Statistics and tropicalization of local field Gaussian measures

Ngoc Mai Tran; Yassine El Maazouz

REVIEW 3 major objections 4 minor 27 references

Statistics and tropicalization of local field Gaussian measures

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Local-field Gaussians admit a complete statistical theory: closed-form MLE, matroidal conditional independence, and a supermodular tropical bijection in dimension two.

desk verdict Genuinely opens a subfield of p-adic statistics; the tropicalization bijection in Theorem 1.3 is misstated without an integrality condition, but the intended result and most of the paper are solid. read the letter →

arxiv 1909.00559 v2 pith:ROH2IARJ submitted 2019-09-02 math.ST stat.TH

Yassine El Maazouz , Ngoc Mai Tran This is my paper · ORCID

classification math.STstat.TH MSC 62H0560E0512J2514T90

keywords localfieldsp-adicGaussianmeasurestropicalizationmaximumlikelihoodconditionalindependenceBruhat-Titsbuildingsupermodularfunctionsmatroids

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 lays foundations for statistics over local fields, such as the p-adic numbers, where Gaussian measures are uniform probability measures on lattices. It establishes that maximum likelihood estimation has a closed-form solution, that conditional independence among coordinates is governed by a matroid over the residue field, and that in dimension two the valuation, or tropicalization, of a Gaussian is exactly a tropical polynomial with supermodular coefficients on the unit square. These results matter because they reduce statistical questions over exotic fields to concrete discrete geometry: lattices, matroids, and piecewise-linear functions.

What carries the argument

The load-bearing object is the lattice $L \subset K^d$, the analogue of the covariance matrix: every non-degenerate Gaussian is uniform Haar measure on $L$. Three tools carry the argument. The Hermite normal form writes any lattice as the image of $O_K^d$ under a lower-triangular matrix with diagonal entries $\pi^a, \pi^b$ and off-diagonal entry $\pi^c x$, giving canonical integer parameters. The valuation map $\mathrm{val}: K^d \to \mathbb{Z}^d$ tropicalizes the measure, and the tail probability $P(V \ge v)$ counts lattice points in $L \cap \pi^v O_K^d$. The proof of Theorem 1.3 conditions on $V_1 \ge v_1$, uses the conditional law on an explicit sublattice $L_{v_1}$, and reads off a tail formula that matches the tropical polynomial; the Bruhat-Tits building of $\mathrm{SL}_d(K)$ is introduced as the parameter space counterpart of the positive-semidefinite cone.

What would settle it

Take $K=\mathbb{Q}_2$ and the lattice $L$ generated by $(1,1)$ and $(0,2)$ over $\mathbb{Z}_2$ (Hermite form with $a=0,b=1,c=0$). Enumerate the points of $L$ with $\mathrm{val}(X_1)\ge 0$ and count the fraction with $\mathrm{val}(X_2)\ge v_2$ for $v_2=0,1,2$; if the fractions are not $q^{-\max(v_2-\min(b,c+\max(v_1-a,0)),0)}$ with $v_1=0$, the asserted conditional law is false.

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Extended reading notes

Core claim

The central claim is a bijection in dimension two. For a non-degenerate Gaussian $X$ in $K^2$ with lattice $L$, define $\phi_L(v) = -\log_q P(V \ge v)$ for $v \in \mathbb{Z}^2$. The paper proves that $\phi_L$ equals the restriction to $\mathbb{Z}^2$ of the tropical polynomial $P(v) = \max(0, v_1-a, v_2-c, v_1+v_2-a-b)$, and that every polynomial of this form with $c_{00}=0$ and $c_{00}+c_{11} \ge c_{01}+c_{10}$ arises from a lattice. The same framework yields Theorem 1.1, that the MLE lattice is the $O_K$-span of the data, and Theorem 1.2, that conditional independence statements are exactly bases of an $\mathbb{F}_q$-realizable matroid.

Load-bearing premise

The bijection in Theorem 1.3 rests on the unproved assertion that, conditional on $V_1 \ge v_1$, the Gaussian is uniform on the explicit sublattice $L_{v_1}$; if that conditional law were not uniform, the tail formula and the bijection would fail.

