup for toric models (0.06 seconds versus over 3600 seconds).

Significance. If correct, the package is a useful and reproducible software contribution to algebraic statistics: it makes the construction of likelihood ideals and ML-degrees accessible for a range of models, implements previously published algorithms, and is the only Macaulay2 package of its kind for discrete models, complementing GraphicalModelsMLE. The reported toric speedup is potentially valuable. However, the correctness guarantee for the headline toric shortcut is explicitly conditional, and the paper does not specify how the implementation handles the non-prime case, so the advertised speedup is not established for the general toric input class.

major comments (3)
  1. [Section 2, Remark 2.1 and Algorithm 1] The central correctness claim for the toric speedup rests on an unverified saturation shortcut. Algorithm 1 as printed saturates by (sum p_i)(prod p_i), but Remark 2.1 states that the implementation saturates only by the single hyperplane sum p_i, and that correctness is guaranteed only if the output is prime. The paper does not describe any primality check, fallback to full saturation, or a proof that the output is prime for the toric models considered. For any toric model where the shortcut output is not prime, computeLC would silently return an ideal possibly containing extraneous components, and the ML-degree computed from it would be unreliable. The 0.06-second versus 3600-second comparison therefore demonstrates only the speed of the shortcut branch, not its correctness for the general toric case. This is a load-bearing gap that must be addressed, either by proving primality of the saturated ideal for all toric inputs, by adding a fallback to full saturation, or by explicitly restricting the claim and documenting the condition under which the shortcut is applied.
  2. [Section 2, paragraph on MLdegree] The description "MLdegree is then implemented by taking the fiber of a generic map from the likelihood correspondence down to the model" is not a fully specified algorithm. It does not state what map is taken, how the generic fiber is constructed and solved, or what correctness conditions apply. Because MLdegree is a headline function of the package, this underspecification prevents a reader from reproducing the computation or verifying that the returned degree is indeed the ML-degree of the input model.
  3. [Section 2, Algorithm 1 versus Remark 2.1] There is a direct inconsistency between the pseudocode in Algorithm 1, which saturates by (sum p_i)(prod p_i), and the implementation description in Remark 2.1, which saturates only by sum p_i. This makes it unclear which algorithm the package actually runs and which correctness statement the user should rely on. The pseudocode should match the code, or the discrepancy should be explained and reconciled.
minor comments (5)
  1. [Abstract] The abstract contains a duplicated word: "a variety that that ties together data" should read "a variety that ties together data". The phrase "a statistcal model" is also a typo for "a statistical model".
  2. [Example 1.1] The displayed ideal output for computeLC on the Hardy-Weinberg curve is hard to read because the monomials are broken across lines without clear alignment. Please reformat the Macaulay2 session output for clarity.
  3. [Section 1.2.3 and Example 1.5] The graph in Example 1.5 is described as "a b c" with an implied edge structure, but the ASCII rendering is ambiguous. A simple figure or explicit edge list would help the reader verify the model.
  4. [Section 2] The reference to the general Lagrange-multiplier algorithm as "[HKS04, Algorithm 6]" is appropriate, but the paper should also state explicitly which parts of the implementation follow BCF23 and which follow HKS04, so that users know the provenance of each branch.
  5. [Data Availability] The footnote pointing to the GitHub repository is helpful; consider adding a version number or commit hash so that the exact code tested in the paper can be identified in the future.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the package implements independently grounded algorithms; the toric saturation caveat in Remark 2.1 is a correctness limitation, not a circular step.

full rationale

The paper's claims are about a Macaulay2 implementation, not a new derivation. computeLC calls Algorithm 1 from the authors' prior work [BCF23] or HKS04's Algorithm 6; the likelihood ideal is constructed by standard operations (toric ideal plus 2x2 minors plus saturation), and MLdegree is read off from a fiber. No fitted parameter is renamed as a prediction, and no equation is defined in terms of its own target. The only flagged limitation is Remark 2.1, which states that the implementation saturates only by V(sum p_i) and that 'correctness is guaranteed if the output is prime.' That is an unverified algorithmic shortcut that could affect the headline timing comparison, but it is not circular: the claimed output does not reduce by construction to the input ideal or to the cited theorem. Self-citations to [BCF23] are used for the toric algorithms, but the package code is runnable and the underlying ideals come from standard toric/Groebner constructions plus external [HKS04], so this is independent support rather than a self-citation chain. No known result is merely renamed. Therefore the paper has no significant circularity.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

No free parameters or invented entities. The central claim depends on standard algorithms from the cited literature and on one unproved sufficiency condition (saturation by the sum hyperplane) flagged by the authors.

assumptions (3)
  • domain assumption Algorithm 6 of HKS04 computes the likelihood ideal for arbitrary ideals via Lagrange multipliers.
    The package implements this algorithm for non-toric models; the text cites HKS04 but does not prove it.
  • domain assumption Theorem 3.2 and Corollary 3.5 of BCF23 give a Grobner basis for complete and joint independence models.
    Used in Algorithm 1 for graphical models whose components are complete graphs.
  • ad hoc to paper Saturation by the single hyperplane sum p_i suffices for the likelihood ideal computation, provided the output is prime.
    Remark 2.1 states correctness is guaranteed only if the output is prime; this is an assumption of the implementation.

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Cite this review

Pith. "Pith review of LikelihoodGeometry: Macaulay2 Package." pith.science (2026). https://pith.science/paper/WXN2FSHE

@misc{pith2026241111165,
  author       = {Pith},
  title        = {Pith review of: LikelihoodGeometry: Macaulay2 Package},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WXN2FSHE}},
  note         = {Machine review of arXiv:2411.11165}
}
abstract

This note introduces the $\texttt{LikelihoodGeometry}$ package for the computer algebra system $\textit{Macaulay2}$. This package gives tools to construct the likelihood correspondence of a discrete algebraic statistical model, a variety that that ties together data and their maximum likelihood estimators. This includes methods for constructing and combining popular statistical models and calculating their ML-degree.

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Reference graph

Works this paper leans on

2 extracted references · 1 canonical work pages

  1. [1]

    Améndola, L

    [AGPH+22] C. Améndola, L. D. García Puente, R. Homs, O. Kuznetsova, and H. J Motwani, Computing maximum likelihood estimates for gaussian graphical model s with macaulay2, Journal of Software for Algebra and Geometry 12 (2022), no. 1, 1–10. ↑1 [AKK19] C. Améndola, D. Kosta, and K. Kubjas, Maximum likelihood estimation of toric fano varieties , Algebraic S...

    arXiv 2022

  2. [2018]

    Tesler, Algebraic Statistics for Computational Biology , Briefings in Bioinformatics 8 (2007), no

    ↑3 [Tes07] G. Tesler, Algebraic Statistics for Computational Biology , Briefings in Bioinformatics 8 (2007), no. 2, 138–139. ↑2, 3 Department of Mathematics, United States Na v al Academy, Annapolis, MD Email address : barnhill@usna.edu Department of Mathematics, Auburn University, Auburn, AL Email address : jdcobb3@gmail.com Department of Mathematics, Mic...

    work page 2007

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