importance sampling

This paper addresses the optimal design of proposal densities in Monte Carlo importance sampling, particularly under challenging settings involving dynamic updates and noise—such as in Approximate Bayesian Computation (ABC) and policy evaluation in reinforcement learning. It provides the first unified theoretical analysis of the applicability boundaries of multiple optimality criteria—including minimum variance and KL-divergence minimization—while establishing a cross-framework evaluation framework that jointly ensures theoretical guarantees and empirically comparable performance. Methodologically, the work integrates variational inference, sequential importance resampling, and annealed posterior modeling to propose a multi-proposal adaptive mechanism. Key contributions are: (1) necessary and sufficient conditions for proposal optimality across frameworks, with convergence guarantees; (2) systematic empirical validation of trade-offs among model selection accuracy, noise robustness, and computational efficiency in adaptive proposal design; and (3) an open-source empirical benchmark enabling reproducible, standardized comparison of future proposal mechanisms.

To address the insufficient accuracy of surrogate models in estimating probabilities of rare events—such as system failure—under uncertainty propagation, this paper proposes a heuristic sampling strategy for experimental design in surrogate model training, based on the Highest Density Region (HDR) of the input parameter distribution. By generating samples uniformly within the HDR, the method significantly improves the accuracy of failure probability estimation compared to conventional random sampling over the full parameter space. Performance is rigorously evaluated using three complementary metrics: leave-one-out error, relative mean squared error (RMSE), and failure probability estimation error. Experimental results across multiple benchmark cases consistently demonstrate that the proposed approach reduces relative RMSE by an average of 32% and lowers failure probability estimation error by one order of magnitude. These improvements are particularly beneficial for tail-probability–sensitive applications, including reliability analysis and Bayesian inverse problems.

This work addresses the challenge of risk prediction when acquiring true outcomes is prohibitively expensive, allowing labels for only a subset of samples. The authors propose a surrogate-assisted optimal sampling framework that, under a fixed annotation budget, leverages covariates, surrogate variables, and an initial estimator to construct a sampling strategy minimizing the expected out-of-sample cross-entropy loss. Coupled with an inverse probability weighted cross-entropy estimator for model training, this approach achieves—without requiring access to true responses during design—theoretically optimal sampling. It uniquely guarantees predictive optimality, robustness to surrogate misspecification, and stability in settings with rare outcomes. Both theoretical analysis and empirical experiments demonstrate that the method significantly outperforms existing approaches, particularly when the surrogate is imperfect or the event of interest is rare.

For expensive black-box target density sampling, this paper proposes an active sampling method framed within the multi-armed bandit (MAB) paradigm. Unlike conventional approaches that optimize a proposal distribution, our method models sample location selection as a sequential decision-making process, jointly leveraging a Gaussian process surrogate model and space-filling criteria to adaptively select evaluation points with maximal information gain. To our knowledge, this is the first work to directly apply MAB for sampling point scheduling—bypassing explicit distribution modeling and substantially reducing the number of target function evaluations. Experiments demonstrate superior performance over state-of-the-art importance sampling methods on multimodal and heavy-tailed distributions. In Bayesian inference tasks, our approach achieves higher approximation accuracy with significantly fewer evaluations.

This work addresses the high computational complexity in estimating probabilistic values—such as Shapley values—arising from the exponential number of coalition evaluations. By analyzing existing Monte Carlo estimators through a unified perspective, the study uncovers a shared first-order error structure characterized by an augmented inverse probability weighting influence term and derives an explicit mean squared error expression. Building on this insight, the authors propose the Efficiency-Aware Surrogate-adjusted Estimator (EASE), which jointly optimizes the sampling strategy and surrogate function to enhance estimation accuracy. Empirical evaluations demonstrate that EASE consistently outperforms current state-of-the-art methods across diverse probabilistic value estimation tasks, achieving substantial improvements in both statistical efficiency and estimation precision.

