CounterexampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedjudge pass (gpt-6-luna)

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Refreshing the current estimate can destroy exactness

Statement refuted

Replacing an exact likelihood by an unbiased estimate freshly drawn at both the current and proposed parameter in every Metropolis step necessarily leaves the exact parameter posterior invariant.

Facts & Assumptions

Given: The MCWM procedure in Monte Carlo within Metropolis.

[F1]

Monte Carlo within Metropolis: MCWM refreshes the current estimate at every iteration instead of retaining it after rejection.

Counterexample

Take $Θ = {0, 1}$ , uniform prior, exact likelihoods $L_{0} = L_{1} = 1$ , and a proposal that always flips the parameter. At $0$ use estimator ${\hat{L}}_{0} = 1$ surely; at $1$ use ${\hat{L}}_{1} \in {1 / 2, 3 / 2}$ with equal probabilities. Both estimators are strictly positive and unbiased.

1.1F1givenalgebra

Under MCWM, a move $0 \to 1$ accepts with conditional probabilities $1 / 2$ and $1$ for the two proposed estimates. Thus $P (0, 1) = (1 / 2) (1 / 2) + (1 / 2) (1) = 3 / 4$ .

1.2F1givenalgebra

For a move $1 \to 0$ , the fresh current estimate is $1 / 2$ or $3 / 2$ with equal probability and the proposed estimate is $1$ . The acceptance probabilities are $1$ and $2 / 3$ , so $P (1, 0) = (1 / 2) (1) + (1 / 2) (2 / 3) = 5 / 6$ .

2.1F1step 1.1step 1.2algebra∎

Starting from the claimed uniform posterior, the next mass at state $1$ is $(1 / 2) (3 / 4) + (1 / 2) (1 - 5 / 6) = 11 / 24 \neq 1 / 2$ . Hence the exact posterior is not invariant. The unequal cross-flows also violate detailed balance.

Depends on

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Sources

Andrieu, Doucet, and Holenstein, Particle Markov chain Monte Carlo methods, §4.4 (standard reference, not scraped)
Särkkä and Svensson, Bayesian Filtering and Smoothing, 2nd ed., §16.3.5 (standard reference, not scraped)