PropositionStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-6-luna)verified 2026-10-22 (gpt-6.1-sol)

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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.

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Posterior-mean plug-in measurement scores need not be exact

Statement

Let $X_{0}$ take values $- 1$ and $1$ with equal probability, let $X_{σ} = X_{0} + σ Z$ with $σ > 0$ and standard Gaussian noise independent of $X_{0}$ , and let $Y = X_{0} + τ W$ for $W \sim N (0, 1)$ independent of $(X_{0}, Z)$ , with $τ > 0$ . At the observation $y = 0$ , the exact measurement density $r_{σ} (0 ∣ x) = E [φ_{τ} (- X_{0}) ∣ X_{σ} = x]$ is constant in $x$ , so its score is zero. The exact posterior mean is $m_{σ} (x) = E [X_{0} ∣ X_{σ} = x] = \tanh (x / σ^{2})$ , where $\tanh (t) = (e^{2 t} - 1) / (e^{2 t} + 1)$ . The plug-in density $φ_{τ} (- m_{σ} (x))$ has score $- m_{σ} (x) m_{σ}^{'} (x) / τ^{2}$ , which is nonzero for $x \neq 0$ . Thus evaluating the measurement likelihood at the exact posterior mean does not in general recover the exact measurement score; this example does not rule out equality in special models.

Statement references

Conditional-score and Tweedie identities for Gaussian smoothing

Diffusion prior and measurement scores

Daras et al., §2.4.4, Eq. 2.20, and §3.1.4, Eq. 3.8

Chung et al., §3.1, Eqs. 7–15 and Theorem 1

Facts & Assumptions

Given: The two-point prior, the independent Gaussian noising and observation noises, and $σ, τ > 0$ .

[A1]

$P (X_{0} = 1) = P (X_{0} = - 1) = 1 / 2$ and $X_{σ} = X_{0} + σ Z$ with $Z$ standard Gaussian independent of $X_{0}$ .

[A2]

The observation noise $W \sim N (0, 1)$ is independent of $(X_{0}, Z)$ , and $Y = X_{0} + τ W$ .

[A3]

The Gaussian density is $φ_{τ} (v) = (\sqrt{2 π} τ)^{- 1} \exp (- v^{2} / (2 τ^{2}))$ .

[F1]

Under additive Gaussian noising, the posterior mean is $m_{σ} (x) = x + σ^{2} \nabla \log p_{σ} (x)$ . Conditional-score and Tweedie identities for Gaussian smoothing

[F2]

The exact measurement score is $g_{σ} (x, y) = \partial_{x} \log r_{σ} (y ∣ x)$ , where $r_{σ} (y ∣ x)$ averages the clean-state likelihood over the conditional law of $X_{0}$ given $X_{σ} = x$ . Diffusion prior and measurement scores

Proof

Given: The hypotheses in the Statement.

1.1A1F1algebra

For $x \in R$ , Bayes' formula gives $\frac{P (X_{0} = 1 ∣ X_{σ} = x)}{P (X_{0} = - 1 ∣ X_{σ} = x)} = \frac{φ_{σ} (x - 1)}{φ_{σ} (x + 1)} = \exp (2 x / σ^{2})$ . Therefore the posterior mean is $(e^{2 x / σ^{2}} - 1) / (e^{2 x / σ^{2}} + 1) = \tanh (x / σ^{2})$ , agreeing with the conditional-mean version in [F1].

1.2A1A2A3F2algebra

At $y = 0$ , both possible clean states have the same likelihood density: $φ_{τ} (0 - 1) = φ_{τ} (0 + 1) = φ_{τ} (1)$ . Averaging these equal values over the conditional law from [F2] gives $r_{σ} (0 ∣ x) = φ_{τ} (1)$ for every $x$ , so its logarithmic derivative and the exact measurement score are zero.

2.1A3step 1.1algebra

The plug-in log density is $\log φ_{τ} (- m_{σ} (x)) = - \log (\sqrt{2 π} τ) - m_{σ} (x)^{2} / (2 τ^{2})$ . Differentiating the quotient expression in step 1.1 gives $m_{σ}^{'} (x) = 4 e^{2 x / σ^{2}} / (σ^{2} (e^{2 x / σ^{2}} + 1)^{2}) > 0$ . Hence the plug-in score is $- m_{σ} (x) m_{σ}^{'} (x) / τ^{2}$ , which is nonzero whenever $x \neq 0$ .

3.1step 1.2step 2.1given∎

The exact score in step 1.2 and the plug-in score in step 2.1 disagree for every nonzero $x$ . Thus the exact posterior mean does not, in general, make likelihood-at-the-mean guidance equal to the exact measurement score. The calculation permits special cases of equality but refutes a general identity.

Depends on

Used by

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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)
Hyungjin Chung et al., Diffusion Posterior Sampling for General Noisy Inverse Problems (2023), arXiv:2209.14687 (standard reference, not scraped)