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.

Two simulated square summaries give a random singular covariance

Example

Let the summary vector $Z = (Z^{(1)}, Z^{(2)})$ be uniform on the four points ${(- 1, - 1), (- 1, 1), (1, - 1), (1, 1)} \subset R^{2}$ . For one parameter value, simulate $Z_{1}, Z_{2}$ independently from this law and use the two-simulation covariance formula of Finite-simulation plug-in synthetic-likelihood evaluation.

Facts & Assumptions

Given: Two independent draws from the stated four-point summary law.

[A1]

AC is inherited through the procedure’s Gaussian-density branch; it is not used in the finite probability calculations. The Axiom of Choice

[F1]

IID means a common law and mutual independence. Identical distribution and IID families

[F2]

The covariance matrix uses centered second moments; the finite plug-in covariance is the Bessel-corrected matrix in the algorithm. Moments, variance, and covariance on a probability space Finite-simulation plug-in synthetic-likelihood evaluation

[F3]

For any two two-dimensional vectors, that sample covariance is not positive definite. A two-simulation covariance in two dimensions is singular

[F4]

The algorithm returns unavailable when its plug-in covariance is not positive definite. Finite-simulation plug-in synthetic-likelihood evaluation

[F5]

The ideal synthetic likelihood uses the population summary mean and covariance, when the latter is positive definite. Ideal and finite-simulation synthetic likelihood

Verification

1.1F2F5givenalgebra

Each coordinate has mean zero and variance one, while the cross moment is zero. Hence the population summary covariance is $I_{2}$ , positive definite.

1.2F1F2givenalgebra

By independence and the four equally likely support points, $P (Z_{1} = Z_{2}) = 4 (1 / 4)^{2} = 1 / 4$ . On this event the sample covariance is the zero matrix.

2.1F1F2step 1.2algebra

The event $Z_{1} = (1, 1)$ and $Z_{2} = (- 1, - 1)$ has probability $1 / 16$ . On it, ${\overset{ˉ}{Z}}_{2} = (0, 0)$ and ${\hat{Σ}}_{2} = (\begin{array}{cc} 2 & 2 \\ 2 & 2 \end{array})$ , which differs from the zero matrix in step 1.2. Thus the sample covariance is a nonconstant random matrix.

3.1A1F3F4step 1.1algebra∎

By [F3], every two-simulation covariance here is not positive definite, so [F4] makes the finite plug-in synthetic likelihood unavailable for every outcome despite the positive-definite population covariance in step 1.1.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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Sources

David T. Frazier et al., Bayesian Inference Using Synthetic Likelihood: Asymptotics and Adjustments, Journal of the American Statistical Association 118(544), 2821–2832 (2023) (standard reference, not scraped)
Victor M.-H. Ong et al., Variational Bayes with Synthetic Likelihood, Statistics and Computing 28(4), 971–988 (2018) (standard reference, not scraped)