AlgorithmAlgorithm: Literature-sourcedProof: Not applicablePipeline-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.

Finite-simulation plug-in synthetic-likelihood evaluation

Definition

Assume AC only as required by the synthetic-likelihood definition and its Gaussian-density supplier (The Axiom of Choice Ideal and finite-simulation synthetic likelihood A positive-definite multivariate normal law has its Gaussian density). IID means common law and mutual independence as in Identical distribution and IID families. This algorithm states the vector sample mean and covariance directly; covariance uses centered second moments as in Moments, variance, and covariance on a probability space.

Inputs: A parameter $θ$ , observed data $y$ , a summary statistic $S : X \to R^{d}$ satisfying the contract of Ideal and finite-simulation synthetic likelihood, an integer simulation count $m \geq 2$ , and a simulator that returns $m$ mutually independent observations with common law $P_{θ}$ .

Output: Either the finite-simulation plug-in synthetic-likelihood value at $S (y)$ , or the explicit result unavailable.

Simulate $Y_{1}, \dots, Y_{m}$ and compute $S_{i} : = S (Y_{i})$ . For each coordinate $j$ , compute ${\overset{ˉ}{S}}_{m, j} : = m^{- 1} \sum_{i = 1}^{m} S_{i, j}$ .
Set ${\overset{ˉ}{S}}_{m} : = ({\overset{ˉ}{S}}_{m, 1}, \dots, {\overset{ˉ}{S}}_{m, d})^{T}$ and compute ${\hat{Σ}}_{m} : = (m - 1)^{- 1} \sum_{i = 1}^{m} (S_{i} - {\overset{ˉ}{S}}_{m}) (S_{i} - {\overset{ˉ}{S}}_{m})^{T} .$
If ${\hat{Σ}}_{m}$ is not positive definite, stop and return unavailable. Do not add a ridge term or otherwise modify the covariance.
If ${\hat{Σ}}_{m}$ is positive definite, take its unique Cholesky factor $L$ with positive diagonal by A matrix admits a Cholesky factorisation with positive diagonal exactly when it is Hermitian positive definite, and that factor is unique. Return the plug-in Gaussian density $(2 π)^{- d / 2} \det ({\hat{Σ}}_{m})^{- 1 / 2} \exp ⁣ [- \frac{1}{2} (S (y) - {\overset{ˉ}{S}}_{m})^{T} {\hat{Σ}}_{m}^{- 1} (S (y) - {\overset{ˉ}{S}}_{m})]$ from Ideal and finite-simulation synthetic likelihood and A positive-definite multivariate normal law has its Gaussian density.

This algorithm evaluates one parameter value. It does not optimize over $θ$ and asserts no unbiasedness, convergence, consistency, posterior calibration, or optimizer property.

Depends on

Used by

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)
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)