Editorial extensions

If this is right

Maximum likelihood estimation over local fields is solved in closed form: the fitted lattice is the $O_K$-span of the data, and when the data has full rank it is the unique full-dimensional maximizer.
Conditional independence among coordinates of a $K$-Gaussian is a matroidal property: the maximal conditionally independent sets form the bases of a matroid representable over the residue field $\mathbb{F}_q$, so checking independence reduces to linear algebra over a finite field.
In dimension two the family of tropicalized Gaussians coincides with the family of tropical polynomials supported on the unit square with supermodular coefficients; the case $c=b$ gives independent coordinates and a product of shifted geometric distributions for the valuation vector.
The Bruhat-Tits building gives a geometric parameter space for non-degenerate Gaussians, with adjacency described by explicit matrix conditions, so statistical models such as conditional-independence families can be studied as subsets of the building.
If the paper's Conjecture 5.1 is correct, the same supermodular-polynomial picture holds in every dimension $d$, giving a finite parametrization of $K$-Gaussians by $2^d - 1$ integer coefficients.

Reading between the lines

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

A direct corollary of the d=2 bijection that the paper leaves implicit: the tropicalized Gaussian family has exactly three integer parameters $(a,b,c)$ with $c \le b$, so fitting to valuation-only data is a finite combinatorial search rather than a continuous optimization problem.
Because the Bruhat-Tits building $B_2(K)$ is a tree, the Gaussian models in $K^2$ sit on a tree; asking whether each conditional-independence model $M_{I,J}$ is a geodesically convex subtree would connect Theorems 1.2 and 1.3 and could lead to fast model-selection algorithms.
The proof idea of conditioning on one coordinate at a time suggests a recursive algorithm for $\phi_L$ in higher dimensions based on the independence lattice $\mathrm{ind}(L)$; testing Conjecture 5.1 by computing $\phi_L$ on random lattices for $d=3$ would be a direct computational check.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The short version: this paper deserves a serious referee. It opens a new subarea — statistics for local-field Gaussians — and the main results are mostly right. But Theorem 1.3, the headline characterization, is misstated: the coefficients c10, c01, c11 in (1.1) have to be integers for the converse to hold. With real coefficients the 'if' direction is false; a tropical polynomial like max(0, v1-1/2, v2-1/2, v1+v2-1) is supermodular but its restriction to Z^2 is not integer-valued and no lattice tail distribution takes non-integer exponents. The proof only constructs a lattice when the coefficients are integers, and the authors' own Conjecture 5.1 says 'integer supermodular coefficients'. So this is a statement-level bug, not a deep flaw: fix Theorem 1.3 to require integer coefficients (or integer-valued phi) and the bijection goes through.

What is actually new: Theorem 1.1 gives the MLE lattice as span_{O_K}(X), with a clean proof using the minimal Haar measure; Theorem 1.2 shows conditional independence is encoded by an F_q-realizable matroid, which is a genuinely non-archimedean phenomenon (there is no analogue over R); the tropicalization bijection in dimension 2, once corrected, is a nice exact result. The paper also introduces the Bruhat-Tits building as a parameter space, with explicit adjacency computations. The proofs of 1.1 and 1.2 are complete; the SVD/Smith normal form background is used properly.

Soft spots: Theorem 1.2 is stated without the full-rank assumption; the proof requires Hermite normal form for a lattice of full rank d. The degenerate case may be salvageable by conditioning on the support subspace, but the theorem as printed promises more than is proved. The conditioning step in the proof of Theorem 1.3 — 'the conditional distribution of X given V1 >= v1 is Gaussian on the sublattice L_{v1}' — is asserted in one sentence. It is actually true; you can verify it by writing X = A Y and conditioning on Y1. But it should be spelled out; it's the most subtle step in the paper. The paper's citation of [Tra18] for d=1 is fair; the d=2 case is new. I don't see a circularity problem.

Who is this for? Anyone working in p-adic probability, ultrametric analysis, or tropical statistics. It is a foundation paper; the significance is contained but real. It should go to peer review, and the referee report will need to insist on the integrality fix and the full-rank qualification. After that, I'd take Theorem 1.3 as correct.

Referee Report

3 major / 4 minor

Summary. The paper develops foundations for statistical inference for Gaussian measures over non-Archimedean local fields, using lattices as the analogue of covariance matrices. Its main results are: (1) uniqueness of the maximum likelihood lattice for full-rank data (Theorem 1.1); (2) representation of conditional independence among coordinates of a local-field Gaussian vector by an F_q-realizable matroid (Theorem 1.2); (3) a bijection, in dimension two, between tropicalizations of Gaussian measures and supermodular functions on the discrete cube (Theorem 1.3); and (4) a discussion of the Bruhat-Tits building as a parameter space for Gaussian measures, with conditional-independence models formulated as an open problem. The proofs rely on Evans' characterization of local-field Gaussians, van Rooij's orthonormality criterion, and Smith normal form / Hermite normal form decompositions of lattices.

