Post-stratified importance sampling

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.

Variance reduction techniques improve the efficiency of drawing conclusions from data. Recent proposals for variance-reduction use two complementary techniques to reduce variance: incorporating machine learning predictions into estimators and weighted sampling. A popular weighted sampling algorithm is Poisson Sampling (PS), which independently flips a biased coin at each unit and includes all heads in the sample. A fundamental flaw with PS is that it has a random sample size, which presents practical challenges and introduces variance into the estimator. We show that a fixed-size alternative, known in survey sampling as Conditional Poisson Sampling (CPS), is a drop-in replacement for PS on general multivariate Z-estimation problems, which include convex M-estimation considered in prior work. Notably, we show that the asymptotic variance of CPS is no worse than that of PS, and can be substantially better in realistic scenarios where predictions are imperfect. We illustrate these advantages on both synthetic and real datasets, and release a Python package implementing CPS. In light of our contributions, there is no need to suffer a variable sample size.

Importance-weighted variational inference approximates intractable densities by optimizing multi-sample Monte Carlo objectives whose tightness increases with the number of importance samples $N$. When reparameterization is unavailable, optimization relies on score-function (REINFORCE) gradient estimators. In this paper, we provide a comprehensive analysis of score-function gradient estimators for importance-weighted objectives by characterizing their asymptotic expectation and variance. We establish that standard variational inference for Monte Carlo objectives (VIMCO) gradient estimators suffer from a vanishing signal-to-noise ratio (SNR) as $N$ increases, which arises from a suboptimal control variate choice and can impair optimization. To address this issue, we derive a general mechanism for constructing VIMCO gradient estimators that avoid SNR collapse. We identify the High-Effective Sample Size optimal variance importance sampling estimator as belonging to this category of estimators and establish its $\sqrt{N}$ SNR scaling. This estimator does not, however, have zero variance at optimality and achieving its $\sqrt{N}$ SNR rate may require prohibitively large $N$ when the importance weights are degenerate. We overcome these limitations by introducing VIMCO gradient estimators that attain zero variance at optimality and support importance weight tempering. Experiments with discrete latent-variable and state-space models validate the theoretical predictions and illustrate the benefits of our estimators.