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One-dimensional Discrete Models of Maximum Likelihood Degree One

Carlos Améndola, Viet Duc Nguyen, Janike Oldekop

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 unit spheres. In this way, we present a novel link between Cauchy-Riemann geometry and algebraic statistics.

One-dimensional Discrete Models of Maximum Likelihood Degree One

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 unit spheres. In this way, we present a novel link between Cauchy-Riemann geometry and algebraic statistics.

Paper Structure

This paper contains 7 sections, 28 theorems, 56 equations, 11 figures, 1 table.

Table of Contents

  1. From CR Geometry to Algebraic Statistics
  2. On the Finiteness of Fundamental Models
  3. Reduced Models and Newton Diagrams
  4. Composition of Fundamental Models
  5. Sharp Models
  6. Enumerating Fundamental Models
  7. Outlook

Key Result

Theorem 1

The degree of a model parameterized by $p: [0,1] \to \Delta_n, \, t \mapsto (c_i t^{\nu_i} (1-t)^{\mu_i})_{i=0}^n$ for suitable non-negative exponents $\nu_i$ and $\mu_i$ and positive real scalings $c_i$ is at most $2n-1$.

Figures (11)

  • Figure 1: All subdiagrams for an entry to be a sink. The sink is the top-right entry.
  • Figure 2: All subdiagrams for an entry to be a source. The source is the top-right entry.
  • Figure 3: Regions of 0's and N's expanding to the north or east result in additional sources.
  • Figure 4: Regions of 0's and N's with P's in its interior result in additional sources.
  • Figure 5: Non-positive regions touching both axes result in additional sources.
  • ...and 6 more figures

Theorems & Definitions (68)

  • Theorem
  • Example 1.1
  • Definition 2.1
  • Theorem 2.2
  • proof : Proof idea
  • Corollary 2.3
  • proof
  • Theorem 2.4
  • proof
  • Corollary 2.5
  • ...and 58 more