Significance. The paper addresses a novel and worthwhile topic: statistics over local fields has not been systematically developed, and the lattice-based perspective is natural and potentially influential. The MLE and conditional-independence characterizations are clean and, modulo the qualification discussed below, largely supported by the proofs. The proposed bijection between tropicalized two-dimensional Gaussians and supermodular functions is an appealing bridge between probability and tropical geometry, and the explicit use of Hermite normal form and Bruhat-Tits buildings gives the paper concrete computational content. However, the statement of Theorem 1.3 is currently false without an integrality condition, which is a load-bearing error in the headline result. The conditional-distribution step in its proof also needs to be justified explicitly. With these corrections and a suitable non-degeneracy assumption for Theorem 1.2, the paper would make a solid contribution.

major comments (3)

[Section 3.2, Theorem 1.3] Theorem 1.3 as stated asserts that every real supermodular tropical polynomial of the form (1.1) with c00=0 and c00+c11 >= c01+c10 restricts to phi_L for some lattice L. This is false. Lemma 3.2 shows that phi_L is integer-valued, but P_0(v)=max(0, v1-1/2, v2-1/2, v1+v2-1) satisfies the stated inequalities while P_0(1,0)=1/2, so its restriction to Z^2 is not integer-valued and cannot equal -log_q P(V>=v) for any Gaussian lattice. The correct statement requires integer coefficients, exactly as given in Conjecture 5.1. The theorem as printed overstates the bijection and must be corrected.
[Section 3.2, proof of Theorem 1.3] The proof depends on the assertion that the conditional distribution of X given V1>=v1 is the Gaussian distribution on the sublattice L_{v1} displayed in the proof. This step is asserted without proof, and it is the basis for the tail formula for V2 and hence for the full expression of phi_L. The assertion is in fact plausible: the event V1>=v1 forces the first lattice coordinate into pi^{max(v1-a,0)} O_K, producing L_{v1}; however, the paper should include this argument explicitly, since the bijection in Theorem 1.3 fails if this conditional law is not exact.
[Theorem 1.2] Theorem 1.2 is stated for an arbitrary Gaussian vector X in K^d, but the proof invokes Lemma 2.2 to obtain a Hermite normal form for the support lattice L, and Lemma 2.2 requires L to have full rank d. For a degenerate Gaussian, the theorem needs either an explicit non-degeneracy assumption or a separate reduction to the support subspace. As written, the statement and proof do not match on this point.

minor comments (4)

[Section 3.2, proof of Theorem 1.3] The text reads 'm01 = c' instead of 'c01 = c' when reparametrizing the coefficients of the tropical polynomial.
[Example 3.1] The displayed conditional independence statements list 'X3 independence X4 | X1' twice; the third statement should presumably be 'X2 independence X3 | X1'.
[Section 4, Proposition 4.2] The condition 'not all epsilon_i are 0 or 1' is ambiguous; it should read 'not all epsilon_i are 0 and not all epsilon_i are 1'.
[Throughout] There are several typographical errors, including 'semidefinite', 'simlimarity', and 'rader', which should be corrected in a final revision.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity; the main results are derived from external benchmark theorems and a direct lattice computation, with the sole self-citation used as background.

full rationale

The derivation chain is self-contained relative to external results. Theorem 1.1 follows from the measure formula λ(L)=q^{-val(det(A))} and inclusion of lattices, both consequences of Evans' characterization and Smith normal form. Theorem 1.2 is a direct application of Evans' independence-orthogonality lemma and van Rooij's mod-π linear independence criterion. Theorem 1.3 is a direct computation from the Hermite normal form of the lattice; the tail function is read off the valuation of the slice L ∩ π^v O_K^2 via Lemma 3.1, and the converse constructs a lattice from the integer coefficients, as stated in Conjecture 5.1. The only self-citation ([Tra18], shifted geometric distributions for d=1) appears in the motivation and is not used in any proof. The paper does not fit parameters to data and then rename the fit as a prediction; nor does it import a uniqueness theorem from the authors' prior work. The statement-level integrality gap in Theorem 1.3 (real coefficients violate the 'if' direction) and the unproved conditional-distribution assertion in Section 3.2 are correctness concerns, not circular reductions.

