The triangle of fundamental one-dimensional discrete models of maximum likelihood degree one and its reflection-symmetric refinement

Abstract

A one-dimensional discrete statistical model of maximum likelihood degree one is, after Bik and Marigliano, the image of a parameterisation t ↦ (cᵢ t^(νᵢ)(1-t)^(μᵢ))ᵢ₌₀ⁿ of the probability simplex Δₙ; it is fundamental when the exponent pairs (νᵢ,μᵢ) determine the scalings cᵢ. Améndola, Nguyen and Oldekop proved that the degree of such a model is at most 2n-1, tabulated the number T(n,d) of fundamental models in Δₙ of degree d for n ≤ 7, and conjectured that T(n,2n-2)=2Σₖ₌₁ⁿ⁻¹aₖaₙ₋ₖ with aₖ=T(k,2k-1). We introduce the refinement Tˢʸᵐ(n,d), the number of those models whose support is invariant under the reflection (ν,μ) ↦ (μ,ν), i.e. under the reparameterisation t ↦ 1-t; we prove that the reflection preserves fundamentality for every support, so that T(n,d)equivTˢʸᵐ(n,d) (mod 2), and we compute Tˢʸᵐ(n,d) for n ≤ 6 and on the main diagonal up to Tˢʸᵐ(8,8)=323. The values are in no index we could find. On the almost sharp cells d=2n-2 we find Tˢʸᵐ(n,2n-2)=0 for 3 ≤ n ≤ 6: no almost sharp fundamental model is its own reflection, which is the step left implicit in the published proof of the lower bound. Rows n ≤ 4 of both triangles, including the value T(3,3)=12 that corrects an entry of Lebl and Lichtblau, are verified in the Lean 4 kernel; rows 5–7 and the value T(8,8)=1 279 349, printed only in a figure of the source, are recomputed independently by an exact search, whose measured cost we use to price the remaining open cells.

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