This work addresses the challenge of estimating small failure probabilities under stochastic inputs in computationally expensive deterministic simulations. We propose a two-stage adaptive budget allocation framework: in Stage I, a Gaussian process surrogate is sequentially trained using a contour-localization strategy; in Stage II, remaining simulation budget is greedily allocated to critical regions—guided by classification entropy—to perform high-fidelity evaluations. A hybrid Monte Carlo estimator is then constructed by integrating surrogate predictions with observed high-fidelity responses. Our method introduces the first “exploration–exploitation decoupled” budget allocation paradigm, overcoming reliability limitations inherent in pure surrogate-based Monte Carlo and importance sampling. Experiments across multiple benchmark functions and an airfoil flow simulation demonstrate that the approach achieves significantly improved accuracy and robustness using only several hundred high-fidelity evaluations.

This study addresses the challenge of sample efficiency in estimating small failure probabilities when the limit state function is computationally expensive. To this end, the authors propose a single-stage joint training framework that simultaneously constructs a Gaussian process surrogate model and an optimal importance sampling density. Departing from conventional two-stage approaches, the method leverages shared observational data and employs kernel density estimation to adaptively generate the sampling distribution—termed KDE-AIS—which asymptotically converges to the zero-variance optimal density in total variation distance. Numerical experiments demonstrate that KDE-AIS achieves higher estimation accuracy with fewer calls to the true model compared to existing methods, including Gaussian process-based adaptive importance sampling schemes.

This work addresses the limitation of traditional importance sampling, which focuses on estimating rare-event probabilities, whereas many applications prioritize minimizing the time to hit a set of rare trajectories. The authors propose a new paradigm centered on guesswork from information theory, defining discovery quality as the description length—i.e., surprisal—of a trajectory under a nominal model. They formulate an optimization objective \( H(Q) + D(Q\|P) \) to design sampling strategies that blend stochastic exploration with systematic search. Under i.i.d. assumptions, large deviations theory is employed to analyze type-defined rare sets, showing that this objective not only characterizes discovery optimality but also serves as a lexicographic tiebreaker among strategies achieving identical hitting times under a fixed computational budget.

In many randomized trials, outcomes such as essays or open-ended responses must be manually scored as a preliminary step to impact analysis, a process that is costly and limiting. Model-assisted estimation offers a way to combine surrogate outcomes generated by machine learning or large language models with a human-coded subset, yet typical implementations use simple random sampling and therefore overlook systematic variation in surrogate prediction error. We extend this framework by incorporating stratified sampling to more efficiently allocate human coding effort. We derive the exact variance of the stratified model-assisted estimator, characterize conditions under which stratification improves precision, and identify a Neyman-type optimal allocation rule that oversamples strata with larger residual variance. We evaluate our methods through a comprehensive simulation study to assess finite-sample performance. Overall, we find stratification consistently improves efficiency when surrogate prediction errors exhibit structured bias or heteroskedasticity. We also present two empirical applications, one using data from an education RCT and one using a large observational corpus, to illustrate how these methods can be implemented in practice using ChatGPT-generated surrogate outcomes. Overall, this framework provides a practical design-based approach for leveraging surrogate outcomes and strategically allocating human coding effort to obtain unbiased estimates with greater efficiency. While motivated by text-as-data applications, the methodology applies broadly to any setting where outcome measurement is costly or prohibitive, and can be applied to comparisons across groups or estimating the mean of a single group.

This study addresses the problem of efficient data acquisition under a fixed budget when multiple data sources exhibit heterogeneous sampling costs and distributional shifts relative to the target population. The authors propose a sampling strategy that quantifies source–target divergence via χ² divergence and maximizes the effective sample size, coupled with a post-stratification estimator to achieve minimax optimal estimation risk for both the overall and subgroup conditional means. This approach is the first to establish theoretically optimal risk bounds simultaneously for population-level and conditional mean estimation under such constraints. Furthermore, the framework naturally extends to predictive settings, where it minimizes excess risk in downstream learning tasks.