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

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.

Finite exponential Gibbs family for inverse inference

Definition

Let $X$ be a finite nonempty set, let $k \in N$ with $k \geq 1$ , let $w : X \to (0, \infty)$ be a positive reference weight, and let $T : X \to R^{k}$ . Here $R^{k}$ consists of functions $k \to R$ , with $k = {0, \dots, k - 1}$ . On this page use the one-based coordinate labels $v_{j} : = v (j - 1)$ for $1 \leq j \leq k$ and $v \in R^{k}$ ; in particular $T_{j} (x) : = T (x) (j - 1)$ . Thus the Euclidean inner product is $θ \cdot T (x) : = \sum_{j = 1}^{k} θ_{j} T_{j} (x) = \sum_{i < k} θ (i) T (x) (i)$ . For $θ \in R^{k}$ , define the partition sum and log-partition function by $Z (θ) : = \sum_{x \in X} w (x) e^{θ \cdot T (x)}, A (θ) : = \log Z (θ),$ and define the finite exponential Gibbs family by $π_{θ} (x) : = w (x) e^{θ \cdot T (x) - A (θ)} (x \in X) .$ All objects here are mathematical; $θ \cdot T (x)$ is a real number. The representation is called minimal when there is no nonzero $a \in R^{k}$ for which $a \cdot T (x)$ is constant over all $x \in X$ .

For a finite set $S \subseteq R^{k}$ , its convex hull is the set of sums $\sum_{s \in S} λ_{s} s$ with $λ_{s} \geq 0$ and $\sum_{s \in S} λ_{s} = 1$ . Its affine hull is the set of sums of the same form with real coefficients $λ_{s}$ whose sum is $1$ . The relative interior of a convex set is its interior in its affine hull with the Euclidean subspace topology. These definitions will be used for the sufficient-statistic support $T (X)$ .

Definition references

The real exponential function and the number $e$ by a power series

The natural logarithm as the inverse of the exponential function

The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$

Facts & Assumptions

Given: A finite nonempty $X$ , positive weights $w (x)$ , and a map $T : X \to R^{k}$ .

[F1]

Dominated exponential normalization and covariance: on a sigma-finite measure space, if a positive exponential weight has one integrable envelope on each compact parameter set for its zeroth, first, and second moments, then its partition function is finite, strictly positive, and $C^{2}$ , its normalized density is a probability density, and the gradient and Hessian of the log partition function are the mean and covariance of the statistic.

[F2]

Counting measure on finite $X$ is sigma-finite, and a function on $X$ is integrable exactly when its finite sum is finite.

[F3]

The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ : $θ \cdot T (x)$ is the ordinary Euclidean inner product, a finite sum of coordinate products.

[F4]

A continuous real-valued function on a nonempty compact metric space is bounded and attains a greatest and a least value: a continuous real-valued function on a nonempty compact metric space attains a finite maximum and minimum.

[F5]

The real exponential function and the number $e$ by a power series: $\exp$ is the real exponential power-series function defined for every real input.

[F6]

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

[F7]

Cauchy-Schwarz $∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥_{2} ∥ y ∥_{2}$ with its equality case, the triangle inequality for $∥ \cdot ∥_{2}$ , the parallelogram law and polarisation: the Euclidean norm is a norm and satisfies $∣ a \cdot b ∣ \leq ∥ a ∥_{2} ∥ b ∥_{2}$ .

[F8]

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

[F9]

The exponential function is strictly increasing: $\exp$ is strictly increasing on $R$ .

[F10]

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

Proof

Given: The finite family and the notation in the Definition.

1.1givenF1F2F3F4F5F7F8F9

For every nonempty compact $C \subseteq R^{k}$ and every $x \in X$ , [F3, F7] give $∣ η \cdot T (x) - η^{'} \cdot T (x) ∣ \leq ∥ η - η^{'} ∥_{2} ∥ T (x) ∥_{2}$ ; hence $η \mapsto η \cdot T (x)$ is continuous. It attains a finite maximum $M_{C} (x)$ on $C$ by [F4]. Define $G_{C} (x) : = w (x) \exp (M_{C} (x)) (1 + \sum_{j = 1}^{k} ∣ T_{j} (x) ∣^{2}) .$ The exponential is finite and positive at this real input by [F5, F8], and the sum is finite, so $G_{C}$ is finite and nonnegative at every $x$ . It is integrable under counting measure by [F2]. Since $η \cdot T (x) \leq M_{C} (x)$ for every $η \in C$ , monotonicity of $\exp$ [F9] shows that $G_{C}$ dominates $w (x) e^{θ \cdot T (x)} (1 + \sum_{j} ∣ T_{j} (x) ∣^{2})$ for every $θ \in C$ . Thus the counting-measure model satisfies the envelope hypotheses of [F1]. Applying [F1] gives $0 < Z (θ) < \infty$ , $A \in C^{2} (R^{k})$ , and the normalized probability mass function $π_{θ}$ .

2.1F1F5F6F8F10step 1.1algebra∎

Since every summand in $Z (θ)$ is positive and finite [F5, F8], each $π_{θ} (x) > 0$ . Also $\exp (A (θ)) = Z (θ)$ by [F6], so $e^{- A (θ)} = 1 / Z (θ)$ by [F8]. By [F10], $π_{θ} (x) = w (x) e^{θ \cdot T (x)} e^{- A (θ)}$ , which is exactly the normalized density supplied by [F1]. Summing over finite $X$ gives $\sum_{x \in X} π_{θ} (x) = e^{- A (θ)} Z (θ) = 1$ . Hence the Definition gives a probability vector with full support for every finite $θ$ .

Finite exponential Gibbs family for inverse inference

Definition

Definition references

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources