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Validity of effective sample size for importance-weighted posterior samples

Determine whether the effective sample size (ESS) can be correctly defined and estimated for posterior samples that are reweighted by importance weights to correct the Kim–Shephard–Chib Gaussian off-set mixture approximation to the exact stochastic volatility model, either by expressing ESS as a function of weighted expectations or by computing ESS from importance-resampled samples, and establish the conditions under which such ESS estimates are valid.

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Background

The paper uses importance weights to correct approximation error in the Kim–Shephard–Chib (KSC) Gaussian off-set mixture approach to stochastic volatility, and applies these weights to rank statistics within the Simulation Based Calibration framework.

However, the authors omit efficiency (ESS) calculations for the reweighted samples because the appropriate definition and computation of ESS under importance reweighting or importance-resampling is uncertain in this context.

References

Efficiency estimates of the re-weighted posterior samples are omitted. It was unclear at the time of writing whether the ESS could be written as a function of the expectation or whether the ESS from re-sampled posterior samples using the importance weights are valid.

— Comparing MCMC algorithms in Stochastic Volatility Models using Simulation Based Calibration (2402.12384 - Wee, 28 Jan 2024) in Section 5.3, Correcting approximation error using importance weights (footnote)