ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generated

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.

A Gaussian synthetic likelihood for a Bernoulli summary is not exact

Example

Let $Θ = (0, 1)$ , let $Y ∣ p$ be Bernoulli with $P_{p} (Y = 1) = p$ and $P_{p} (Y = 0) = 1 - p$ , and use the identity summary $S (Y) = Y$ .

Facts & Assumptions

Given: The Bernoulli model and statistic above, with $p \in (0, 1)$ .

[A1]

AC is assumed only to use the Gaussian synthetic-likelihood density interface. The Axiom of Choice

[F1]

The synthetic likelihood uses the Gaussian density with the exact summary mean and covariance when that covariance is positive definite. Ideal and finite-simulation synthetic likelihood

[F2]

A positive-definite multivariate-normal law has the stated Lebesgue density. A positive-definite multivariate normal law has its Gaussian density

[F3]

A statistic is a measurable map, and its law is the pushforward law. Statistics and estimators Law or distribution of a random element

[F4]

Means and variances are expectations and centered second moments. Expectation of a nonnegative or integrable random variable Moments, variance, and covariance on a probability space

Verification

1.1F1F2F3F4givenalgebra

The identity summary has mean $p$ and variance $p (1 - p) > 0$ , so its ideal Gaussian synthetic likelihood is the density $g_{p} (x) = (2 π p (1 - p))^{- 1 / 2} \exp ⁣ (- \frac{(x - p)^{2}}{2 p (1 - p)})$ .

2.1A1F1F2step 1.1algebra

For $x \in (1 / 4, 3 / 4)$ and $p \in (0, 1)$ , $∣ x - p ∣ < 1$ . Hence $g_{p} (x) \geq c_{p} : = (2 π p (1 - p))^{- 1 / 2} \exp ⁣ (- \frac{1}{2 p (1 - p)}) > 0$ . The Gaussian law therefore assigns $(1 / 4, 3 / 4)$ probability at least $c_{p} / 2 > 0$ , since this interval has Lebesgue length $1 / 2$ .

3.1F3step 2.1algebra∎

The true summary law is supported on ${0, 1}$ , so it assigns $(1 / 4, 3 / 4)$ probability zero. It differs from the Gaussian law in step 2.1. Thus the Gaussian synthetic likelihood is not the exact likelihood of this non-Gaussian summary model.

Depends on

Used by

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Dependency tree · two levels

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Sources

Victor M.-H. Ong et al., Variational Bayes with Synthetic Likelihood, Statistics and Computing 28(4), 971–988 (2018) (standard reference, not scraped)
Christopher C. Drovandi et al., Approximating the Likelihood in Approximate Bayesian Computation, Handbook of Approximate Bayesian Computation (2020) (standard reference, not scraped)