ExampleConstruction: AI-generatedVerification: 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.

Diffusion inverse solver families and their target laws

Example

Let $d \geq 1$ , let $μ_{0}$ be a probability measure on $R^{d}$ dominated by a sigma-finite measure $λ$ , with density $ρ_{0} = d μ_{0} / d λ$ . Fix an observation vector $y \in R^{p}$ and a nonnegative measurable likelihood function $ℓ_{y} : R^{d} \to [0, \infty)$ . Assume $0 < Z_{y} = \int ℓ_{y} (x) ρ_{0} (x) λ (d x) < \infty$ and define the posterior probability measure by $π_{y} (d x) = Z_{y}^{- 1} ℓ_{y} (x) ρ_{0} (x) λ (d x)$ . For the latent-optimization comparison, fix a Borel measurement map $A : R^{d} \to R^{p}$ . The following diffusion-inverse solver families have different output laws and exactness conditions.

Explicit-score guidance uses the exact conditional score $c_{σ_{k}} (x, y) = s_{σ_{k}} (x) + g_{σ_{k}} (x, y)$ at each step $t_{k}$ of a finite schedule $0 = t_{0} < t_{1} < \dots < t_{N}$ , where a declared map $σ : {t_{1}, \dots, t_{N}} \to (0, \infty)$ gives $σ_{k} = σ (t_{k})$ and the forward noising convention is $X_{σ_{k}} = X_{0} + σ_{k} Z$ with $X_{0} \sim μ_{0}$ and independent $Z \sim N (0, I_{d})$ . This branch requires the fixed- $y$ likelihood density to be bounded, strictly positive and Borel relative to a fixed sigma-finite observation measure, and the observation variable to be conditionally independent of the Gaussian noising noise given $X_{0}$ , as required by the conditional-score identity. For a finite integer $N \geq 1$ , let a supplied posterior path probability $Π_{y}$ on $(X_{0}, \dots, X_{N})$ have $X_{0}$ -marginal $π_{y}$ and terminal marginal $ν_{N}$ , and let $R_{k}$ be Borel conditional probability kernels of $X_{k - 1}$ given $X_{k}$ under $Π_{y}$ . A reverse-kernel sampler then returns $π_{y}$ when it starts from $ν_{N}$ , uses each $R_{k}$ with no projection, and initialization, corrections and transition sampling all succeed almost surely within the declared budgets. If failures are possible, exact kernels alone do not ensure that the law conditional on completed status is the posterior. A learned score, plug-in likelihood, or other surrogate does not supply that certificate by itself.
Variational inference selects a probability density $q$ with respect to $λ$ , with $q ≪ π_{y}$ , and optimizes the evidence lower bound (ELBO) $ELBO (q) = \int q (x) \log (ℓ_{y} (x) ρ_{0} (x) / q (x)) λ (d x)$ when the logarithmic integrals are finite, using the convention that the integrand is zero where $q = 0$ . Its output law is the selected $q$ ; it equals the posterior only when $q = π_{y}$ as probability measures, equivalently when their densities agree $λ$ -almost everywhere.
Latent optimization specifies a latent domain $Z$ , a generator map $G : Z \to R^{d}$ , and a deterministic finite-iteration procedure that returns $\hat{z}$ using the data-residual objective $∥ y - A (G (z)) ∥^{2}$ , without assuming global minimization. Its output law is the point mass $δ_{G (\hat{z})}$ . If $π_{y}$ is not a point mass, this output law is not a posterior draw. A maximum-a-posteriori (MAP) estimate here means a maximizer of the chosen posterior density $Z_{y}^{- 1} ℓ_{y} ρ_{0}$ with respect to $λ$ ; the measurement-residual objective alone does not establish that claim.
Markov chain Monte Carlo (MCMC) uses a probability kernel $K_{y}$ on the Borel space $R^{d}$ , and sequential Monte Carlo (SMC) produces a finite weighted particle measure ${\hat{π}}_{J} = \sum_{j = 1}^{J} w_{j} δ_{x_{j}}$ for an integer $J \geq 1$ , with $w_{j} \geq 0$ and $\sum_{j} w_{j} = 1$ , where $δ_{x}$ is unit mass at $x$ . If $π_{y}$ is invariant for $K_{y}$ and a Markov chain starts with law $π_{y}$ , every finite-step marginal remains $π_{y}$ ; invariance alone does not make every initial law exact after finitely many steps. A realized weighted particle measure equals the posterior only when the two measures are equal; every finite empirical measure differs from a non-atomic posterior, meaning one that assigns zero mass to each singleton.

