ExampleConstruction: Literature-sourcedVerification: Literature-sourcedPipeline-generatedverified 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.

Posterior summaries close to DRAM with low independent-MH acceptance

Example

Assume the Axiom of Choice for the general density-based Metropolis–Hastings interface. In the stationary-diffusion inverse-problem experiment with $m = 256$ observations, noise variance $γ^{2} = 10^{- 5}$ , and $n_{K L} = 128$ prior modes, use the baseline conditional-flow-matching MLP with 800 hidden units. The reported flow-matching independent-Metropolis–Hastings acceptance rates for four different true parameters are respectively $0.66 %$ , $11.21 %$ , $5.97 %$ , and $5.47 %$ . In this same experiment, the reported posterior means and 95% bands agree with the DRAM reference to between $0.8 %$ and $1.8 %$ in relative $L^{2}$ . The first truth is a prior-predictive tail observation: its normalized radius is $2.53$ , the median training radius is $0.85$ , and only $0.12 %$ of the two million training observations have a larger radius. This reported case shows that close posterior means and bands do not by themselves imply high independent-Metropolis–Hastings acceptance or establish full posterior accuracy.

Example references

A density-based Metropolis–Hastings correction accepts according to the common-density target/proposal ratio.

Blechschmidt et al., §4.2.1–§4.2.2, Fig. 6 and Table 1

Roberts and Rosenthal, §2.1 and Proposition 2

Facts & Assumptions

Given: The reported setting and empirical quantities in the Statement.

[A1]

Assume AC for the general common-density Metropolis–Hastings construction. The Axiom of Choice, The common-density Metropolis–Hastings ratio

[F1]

In the stated $m = 256$ , $γ^{2} = 10^{- 5}$ , $n_{K L} = 128$ setting, Table 1 reports baseline MLP-800 acceptance rates $(0.66, 11.21, 5.97, 5.47) %$ for the four truths. [Blechschmidt et al., §4.2.1, Fig. 6 and Table 1]

[F2]

In the same Fig. 6 setting, the paper reports that the MLP's posterior means and 95% bands match the DRAM reference between $0.8 %$ and $1.8 %$ in relative $L^{2}$ . [Blechschmidt et al., §4.2.1]

[F3]

For the first truth the normalized observation radius is $2.53$ versus median training radius $0.85$ , and the source reports that $0.12 %$ of the two million training observations have a still larger radius. It describes the sine observation as an extreme prior-predictive tail with almost no nearby training data. [Blechschmidt et al., §4.2.2]

[F4]

The acceptance figures are for the paper's independent-MH correction, which uses a flow-matching proposal and a density-based acceptance ratio; the general density-ratio interface is stated in the published MH lemma. The common-density Metropolis–Hastings ratio

Verification

Given: [A1], [F1]–[F4] and the empirical qualifications in the Statement.

1.1F1givenalgebra

The source labels its proposal model as a conditional-flow-matching MLP. The four rates in [F1] are all below $12 %$ , while the fourth is $5.47 %$ and the first is $0.66 %$ . Thus the table itself records low acceptance across all four truths for this baseline configuration; these are the reported finite-experiment rates, not an asymptotic bound.

2.1A1F1F2F3F4step 1.1given∎

By [F2], posterior means and 95% bands are nevertheless close to the DRAM reference in relative $L^{2}$ . By [F3], the first observation lies far into the empirical prior-predictive tail, which accompanies the lowest rate. Under [A1], [F4] identifies the rates as those of the paper's density-based independent-MH correction. Hence this is a concrete reported case where close mean and band summaries coexist with low independent-MH acceptance; it does not prove that every low-acceptance flow has accurate summaries or that these summaries establish full-distribution accuracy.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

25 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

Jan Blechschmidt, Oliver G. Ernst, Moritz Poguntke, and Björn Sprungk, Flow Matching for Fast Posterior Sampling in Bayesian Inverse Problems (standard reference, not scraped)
Gareth O. Roberts and Jeffrey S. Rosenthal, General State Space Markov Chains and MCMC Algorithms (standard reference, not scraped)