ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-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.

Correct invariance can coexist with low effective sample size

Example

Assume AC. Let $E = {- 1, + 1}$ with the discrete sigma-algebra $2^{E}$ , and let the target posterior $π$ be uniform on $E$ . For $0 < ε < 1 / 2$ , define $K_{ε} (x, A) : = (1 - ε) 1_{A} (x) + ε 1_{A} (- x) (x \in E, A \subseteq E),$ and set $ε = 1 / 100$ . The Ionescu--Tulcea construction supplies a time-homogeneous Markov chain on canonical path space with initial law $π$ and transition kernel $K_{ε}$ . Use this chain with its natural filtration and set $f (x) = x$ . Define its lag- $k$ autocorrelation by $ρ_{f} (k) : = \frac{Cov (f (X_{0}), f (X_{k}))}{Var (f (X_{0}))} .$ For every $k \geq 1$ , $ρ_{f} (k) = (49 / 50)^{k}$ . Its integrated autocorrelation time $τ_{f} : = 1 + 2 \sum_{k = 1}^{\infty} ρ_{f} (k)$ is therefore $99$ , so the theoretical functional-specific effective sample size of $N = 10,000$ draws is ${ESS}_{f} (10,000) = \frac{10,000}{99} \approx 101.01.$ The invariant law is exactly the target posterior, while this functional has only about one percent of the nominal independent-sample size. This is the theoretical ESS from the stationary autocorrelation sequence, not an empirical finite-run ESS estimate, and it asserts no central limit theorem.

Example references

The theoretical ESS uses a stationary functional with positive finite variance, absolutely summable autocorrelations, and positive integrated autocorrelation time.

An invariant initial law makes the Markov chain strictly stationary under AC.

Under AC, an initial law and a probability kernel give a canonical path-space Markov chain.

Conditional expectation is linear and preserves expectation.

A bounded measurable factor can be taken outside conditional expectation.

The stationary sample-average variance bound requires its own covariance-summability assumption.

Vehtari et al., §3.2, functional-specific ESS

Facts & Assumptions

Given: The two-state probability kernel, its initial law $π$ , and the bounded functional $f$ in the Example.

[F1]

A probability kernel has probability-measure rows and measurable event evaluations. Measure kernel and probability kernel

[F2]

Invariance means $π K (A) = π (A)$ for every measurable event $A$ . Invariant and stationary distribution for a Markov kernel

[F3]

A time-homogeneous Markov chain has one-step conditional law given by its transition kernel. Time-homogeneous Markov chain with transition kernel

[A1]

AC is assumed for conditional-expectation existence, the Ionescu--Tulcea chain construction, and the invariant-initial-law theorem. The Axiom of Choice

[F4]

If the initial law is invariant, the chain is strictly stationary. Invariant initial law makes a Markov chain stationary

[F5]

The theoretical functional ESS is $N / (1 + 2 \sum_{k \geq 1} ρ_{f} (k))$ when the process is strictly stationary, has positive finite variance, absolutely summable autocorrelations, and positive integrated autocorrelation time; this definition asserts no CLT. Functional-specific effective sample size

[F6]

For a stationary chain with absolutely summable lag covariances, the finite-sample mean-square error is bounded by $N^{- 1} (γ_{0} + 2 \sum_{k \geq 1} ∣ γ_{k} ∣)$ . Bounded posterior functionals and stationary sample error

[F7]

Variance is the expected squared centered value, and covariance is the expected product of centered values. Moments, variance, and covariance on a probability space

[F8]

Under AC, the Ionescu--Tulcea construction gives a canonical path-space chain with the specified initial law and homogeneous transition kernel. Ionescu-Tulcea construction of a Markov chain

[F9]

Conditional expectation is linear and preserves expectation for integrable variables. Basic algebra and order properties of conditional expectation

[F10]

A bounded measurable factor can be taken outside conditional expectation. Taking out what is known

Verification

1.1A1F1F8given

Since $0 < ε < 1 / 2$ , each row assigns the nonnegative masses $1 - ε$ and $ε$ to the two states, and every subset is measurable; thus $K_{ε}$ is a probability kernel [F1]. Under AC [A1], apply [F8] to initial law $π$ and kernel $K_{ε}$ ; this supplies the canonical chain used in the Example.

2.1F1F2step 1.1givenalgebra

For either singleton ${x}$ , $π K_{ε} ({x}) = \frac{1}{2} (1 - ε) + \frac{1}{2} ε = \frac{1}{2} = π ({x})$ ; additivity gives $π K_{ε} = π$ by [F2].

2.2F3F5F7F9F10step 1.1givenalgebra

Put $λ = 1 - 2 ε = 49 / 50$ . On the two-point state space, summing the conditional probabilities in [F3] against the values $+ 1$ and $- 1$ and using linearity [F9] gives $E [X_{n + 1} ∣ F_{n}] = λ X_{n}$ . Since $X_{0} \sim π$ , $E X_{0} = 0$ ; expectation preservation [F9] gives $E X_{n} = 0$ for all $n$ . As $X_{0}$ is bounded and $F_{k}$ -measurable, [F10] and expectation preservation [F9] give $E [X_{0} X_{k + 1}] = λ E [X_{0} X_{k}]$ , and $E [X_{0}^{2}] = 1$ . Induction yields $Cov (X_{0}, X_{k}) = λ^{k}$ and, since $Var (X_{0}) = 1$ , $ρ_{f} (k) = λ^{k}$ .

3.1A1F2F4F7step 2.1given

Since $X_{0} \sim π$ and $π$ is invariant by step 2.1, [A1] permits the invariant-initial-law result [F4], which makes the chain strictly stationary. Its one-time law is symmetric, so $E X_{j} = 0$ and $Var (X_{j}) = 1$ .

4.1F5F6F7step 2.2step 3.1givenalgebra∎

Since the chain is strictly stationary by step 3.1 and has variance one, [F5] applies. Step 2.2 gives $ρ_{f} (k) = λ^{k}$ ; these autocorrelations are absolutely summable because $0 < λ < 1$ , with $\sum_{k \geq 1} λ^{k} = λ / (1 - λ) = 49$ . Thus $τ_{f} = 1 + 2 \cdot 49 = 99 > 0$ and $[F 5]$ gives ${ESS}_{f} (10,000) = 10,000 / 99$ . Since $γ_{k} = ρ_{f} (k)$ here, [F6] applies, but neither it nor the ESS definition supplies a CLT.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

50 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

Aki Vehtari et al., Rank-normalization, folding, and localization: An improved R-hat for assessing convergence of MCMC (standard reference, not scraped)