TheoremStatement: AI-adaptedProof: AI-generatedPipeline-generatedprecheck passverified 2026-10-22 (gpt-6.1-sol)

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Identifiability and covariance in a finite exponential Gibbs family

Statement

For the finite exponential family of the preceding Definition, write $μ_{θ} : = \sum_{x \in X} π_{θ} (x) T (x)$ . Define ${Cov}_{π_{θ}} (T)$ by entries ${Cov}_{π_{θ}} (T)_{a b} : = \sum_{x \in X} π_{θ} (x) (T_{a} (x) - μ_{θ, a}) (T_{b} (x) - μ_{θ, b})$ . The gradient and Hessian mean $\nabla A (θ) : = (\partial_{j} A (θ))_{j = 1}^{k}$ and $\nabla^{2} A (θ) : = (\partial_{i} \partial_{j} A (θ))_{i, j = 1}^{k}$ . A symmetric matrix is positive definite when its quadratic form is positive on every nonzero vector. The map $θ \mapsto π_{θ}$ is injective when $π_{θ} (x) = π_{φ} (x)$ for every $x$ implies $θ = φ$ . The following conditions are equivalent: the representation is minimal; no nonzero $a \in R^{k}$ makes $a \cdot T (x)$ constant on $X$ ; the affine hull of $T (X)$ is $R^{k}$ ; $\nabla^{2} A (θ)$ is positive definite for every $θ$ ; and the map $θ \mapsto π_{θ}$ is injective.

Statement references

Finite exponential Gibbs family for inverse inference

Dominated exponential normalization and covariance

Facts & Assumptions

Given: The finite exponential family, its log-partition function $A$ , and its full-support probability vectors $π_{θ}$ .

[F1]

Finite exponential Gibbs family for inverse inference: $X$ is finite nonempty, $w (x) > 0$ , $π_{θ} (x) > 0$ for every $x, θ$ , and the Definition calls the representation minimal exactly when no nonzero $a$ makes $a \cdot T$ constant on $X$ .

[F2]

Dominated exponential normalization and covariance: under a compact-parameter integrable envelope, $\partial_{a} A = E_{θ} [T_{a}]$ and $\partial_{a} \partial_{b} A = {Cov}_{θ} (T_{a}, T_{b})$ . The finite model satisfies that envelope by step 1.1 of [F1]'s Definition proof and [F1].

[F3]

Rank-nullity: $\dim_{F} V = nullity T + rank T$ : for a linear map with finite-dimensional domain, domain dimension is kernel dimension plus image dimension.

[F4]

The affine hull of a finite set $T (X)$ is $T (x_{0}) + D$ , where $D$ is the span of the finitely many differences $T (x) - T (x_{0})$ ; it is all of $R^{k}$ exactly when $D = R^{k}$ .

[F5]

The natural logarithm as the inverse of the exponential function: $\log$ is the inverse of $\exp$ on $(0, \infty)$ , so $\log (\exp t) = t$ .

[F6]

Order, continuity, range, and the product, quotient, and reciprocal laws for the natural logarithm: $\log$ is strictly increasing and satisfies $\log (a b) = \log a + \log b$ for positive $a, b$ .

[F7]

The exponential addition formula $\exp (x + y) = \exp (x) \exp (y)$ : $\exp (s + t) = \exp (s) \exp (t)$ for real $s, t$ .

[F8]

The exponential is positive and satisfies $\exp (- x) = 1 / \exp (x)$ : $\exp (t) > 0$ for every real $t$ .

Proof

Given: The finite exponential family of the Statement.

1.1givenF1F2

The envelope verified in [F1] lets [F2] apply with counting measure, so $\nabla A (θ) = E_{π_{θ}} T$ and $\nabla^{2} A (θ) = {Cov}_{π_{θ}} (T)$ .

2.1F1step 1.1algebra

For any $a \in R^{k}$ , the quadratic form is $a^{T} \nabla^{2} A (θ) a = \sum_{x \in X} π_{θ} (x) (a \cdot T (x) - E_{π_{θ}} [a \cdot T])^{2}$ . Every weight is positive by [F1], so this sum is zero exactly when $a \cdot T (x)$ has the same value at every $x \in X$ . Thus the Hessian is positive definite for every $θ$ exactly when no nonzero $a$ makes $a \cdot T$ constant, which is minimality by [F1].

2.2F1F5F6F7F8step 1.1algebra

If $π_{θ} = π_{φ}$ as functions on $X$ , positivity allows taking logarithms of their equal positive values. The product law and inverse identity [F5, F6] then give $(θ - φ) \cdot T (x) = A (θ) - A (φ)$ for every $x$ . Under minimality this forces $θ - φ = 0$ . Conversely, if a nonzero $a$ makes $a \cdot T (x) = c$ constant, the exponential addition formula [F7] gives $Z (θ + t a) = \exp (t c) Z (θ)$ . Since $Z (θ) > 0$ and $\exp (t c) > 0$ by [F1, F8], the logarithm laws [F5, F6] give $A (θ + t a) = A (θ) + t c$ . Substitution in the family formula and another use of [F7] show $π_{θ + t a} = π_{θ}$ for every real $t$ ; the parametrization is not injective.

3.1givenF1F3F4algebra∎

Let $D = span {T (x) - T (x_{0}) : x \in X}$ for a fixed $x_{0} \in X$ . If $D$ is proper, choose a basis $v_{1}, \dots, v_{r}$ of $D$ , with $r < k$ , and define $L : R^{k} \to R^{r}$ by $L (a) = (a \cdot v_{1}, \dots, a \cdot v_{r})$ . Rank-nullity [F3] gives $\dim \ker L \geq k - r > 0$ , so some nonzero $a$ is orthogonal to every difference and $a \cdot T$ is constant. Conversely, if such a nonzero $a$ exists, every difference lies in the proper hyperplane $a^{⊥}$ , so $D$ is proper. By [F4], $D = R^{k}$ exactly when the affine hull is $R^{k}$ . This proves the stated equivalence with full affine dimension and completes the identifiability conditions.

Identifiability and covariance in a finite exponential Gibbs family

Statement

Statement references

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources