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A Gaussian synthetic likelihood can change an exact posterior

Statement refuted

An identifiable discrete summary model's Gaussian synthetic likelihood need not preserve the exact posterior, even when the ideal Gaussian uses the exact first two summary moments.

Facts & Assumptions

Given: Let $Θ = {0, 1}$ with equal prior mass and let the summary/data space be ${0, 1, 2, 3}$ with its power-set sigma-algebra. Regard the identity summary as a map into $(R, B (R))$ . Define the two sampling laws by $\begin{array}{crrrr} y & 0 & 1 & 2 & 3 \\ P_{0} (y) & 6 / 24 & 6 / 24 & 6 / 24 & 6 / 24 \\ P_{1} (y) & 5 / 24 & 9 / 24 & 3 / 24 & 7 / 24 \end{array}$ and observe $y = 1$ . Define the exact posterior from this likelihood, and define the synthetic-likelihood posterior by normalizing the ideal Gaussian synthetic likelihood against the same prior.

[F1]

An exact target is $Π_{x} (A) = Z_{x}^{- 1} \int_{A} ℓ_{x} d Π_{0}$ for an explicitly measurable likelihood and finite positive evidence. Exact inferential targets and approximation chains

[F2]

A statistic is a measurable map to a declared measurable target, and the ideal synthetic likelihood is the Gaussian density at the observed summary using the exact summary mean and covariance when that covariance is positive definite. Statistics and estimators A measurable function between measurable spaces The Borel sigma-algebra of a topological space Ideal and finite-simulation synthetic likelihood

[F3]

The identity estimand is identifiable when equal sampling laws imply equal parameter values. Parametric statistical models Observational equivalence Identifiability of an estimand

[F4]

Under the Axiom of Choice, positive-definite summary covariance supplies the ideal Gaussian density; no equality with the exact summary law is part of that definition. The Axiom of Choice Ideal and finite-simulation synthetic likelihood

[F5]

Total variation is the event-probability supremum. Total variation distance for probability laws

Counterexample

Given: The finite model and prior in Facts & Assumptions.

1.1givenF2F3algebra

Each row is nonnegative and sums to one. The laws differ because $P_{0} ({1}) = 1 / 4$ whereas $P_{1} ({1}) = 3 / 8$ ; hence the identity estimand is identifiable by [F3]. The finite identity summary is measurable because the source and target carry their stated power-set and Borel sigma-algebras.

1.2F2F4givenalgebra

Under $P_{0}$ , $E [S] = (0 + 1 + 2 + 3) / 4 = 3 / 2$ and $E [S^{2}] = (0 + 1 + 4 + 9) / 4 = 7 / 2$ . Under $P_{1}$ , the corresponding weighted sums are $(9 + 6 + 21) / 24 = 3 / 2$ and $(9 + 12 + 63) / 24 = 7 / 2$ . Thus both variances equal $7 / 2 - (3 / 2)^{2} = 5 / 4 > 0$ , so the one-dimensional covariance is positive definite and the ideal Gaussian synthetic likelihood is defined by [F2], [F4].

2.1F1F2step 1.2algebra

The exact likelihoods at $y = 1$ are $1 / 4$ and $3 / 8$ , so the equal-prior exact posterior mass at $θ = 1$ is $(3 / 8) / (1 / 4 + 3 / 8) = 3 / 5$ . The ideal Gaussian synthetic likelihoods coincide because their means and variances coincide. Their common positive value cancels in normalization against the equal prior, so the synthetic-likelihood posterior has mass $1 / 2$ at $θ = 1$ .

3.1F5step 2.1algebra∎

The two posterior measures differ on event ${1}$ by $∣ 3 / 5 - 1 / 2 ∣ = 1 / 10$ , so their total-variation distance is at least $1 / 10$ by [F5]. Thus matching the first two moments does not remove this posterior target bias.

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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, Clara Grazian, Kerrie Mengersen, and Christian Robert, Approximating the Likelihood in Approximate Bayesian Computation, Handbook of Approximate Bayesian Computation (2020) (standard reference, not scraped)
David T. Frazier, David J. Nott, Christopher Drovandi, and Robert Kohn, Bayesian Inference Using Synthetic Likelihood: Asymptotics and Adjustments, 2019 arXiv preprint arXiv:1902.04827 (standard reference, not scraped)