One-dimensional Discrete Models of Maximum Likelihood Degree One

This page contains the source code and explanations related to the computational

results presented in the paper:

Carlos Améndola, Viet Duc Nguyen Janike Oldekop: One-dimensional Discrete Models of Maximum Likelihood Degree One

ARXIV: https://arxiv.org/abs/2507.18686

ABSTRACT: We settle a conjecture by Bik and Marigliano stating that the degree of a one-dimensional discrete model with rational maximum likelihood estimator is bounded above by a linear function in the size of its support, therefore showing that there are only finitely many fundamental such models for any given number of states. We study these models from a combinatorial perspective with regard to their existence and enumeration. In particular, sharp models, those whose degree attains the maximal bound, enjoy special properties and have been studied as monomial maps between complex unit spheres. In this way, we present a novel link between Cauchy-Riemann geometry and algebraic statistics.

Required Software

• Python 3

• Cython

• NumPy

Fundamental Models

A one-dimensional discrete model of ML degree one is a subset of the probability simplex \(\Delta_n\) that can be parameterized as

\[p : [0,1] \to \Delta_n, \qquad t \mapsto (c_it^{\nu_i}(1-t)^{\mu_i})_{i=0}^n,\]

where \(\nu_i\) and \(\mu_i\) are non-negative integers and \(c_i\) are positive real coefficients. Such a model is reduced if the exponent pairs \((\nu_i,\mu_i)\) are pairwise distinct and different from \((0,0)\). A reduced model is fundamental if, given the support \(\{(\nu_i,\mu_i)\mid 0\le i\le n\}\), the scalings \(c_i\) are uniquely determined by the constraint

\[c_0t^{\nu_0}(1-t)^{\mu_0} + c_1t^{\nu_1}(1-t)^{\mu_1} + \ldots + c_nt^{\nu_n}(1-t)^{\mu_n} = 1.\]

Its degree is \(\max \{ \nu_i + \mu_i \mid 0 \le i \le n \}\). According to Section 6 in the paper, fundamental models in \(\Delta_n\) of degree \(d\) exist whenever \(n\le d\le 2n-1\). Additionally, the number of fundamental models is finite by Corollary 2.3 in the paper. We computed the number of fundamental models in \(\Delta_n\) of degree \(d\) for small \(n,d\in\mathbb{N}^+\). The results are shown in Table 1 and Figure 9 in the paper. The main code can be viewed below.

• Support Candidates
• Finding fundamental models
• Visualizing fundamental models in \(\Delta_7\) of degree \(13\)
• References

The source code can be downloaded here:

support_candidates.ipynb

fundamental_models.ipynb

The notebooks make use of a chipsplitting library developed for this project that can be downloaded here:

chipsplitting.zip

A dump of candidate supports can be downloaded from the repository (see the data/ directory). Note that cases for degree \(d \geq 10\) have been excluded due to large file sizes (over 100mb). They can be recomputed by running the support_candidates.ipynb notebook, though this may take several hours.

The dataset of all fundamental models, computed by fundamental_models.ipynb, is also available there (see the fundamental-models/ directory).

---

Project page created: 29/07/2025

Project contributors: Carlos Améndola, Viet Duc Nguyen, Janike Oldekop

Corresponding author of this page: Janike Oldekop, oldekop@math.tu-berlin.de

Software used: The computations were performed using Python 3, with the libraries Cython and NumPy. The code was executed within a Jupyter Notebook environment.

System setup used: The analysis was run on a MacBook Pro (2023) equipped with an Apple M3 Pro processor and 36 GB of RAM, running macOS 15.5.

License for code of this project page: MIT License (https://spdx.org/licenses/MIT.html)

License for all other content of this project page: CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/)

Last updated 29/07/2025.