ExampleConstruction: AI-generatedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-6-luna)verified 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.

An improper counting prior can have finite evidence at one datum

Example

Assume the Axiom of Choice. Let $X = R$ with its usual real Banach norm, and let $N = {0, 1, 2, \dots}$ with its power-set sigma-algebra and counting measure $μ$ . Let $U = D = N \subseteq X$ and take the forward map $F (n) = n$ into data space $R$ with its usual metric and Borel sigma-algebra. The measure $μ$ is sigma-finite and has $μ (N) = + \infty$ . For parameter $n$ , use the proper Gaussian observation law $Y ∣ n \sim N (n, 1)$ , with Lebesgue density

L_{y} (n) = \frac{1}{\sqrt{2 π}} \exp ⁣ (- \frac{(y - n)^{2}}{2}) .

At datum $y = 0$ its evidence is

Z (0) = \frac{1}{\sqrt{2 π}} \sum_{n = 0}^{\infty} e^{- n^{2} / 2} \in (0, \infty),

and the normalized fiberwise update is the probability

Π^{0} ({n}) = \frac{e^{- n^{2} / 2}}{\sum_{j = 0}^{\infty} e^{- j^{2} / 2}} (n \in N) .

Because $μ$ has infinite total mass, this is a fixed-data update only and does not define a prior-predictive probability.

Example references

A sigma-finite improper measure may have a fixed-data normalizer, but not a prior-predictive probability.

A forward map is a function on a nonempty admissible truth set between metric spaces.

A finite positive evidence normalizes a nonnegative likelihood section to a probability measure.

Counting measure assigns cardinality to finite sets and infinite mass to infinite sets.

Integration of a nonnegative function on the natural numbers against counting measure is its nonnegative series.

A sampling probability kernel has pointwise probability sections and measurable evaluations.

A normal law has its location-scale density for positive variance.

The geometric series with ratio of absolute value less than one converges.

Latz, §2.2, Theorem 2.5 and its proof in Appendix B (printed p. 23), full paper

Dashti and Stuart, §3.2, full lecture notes

Facts & Assumptions

Given: The counting measure on $N$ , the Gaussian sampling likelihood $L_{y} (n)$ , the fixed datum $y = 0$ , and AC for the repository's standard-normal definition.

[A1]

AC selects from every family of nonempty sets. The Axiom of Choice

[F1]

Counting measure assigns finite sets their cardinality and infinite sets measure $+ \infty$ . Counting measure on an arbitrary set

[F2]

The counting set function is a measure on the power-set sigma-algebra. Counting measure is a measure

[F3]

Sigma-finiteness means a countable measurable cover by sets of finite measure. Finite, sigma-finite, and semifinite measures

[F4]

Under AC, $N (m, 1)$ is the affine image of the standard normal law and has Lebesgue density $\frac{1}{\sqrt{2 π}} e^{- (y - m)^{2} / 2}$ . Standard normal and normal laws, A positive-variance normal law has its location-scale density

[F5]

AC implies countable choice. AC implies DC implies countable choice

[F6]

On $N$ with counting measure, $\int f d μ = \sum_{n = 0}^{\infty} f (n)$ for nonnegative measurable $f$ . Counting-measure integration is a nonnegative series

[F7]

For $0 < r < 1$ , $\sum_{n = 0}^{\infty} r^{n}$ is finite. For $∣ r ∣ < 1$ , $\sum_{k \geq 0} r^{k} = 1 / (1 - r)$ , and for $∣ r ∣ \geq 1$ the series diverges

[F8]

The exponential is positive, satisfies $e^{- x} = 1 / e^{x}$ , is continuous and strictly increasing, and has $e^{0} = 1$ . The exponential is positive and satisfies $\exp (- x) = 1 / \exp (x)$ , The exponential function is strictly increasing, The real exponential function and the number $e$ by a power series

[F9]

For a sigma-finite measure and measurable likelihood section, evidence is the fixed-data normalizer; for an improper prior it does not define a prior-predictive probability or conditional law. Evidence normalizer and fiberwise posterior update

[F10]

A finite positive evidence normalizes the likelihood section to a probability measure. Fiberwise posterior propriety is equivalent to finite positive evidence