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

The paper contributes no fitted constants or invented entities. It relies on five imported or standard mathematical facts, all from prior literature; these are the axioms the reader must accept. The most domain-specific is Evans' characterization of local-field Gaussians as uniform measures on lattices.

assumptions (5)

domain assumption Gaussian measures on K^d are exactly normalized Haar measures on lattices (Evans, [Eva01b], Theorems 4.4 and 4.6).
This is the foundation of the paper; all three theorems are stated in terms of lattices, so the entire framework depends on importing this characterization without reproof.
domain assumption Independence of linear forms against a standard Gaussian is equivalent to orthogonality (Evans, [Eva01b], Theorem 4.8, restated as Lemma 2.3).
Used in the proof of Theorem 1.2 to translate conditional independence of coordinates into orthogonality of rows, then into linear independence over F_q.
standard math For vectors of norm 1 in K^d, orthonormality is equivalent to linear independence of their reductions modulo π (Proposition 2.2, from [vR78] Exercise 5.A).
This is the bridge from orthogonality to matroid representability over the residue field in Theorem 1.2.
standard math Haar measure on K^d is normalized by λ(O_K^d)=1 and transforms as λ(AS)=|det A| λ(S).
Used in Lemma 3.1 to compute lattice measures as q^{-val(det A)}, the key step in the MLE proof.
standard math Smith normal form / SVD for matrices over K (Proposition 2.4, from [Eva02]) and existence of Hermite normal form (Lemma 2.2).
Provides canonical representatives for lattices, used in Theorems 1.2 and 1.3.

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Pith. "Pith review of Statistics and tropicalization of local field Gaussian measures." pith.science (2026). https://pith.science/paper/ROH2IARJ

@misc{pith2026190900559,
  author       = {Pith},
  title        = {Pith review of: Statistics and tropicalization of local field Gaussian measures},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ROH2IARJ}},
  note         = {Machine review of arXiv:1909.00559}
}

abstract

This paper aims to lay the foundations for statistics over local fields, such as the field of $p$-adic numbers. Over such fields, we give characterizations for maximum likelihood estimation and conditional independence for multivariate Gaussian distributions. We also give a bijection between the tropicalization of such Gaussian measures in dimension 2 and supermodular functions on the 2-dimensional discrete cube. Finally, we introduce the Bruhat-Tits building as a parameter space for Gaussian distributions and discuss their connections to conditional independence statements as an open problem.

Figures

Figures reproduced from arXiv: 1909.00559 by the authors.

**Figure 2.1.** Local depiction of Q2 as a tree. Parallel to Kac’s characterization of classical Gaussians [Kac39], Evans [Eva01b, Definition 4.1] defined the Gaussian measure on Kd for some local field K to be one that is invariant under orthonormal transformations [PITH_FULL_IMAGE:figures/full_fig_p003_2_1.png] view at source ↗

**Figure 3.1.** Description of the polynomial PL in dimension d = 2. The function ϕL is the restriction of this polynomial to Z 2 . Note that, when c = b, the tropical variety of PL is the 1-skeleton of the normal fan of a square. In that case the two entries of X are independent and the probability distribution of V is just a product of two shifted geometric distributions. When c < b, the rows of the lower triangular matrix are no… view at source ↗

**Figure 4.** [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗

Figures from the paper (1 more)

**Figure 5.1.** Figure 5.1: Tropical geometry of the Gaussian measure on [PITH_FULL_IMAGE:figures/full_fig_p014_5_1.png]

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Buildings: theory and applications , volume 248

Peter Abramenko and Kenneth S Brown. Buildings: theory and applications , volume 248. Springer Science & Business Media, 2008

2008

[2] [2]

Tropical polyhedra are equivalent to mean payoff games

Marianne Akian, Stephane Gaubert, and Alexander Guterman. Tropical polyhedra are equivalent to mean payoff games. International Journal of Algebra and Computation , 22(01):1250001, 2012

work page 2012

[3] [3]

A decomposition theorem for L \'evy processes on local fields

Sergio Albeverio and Xuelei Zhao. A decomposition theorem for L \'evy processes on local fields. J. Theoret. Probab. , 14(1):1--19, 2001

work page 2001

[4] [4]

Tropical geometry to analyse demand

Elizabeth Baldwin and Paul Klemperer. Tropical geometry to analyse demand. Unpublished paper.[281] , 2013

work page 2013

[5] [5]

Local fields , volume 3

John William Scott Cassels. Local fields , volume 3. Cambridge University Press Cambridge, 1986

work page 1986

[6] [6]

Representation theory of finite groups and associative algebras , volume 356

Charles W Curtis and Irving Reiner. Representation theory of finite groups and associative algebras , volume 356. American Mathematical Soc., 1966

1966

[7] [7]

Expectation, conditional expectation and martingales in local fields

Steven N Evans and Tye Lidman. Expectation, conditional expectation and martingales in local fields. Electronic Journal of Probability , 12(17):498--515, 2007

work page 2007

[8] [8]

Local field gaussian measures

Steven N Evans. Local field gaussian measures. In Seminar on Stochastic Processes, 1988 , pages 121--160. Springer, 1989

work page 1988