These are law-level distinctions, not performance claims about any trained system.

Example references

Conditional-score and Tweedie identities for Gaussian smoothing

Diffusion prior and measurement scores

Measurement-guided reverse-kernel reconstruction

Invariant and stationary distribution for a Markov kernel

Daras et al., §2.4.4 and §§3.1–3.4

Song et al., §5

Chung et al., §3.1

Prince, Chapter 18, “Diffusion models”

Facts & Assumptions

Given: A proper posterior $π_{y}$ , the exact conditional-score decomposition, a posterior path law and its adjacent reverse conditional kernels when that sufficient exactness condition is considered, a variational density $q$ with finite log integrals, and a Markov kernel $K_{y}$ when the MCMC comparison is considered.

[A1]

$π_{y}$ is the probability measure with density $Z_{y}^{- 1} ℓ_{y} ρ_{0}$ and positive finite evidence specified in the Example.

[A2]

For the exact finite reverse-kernel case, a joint posterior path law $Π_{y}$ on $(X_{0}, \dots, X_{N})$ has marginal laws $Π_{y, k}$ , with $Π_{y, N} = ν_{N}$ and $Π_{y, 0} = π_{y}$ . Its supplied Borel conditional kernels $R_{k}$ satisfy $Π_{y, k - 1} (B) = \int R_{k} (B ∣ x, y) Π_{y, k} (d x)$ for every Borel $B$ . All operations succeed almost surely within their declared budgets in this sufficient exactness case.

[A3]

For the variational calculation, $q$ is a density with respect to $λ$ , $q ≪ π_{y}$ , and the displayed logarithmic integrals are finite.

[A4]

For the latent optimization comparison, the initialization and tie-breaking are fixed, so the returned point $G (\hat{z})$ is deterministic for the fixed observation.

[A5]

In the non-atomic particle comparison, $π_{y} ({x}) = 0$ for every $x$ . Uniform Lebesgue probability on $[0, 1]$ is one such target.

[F1]

The exact conditional score under Gaussian noising is the sum of the prior score and the averaged measurement score. Conditional-score and Tweedie identities for Gaussian smoothing

[F2]

The reverse-kernel algorithm takes a terminal law, corrected probability kernels and optional projections, and makes no posterior-exactness claim by default. Measurement-guided reverse-kernel reconstruction

[F3]

A probability measure $π$ is invariant for a probability kernel $K$ when $\int K (x, A) π (d x) = π (A)$ for every measurable $A$ . Invariant and stationary distribution for a Markov kernel

Verification

Given: The assumptions in the Example and in [A1]–[A5].

1.1A2F1F2construct

In the explicit-score family, [F1] identifies the exact conditional score. Under [A2], the initial sampler state has law $Π_{y, N} = ν_{N}$ . If its state at step $k$ has law $Π_{y, k}$ , applying $R_{k}$ gives $B$ probability $\int R_{k} (B ∣ x, y) Π_{y, k} (d x) = Π_{y, k - 1} (B)$ . Induction down to $k = 1$ therefore gives output law $Π_{y, 0} = π_{y}$ . This applies to [F2] with these kernels and no projection. Almost-sure success makes completed status have probability one, introducing no conditioning bias. The sampled joint path need not have law $Π_{y}$ ; only its one-time marginals are used.

1.2A1A3algebra

Since $q ≪ π_{y}$ and both are probability measures, the Radon–Nikodym density $f = d q / d π_{y}$ has integral one. Convexity of $t \mapsto t \log t$ and Jensen give $D_{K L} (q ∥ π_{y}) = \int f \log f d π_{y} \geq 0$ , with equality only when $f = 1$ $π_{y}$ -almost everywhere. Since $π_{y} (x) = Z_{y}^{- 1} ℓ_{y} (x) ρ_{0} (x)$ , the integrability in [A3] gives $ELBO (q) = \int q \log (Z_{y} π_{y} / q) d λ = \log Z_{y} - \int q \log (q / π_{y}) d λ \leq \log Z_{y}$ , with equality exactly when $q = π_{y}$ . Thus the variational output law is its selected $q$ ; it is the target law in the exact case $q = π_{y}$ , and it differs from the target precisely when the induced probability measures differ, equivalently when the densities disagree on a set of positive $λ$ -measure.

