Beyond Effective Sample Size: Effective Number of Proposals for Adaptive Importance Sampling

2026-08-15

AI Summary

This study addresses the limitation of traditional Effective Sample Size (ESS) in detecting proposal redundancy and collapse within Adaptive Importance Sampling. We propose Effective Number of Proposals (ENP), a novel metric integrating normalized weights with sample similarity to accurately assess non-redundant proposal contributions, thereby overcoming ESS diagnostic failures. Serving as a feedback signal, ENP effectively identifies overlooked proposal degeneracy and guides adaptive update strategies. Experimental results demonstrate that this metric significantly enhances both diagnostic accuracy and sampling quality in Population Adaptive Importance Sampling for complex distribution approximation. By providing a more reliable assessment of proposal diversity, ENP offers a robust alternative to ESS, ensuring more stable and efficient adaptation in high-dimensional inference tasks where standard diagnostics often fail to capture structural deficiencies in the proposal mixture.

Abstract

Population-based adaptive importance sampling (AIS) methods use a set of proposal densities to approximate complex target distributions. Their performance is commonly assessed through effective sample size (ESS) and related weight-based diagnostics, which measure the concentration of normalized importance weights. However, a large ESS only indicates that the normalized sample weights are not strongly concentrated; it does not describe how the proposal components are arranged in the sampling space. In population-based AIS, several proposal components may generate samples in the same region of the target, so the sample weights can appear well balanced even though the effective number of distinct proposal components is small. This letter introduces the effective number of proposals (ENP), a similarity-aware proposal-level diagnostic for population-based AIS. ENP combines the total normalized weight assigned to each proposal with a redundancy measure computed from similarities among target-weighted samples, estimating the number of non-redundant empirical proposal contributions to the approximation. We establish basic effective-number properties and show that ENP detects proposal collapse and duplication missed by standard ESS. We also illustrate its use as a targeted feedback signal for proposal rejuvenation.

Problem

• Adaptive Importance Sampling
• Effective Sample Size
• Proposal Collapse
• Diagnostic Metrics
• Redundancy

Innovation

• Effective Number of Proposals
• Adaptive Importance Sampling
• Similarity-aware Diagnostic
• Proposal Collapse Detection
• Redundancy Measure