Mousavi & Elvira's ENP diagnostic catches proposal collapse that ESS misses

Population-based adaptive importance sampling (AIS) methods generate samples from multiple proposal distributions to approximate difficult target distributions. For years, practitioners have relied on effective sample size (ESS) and similar weight-based diagnostics to gauge performance. But as Ali Mousavi and Victor Elvira argue in their new paper (arXiv:2608.15154, submitted Aug 2026), ESS has a critical blind spot: it only measures how concentrated the normalized importance weights are, not how the proposals themselves are arranged in sample space. When several proposal components land in the same region of the target, the weights can look healthy while the effective number of distinct proposals is tiny. This misleads users into thinking the approximation is robust when it is actually fragile.

The authors introduce the effective number of proposals (ENP), a similarity-aware, proposal-level diagnostic that combines the total normalized weight per proposal with a redundancy measure computed from similarities among target-weighted samples. ENP estimates how many non-redundant empirical proposal contributions exist, making it sensitive to proposal collapse and duplication that standard ESS completely misses. They establish basic effective-number properties for ENP and demonstrate, with illustrative examples, that it reliably detects pathologies in AIS configurations. More importantly, ENP can be used as a direct feedback signal to trigger targeted proposal rejuvenation, guiding when and which components need to be adapted. For statisticians and machine learning engineers running high-dimensional sampling pipelines, ENP offers a practical, principled upgrade over legacy diagnostics, potentially making adaptive samplers more reliable in real-world Bayesian inference and probabilistic programming systems.