1.3A1A4construct

Under [A4], repeated execution of the deterministic latent optimizer returns the single value $G (\hat{z})$ , so its output law is $δ_{G (\hat{z})}$ . If $π_{y}$ is not a point mass, this point mass is not $π_{y}$ . The stated objective contains only the measurement residual; it does not specify the negative log posterior or a global optimizer of that posterior objective, so the given inputs establish no MAP conclusion.

1.4F3algebra

If a Markov chain starts with law $π_{y}$ , one transition has law $π_{y} K_{y} = π_{y}$ by [F3]; induction gives the same marginal after every finite number of transitions. In contrast, on the Borel space $R$ let $π = (δ_{0} + δ_{1}) / 2$ and let $K (x, B) = 1_{B} (x)$ . Then $π K = π$ , but a chain starting at $δ_{0}$ remains $δ_{0}$ after each finite step and is never $π$ .

1.5F2constructalgebra

To check the failure boundary, take $d = p = 1$ , the Borel observation space $Y = R$ , $y = 0$ , and $A (x) = 0$ . Let $μ_{0} = π_{y} = (δ_{0} + δ_{1}) / 2$ , dominated by $λ = δ_{0} + δ_{1}$ with $ρ_{0} \equiv 1 / 2$ . Take $Y \equiv 0$ , whose conditional density is identically one relative to the finite observation measure $δ_{0}$ on $R$ ; thus $ℓ_{y} \equiv 1$ and $Z_{y} = 1$ . Use $N = 2$ , $t_{k} = k$ , terminal law $ν_{2} = δ_{0}$ , zero score, singleton correction spaces $C_{1} = C_{2} = {*}$ , corrections, for every $v \in R$ , $C_{2} (x, v) = *$ and $C_{1} (x, v) = ⊥$ exactly when $x = 1$ (otherwise $*$ ), kernels $Q_{θ, 2} (B ∣ x, v, *) = π_{y} (B)$ and $Q_{θ, 1} (B ∣ x, v, *) = δ_{x} (B)$ , and no projection. Initialize deterministically at zero, sample $Q_{θ, 2}$ with one fair coin, and sample $Q_{θ, 1}$ by returning the current state; these operations have finite budgets. The exact path law is $Π_{y} (d x_{0}, d x_{1}, d x_{2}) = δ_{0} (d x_{2}) π_{y} (d x_{1}) δ_{x_{1}} (d x_{0})$ , so its reverse kernels draw $X_{1}$ from $π_{y}$ and then set $X_{0} = X_{1}$ . Completion occurs exactly when $X_{1} = 0$ , so its probability is $1 / 2$ and its conditional output law is $δ_{0}$ , not $π_{y}$ . Thus allowing the failure branches in [F2] invalidates an unconditional exactness inference from the reverse kernels alone.

2.1A5construct∎

A realized weighted empirical particle measure ${\hat{π}}_{K} = \sum_{j = 1}^{K} w_{j} δ_{x_{j}}$ is supported on the finite set $S = {x_{1}, \dots, x_{K}}$ . By [A5] and finite additivity, $π_{y} (S) \leq \sum_{j = 1}^{K} π_{y} ({x_{j}}) = 0$ , while ${\hat{π}}_{K} (S) = 1$ , so the finite empirical measure is not the target. This does not determine the marginal law of a randomly selected particle; that exactness is a separate algorithm-specific claim.

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Sources

Giannis Daras et al., A Survey on Diffusion Models for Inverse Problems (2024), arXiv:2410.00083 (standard reference, not scraped)
Yang Song et al., Score-Based Generative Modeling through Stochastic Differential Equations (2021), arXiv:2011.13456 (standard reference, not scraped)
Hyungjin Chung et al., Diffusion Posterior Sampling for General Noisy Inverse Problems (2023), arXiv:2209.14687 (standard reference, not scraped)
Simon J. D. Prince, Understanding Deep Learning, Chapter 18: Diffusion Models (2026), pp. 363–387; full course notes (standard reference, not scraped)