ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedjudge pass (gpt-6-luna)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

A Gaussian LDA model and a conditional logistic model can share a decision boundary

Example

Assume AC. In one dimension, take LDA classes $1$ and $2$ with equal priors and conditional laws $X ∣ Y = 1 \sim N_{1} (- 1, 1)$ and $X ∣ Y = 2 \sim N_{1} (1, 1)$ . Recode class $1$ as binary label $0$ and class $2$ as binary label $1$ in a conditional logistic model with $\Pr (Y = 1 ∣ x) = σ (2 x)$ . Both classifiers have tie boundary $x = 0$ and agree on both sides of that boundary; ties select class $1$ (binary label $0$ ). This equality of boundaries for these parameters does not identify the two estimation procedures or equate their coefficients in general.

The number line marks the two Gaussian means and their posterior tie boundary; the score calculation below establishes the boundary.

diagram failed to render:
\begin{tikzpicture}[x=1.25cm,y=1cm]
  \draw[->] (-2.1,0) -- (2.1,0) node[right] {$x$};
  \fill (-1,0) circle (2pt);
  \fill (1,0) circle (2pt);
  \draw[dashed] (0,-0.32) -- (0,0.42);
  \node[below] at (-1,0) {$\mu_1=-1$};
  \node[below] at (1,0) {$\mu_2=1$};
  \node[above] at (0,0.42) {$x=0$};
\end{tikzpicture}

Example references

Common-covariance LDA scores are affine and QDA scores are quadratic

Binary logistic and multiclass softmax conditional models

The Axiom of Choice

Hastie, Tibshirani, and Friedman, ESL, §4.4.5, equation (4.33)

Stanford CS229 Spring 2022 lecture notes, §4.1.3

Facts & Assumptions

Given: AC, equal LDA priors, $μ_{1} = - 1$ , $μ_{2} = 1$ , $Σ = 1$ , the stated class recoding, and the conditional logistic slope $2$ with zero intercept.

[F1]

Under common covariance, the LDA posterior log-score for class $k$ is $x^{T} Σ^{- 1} μ_{k} - \frac{1}{2} μ_{k}^{T} Σ^{- 1} μ_{k} + \log π_{k}$ , up to a class-independent term. Common-covariance LDA scores are affine and QDA scores are quadratic

[F2]

The binary conditional logistic probability is $σ (β_{0} + x β)$ , with $σ (t) = (1 + e^{- t})^{- 1}$ . Binary logistic and multiclass softmax conditional models

[A1]

AC is carried because the LDA Gaussian law and density theorem assume it. The Axiom of Choice

Verification

1.1givenF1A1algebra

For the LDA model, [F1] gives $δ_{2} (x) - δ_{1} (x) = x (1 - (- 1)) - \frac{1}{2} (1^{2} - (- 1)^{2}) + \log (1) = 2 x$ . The two posterior scores tie exactly at $x = 0$ , with class $1$ selected there by the smallest-label rule.

2.1givenF2step 1.1algebra∎

For the logistic model, [F2] gives $\Pr (Y = 1 ∣ x) = σ (2 x)$ . Since $σ^{'} (t) = σ (t) (1 - σ (t)) > 0$ and $σ (0) = 1 / 2$ , this probability is below, equal to, or above $1 / 2$ exactly when $x$ is below, equal to, or above zero. Under the stated recoding, the class decisions agree on each region and ties select class $1$ ; no coefficient-estimation equivalence is claimed.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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Sources

Hastie, Tibshirani, and Friedman, The Elements of Statistical Learning, 2nd ed., corrected 12th printing (standard reference, not scraped)
Stanford CS229 Spring 2022 lecture notes (standard reference, not scraped)