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.

Rescaling loss changes the effective temperature

Example

Let $Θ = {0, 1}$ with its power-set sigma-algebra and prior $π (0) = p$ , $π (1) = 1 - p$ for $0 < p < 1$ . At a fixed observation $x_{0}$ , let $ℓ_{x_{0}} (0) = a$ and $ℓ_{x_{0}} (1) = b$ with $a \neq b$ . For every $w, c > 0$ , replacing this loss section by $c ℓ_{x_{0}}$ at temperature $w$ gives the same update as retaining $ℓ_{x_{0}}$ at temperature $c w$ . At any two distinct positive temperatures, the updates are different.

Facts & Assumptions

Given: The finite parameter space $Θ = {0, 1}$ with its power-set sigma-algebra; an observation $x_{0}$ ; $0 < p < 1$ ; real loss-section values $a \neq b$ ; and positive temperatures $w, c, t_{1}, t_{2}$ with $t_{1} \neq t_{2}$ .

[F1]

A countable discrete measurable space is standard Borel: every nonempty at-most-countable set with its power-set sigma-algebra is standard Borel.

[F2]

Probability measures and probability spaces: a probability measure is a measure of total mass one.

[F3]

Generalized Bayes loss update and temperature: when its normalizer is finite and positive, the update is the measure with density $e^{- t ℓ} / Z_{t}$ relative to the prior.

[F4]

A finite-normalizer loss update is a probability measure: each update with a finite positive normalizer is a probability measure.

[F5]

The exponential function is strictly increasing: the exponential is strictly increasing, hence injective, on $R$ .

[F6]

The exponential is positive and satisfies $\exp (- x) = 1 / \exp (x)$ : $e^{u} > 0$ and $e^{- u} = 1 / e^{u}$ for every real $u$ .

Verification

1.1F1F2F6givenalgebra

The set $Θ$ is nonempty and finite, so [F1] makes its power-set measurable space standard Borel. Since $0 < p < 1$ , the masses $p$ and $1 - p$ are positive and sum to one, so [F2] makes $π$ a probability. For every $t > 0$ , the two real loss values give positive finite weights $e^{- t a}$ and $e^{- t b}$ by [F6]; their prior-weighted sum is therefore finite and positive.

2.1F3F4step 1.1algebra

By [F3]–[F4], the update $q_{t} = G_{t, π, x_{0}}$ has masses $q_{t} (0) = \frac{p e^{- t a}}{p e^{- t a} + (1 - p) e^{- t b}}, q_{t} (1) = \frac{(1 - p) e^{- t b}}{p e^{- t a} + (1 - p) e^{- t b}} .$ Consequently its odds satisfy $\frac{q_{t} (1)}{q_{t} (0)} = \frac{1 - p}{p} e^{- t (b - a)} .$ .

2.2step 1.1F3algebra

For $θ \in Θ$ , the unnormalized weight after replacing $ℓ_{x_{0}}$ by $c ℓ_{x_{0}}$ at temperature $w$ is $π (θ) e^{- w c ℓ_{x_{0}} (θ)} = π (θ) e^{- (c w) ℓ_{x_{0}} (θ)}$ . These are exactly the unnormalized masses for $q_{c w}$ . The two updates have the same finite positive normalizer by step 1.1, so their values on both singletons, and hence on every subset of $Θ$ , agree.

3.1F5step 2.1algebra∎

Because $b - a \neq 0$ and $t_{1} \neq t_{2}$ , the real numbers $- t_{1} (b - a)$ and $- t_{2} (b - a)$ differ. Injectivity in [F5] makes their exponentials different. Multiplication by the fixed positive factor $(1 - p) / p$ preserves that difference, so step 2.1 gives distinct odds. On a two-point probability space the odds determine both singleton masses; therefore $q_{t_{1}} \neq q_{t_{2}}$ .

Depends on

Used by

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Sources

P. G. Bissiri, C. C. Holmes, and S. G. Walker, A general framework for updating belief distributions, JRSS B 78(5), 2016 (standard reference, not scraped)