bibkey: drtonsullivant2007algebraic authors: Mathias Drton and Seth Sullivant year: 2007 title: Algebraic Statistical Models doi: 10.48550/arXiv.math/0703609 url: https://arxiv.org/abs/math/0703609v1 claim: “Discrete conditional independence is characterized by the cross-product quadratic equations on each conditioning fiber; strictly positive categorical probabilities have log probability-ratio coordinates.” strata_touched:

• D5/S3/Estimation/DataProcessing/FiniteSimplexFiberPolytope license: citation-only triage: anchor

Discrete conditioning fibers and positive probability ratios

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The DOI 10.48550/arXiv.math/0703609 resolves to the arXiv abstract page of this preprint. The versioned locator https://arxiv.org/abs/math/0703609v1 shows the title “Algebraic statistical models”, the authors Mathias Drton and Seth Sullivant, and the version 1 date 2007-03-20. The statements below cite that version.

Consumed statements

The consumed primary text is Algebraic Statistical Models, arXiv:math/0703609v1. Section 3.2, Example 16, PDF p. 13, states that $X_1\perp X_2\mid X_3$ is equivalent to the equations

$$p_{i_1{j}_{1}k}{p}_{i_2{j}_{2}k}-p_{i_1{j}_{2}k}{p}_{i_2{j}_{1}k}=0$$

for every pair of values of each of the first two variables and every conditioning value $k$. This is an equation on the joint probabilities, not a requirement that every conditioning event have positive mass. For a positive-mass binary fiber its normalized table is independent exactly when its determinant is zero. At zero mass every nonnegative cell is zero, and no conditional distribution is asserted.

Example 4, PDF pp. 4–5, uses the natural coordinates ${\eta }_x=\log (p_x/p_m)$ for strictly positive categorical probabilities. It identifies the full interior of the probability simplex with these coordinates. It does not extend a finite ratio parameterization to cells of zero probability.

The determinant and native guard volume uses Example 16 for the conditional binary table on $z=0$ and Example 4 for positive ratios relative to the empty cell. Its conditional covariance, separator normalization, odds ratio and target residual are elementary finite-table consequences shown there. The paper supplies the classical statistical ingredient; the native high-to-low FIB reader, its actual reply archive and continuing guard task are supplied by the project. No source acquisition, native instrument, physical or formal certification is attributed to this reference.