The Fisher score on the closed simplex

We extend classical analytic tools for finite-state statistical models to allow zero probabilities. Using methods from algebraic statistics and information geometry, we develop a framework in which a smooth statistical model could hit the boundary of the simplex, for example, in contingency tables with non-structural zeros. The central object of our approach is the vector bundle whose fibres are the $p$-contrasts associated to each probability distribution $p$. In this framework, Fisher score and other key statistical concepts, such as entropy for one-dimensional statistical models, admit an algebraic representation also on the boundary of the simplex.

💡 Research Summary

The paper “The Fisher score on the closed simplex” develops a rigorous mathematical framework that extends the classical differential‑geometric tools of information geometry to probability simplices that include zero‑probability cells. The authors combine concepts from algebraic statistics (contrast spaces, monomial varieties) with the affine‑space viewpoint of information geometry to treat one‑parameter exponential families whose trajectories may touch or lie on the boundary of the simplex.

First, the authors formalize the probability simplex Δ(Ω) for a finite outcome set Ω and introduce the contrast space C(Ω), the linear subspace of zero‑sum vectors. For each distribution p∈Δ(Ω) a p‑contrast is any random variable u with zero expectation under p. Theorem 2.1 characterizes the tangent bundle TΔ(Ω) as the set of pairs (p, v) where v∈C(Ω) and the support of v is contained in the support of p. This condition guarantees that a differentiable curve passing through a boundary point cannot move in directions that would assign positive mass to a previously zero cell. Consequently, the tangent space on each face of the simplex is a coordinate sub‑space C_I(Ω) spanned by differences of unit vectors belonging to the face’s index set I. The paper illustrates this construction with detailed examples for 2×2 contingency tables, showing how the algebraic ideal generated by products of coordinates encodes the various faces.

Next, the Fisher score is revisited. In the interior of the simplex the score of a smooth one‑parameter model γ(t) is the derivative of the log‑density centered by its expectation: s_p(γ(t)) = d/dt log γ(t) − E_p