[F11]

A proper Bayesian model requires a probability prior; an infinite counting measure is not such a prior. Bayesian model as a prior and sampling kernel

[F12]

The function-space propriety theorem assumes a Borel probability on a separable real Banach parameter space, so it does not apply to this infinite counting measure on $N$ . Function-space posterior propriety from a common lower envelope

[F13]

A probability kernel has probability sections at every parameter and measurable evaluations. Measure kernel and probability kernel

[F14]

A forward map is defined on a nonempty admissible truth set between metric parameter and data spaces. Inverse-problem models and solution classes

[F15]

A function on $(N, P (N))$ is measurable because every preimage is a subset of $N$ . A measurable function between measurable spaces

[F16]

$N$ is countably infinite by its identity bijection, and no finite set is equinumerous with $N$ . Finite, countably infinite, countable, uncountable, The pigeonhole principle on $N$

Verification

Given: The hypotheses and conventions in the Statement.

1.1F1F2F3F16givenconstruct

For each $m \in N$ , the initial set $E_{m} = {0, 1, \dots, m}$ has counting measure $m + 1 < \infty$ by [F1] and [F2], and $⋃_{m} E_{m} = N$ . Thus $μ$ is sigma-finite by [F3]. By [F16], $N$ is infinite, so [F1] gives $μ (N) = + \infty$ .

1.2F4F6F8F15givenalgebra

The counting-integral formula [F6] gives $Z (0) = \int_{N} \frac{e^{- n^{2} / 2}}{\sqrt{2 π}} μ (d n) = \frac{1}{\sqrt{2 π}} \sum_{n = 0}^{\infty} e^{- n^{2} / 2}$ . Its $n = 0$ term is $1 / \sqrt{2 π} > 0$ by [F8].

2.1A1F4F5F11F13F14F15step 1.1given

AC implies countable choice by [F5], so the normal-density interface [F4] applies: for each $n$ , $N (n, 1)$ is a probability law with the displayed Gaussian density. Since the parameter sigma-algebra is $P (N)$ , every evaluation $n \mapsto N (n, 1) (B)$ is measurable, and the likelihood section $n \mapsto L_{y} (n)$ is measurable by [F15]; thus the sampling family satisfies [F13]. The map $F (n) = n$ on the nonempty admissible set $U = N$ satisfies the forward-map interface [F14]. By [F11] this is not a proper Bayesian model with prior $μ$ , because [step 1.1] shows the prior mass is infinite.

2.2F7F8step 1.2givenalgebra

For every $n \geq 1$ , $n^{2} - n = n (n - 1) \geq 0$ , since $n$ and $n - 1$ are nonnegative; hence $n^{2} \geq n$ . Monotonicity of the exponential [F8] gives $e^{- n^{2} / 2} \leq e^{- n / 2}$ . Let $r = e^{- 1 / 2}$ . By [F8], $0 < r < 1$ , and [F7] makes $\sum_{n = 1}^{\infty} r^{n}$ finite. Comparison of nonnegative partial sums now bounds the evidence series in step 1.2 by $1 + \sum_{n = 1}^{\infty} r^{n} < \infty$ . Hence $0 < Z (0) < \infty$ .

3.1F9F10F11F12step 1.1step 1.2step 2.1step 2.2∎

Applying [F10] to $L_{0}$ and the sigma-finite measure $μ$ gives the probability with singleton masses $L_{0} (n) / Z (0) = e^{- n^{2} / 2} / \sum_{j} e^{- j^{2} / 2}$ , as stated. Its fixed-data meaning and lack of a prior-predictive probability follow from [F9]; [F11] confirms that no proper Bayesian model is supplied by this prior. The function-space propriety theorem [F12] does not apply because its prior is a probability, whereas [step 1.1] gives $μ (N) = + \infty$ .

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

122 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

M. Dashti and A. M. Stuart, The Bayesian Approach to Inverse Problems (full lecture notes, arXiv:1302.6989) (standard reference, not scraped)
J. Latz, On the Well-posedness of Bayesian Inverse Problems, SIAM/ASA Journal on Uncertainty Quantification 8 (2020) (standard reference, not scraped)