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

Ali Mousavi; Victor Elvira

stat.MLAug 15, 2026

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

Authors: Ali Mousavi, Victor Elvira

Organizations: Department of Computer Engineering, Islamic Azad University, Neyshabur Branch, Neyshabur, Iran · School of Mathematics, University of Edinburgh, United Kingdom

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.

Figures & tables

Sc.	ESS ${}_{2}/S$	Per./ $S$	ESS ${}_{\infty}/S$	ENP ${}_{\rho}^{(2,2)}$
Sc.	ESS ${}_{2}/S$	Per./ $S$	ESS ${}_{\infty}/S$	$\sigma_{Z}=1.5$	$\sigma_{Z}=2.5$	$\sigma_{Z}=4.0$
A	0.794	0.860	0.196	4.998	5.000	4.961
B	0.733	0.794	0.536	1.005	1.000	1.002
C	0.790	0.850	0.280	2.005	2.000	2.001

TABLE I: Controlled diagnostic scenarios on the two-dimensional five-modal target. The rates ESS

{}_{2}/S

, Per./

S

, and ESS

{}_{\infty}/S

are normalized by the total number of samples

S=NK

. ENP is reported for different similarity bandwidths

\sigma_{Z}

Method	ESS ${}_{2}/S$	ENP	Coverage	MSE	Trig.
Standard DM-PMC	0.618	4.47	3.60	46.48	0.00
ESS-triggered	0.459	5.93	4.75	13.46	22.00
ENP-triggered	0.603	6.03	4.90	2.87	1.15

TABLE II: Diagnostic-triggered proposal rejuvenation from collapsed initialization. Coverage is the number of reached target modes; MSE is computed for the self-normalized target-mean estimate.

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Amortised adaptive importance sampling

Arran Carter, Víctor Elvira, Esmeralda S. Whitammer

University of Edinburgh

Importance Sampling Adaptive Sampling Amortized Inference

Open paper AskPDF arXiv

Monte Carlo sampling algorithms aim to produce samples from a target distribution with a given unnormalised probability density function. The high cost of running Monte Carlo algorithms, with many simulation steps or particles needed to obtain a good empirical approximation to the target, can be reduced by introducing learnable components into the sampler -- a form of amortisation. Once these components are trained, amortised samplers can produce high-quality samples with less computation at sampling time than training-free algorithms would require. We propose an amortised sampler built upon the framework of adaptive importance sampling -- a hierarchical importance sampling procedure where the proposal is iteratively adapted. Our amortised adaptive importance sampling algorithm fits a lightweight neural network that adapts the proposal using information obtained over the course of sampling. By training our sampler over many different targets taken from the same family of distributions, we obtain a sampler that can adapt to a specific target at sampling time without a priori access to the target's parameters. We show that our procedure can efficiently sample from targets within the family the sampler was trained on, giving improvements over target-specific amortised samplers and training-free adaptive importance sampling algorithms. We also explore the potential for generalisation, where the trained sampler is used to sample from targets outwith its training family.

Abstract

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