Optimality in importance sampling: a gentle survey

A new survey by researchers Fernando Llorente and Luca Martino provides an exhaustive review of optimality in importance sampling, a critical component of Monte Carlo methods where the choice of proposal density determines estimator performance [1][3]. The paper, posted to arXiv and last revised in June 2026, frames the notion of optimality as fundamental to designing adaptive procedures that automatically tune the proposal density within Monte Carlo schemes [1][2]. The authors adopt a minimum-variance criterion as their central measure of efficiency, defining optimality as proposal densities that minimize the variance of the importance sampling estimator under consideration [4]. This definition, they note, is mathematically tractable and unifies many results across importance sampling theory [4]. The survey systematically analyzes several frameworks. It covers the approximation of the marginal likelihood for model selection, the use of multiple proposal densities—including applications in computer graphics for global illumination and physically based rendering—and sequences of tempered posteriors with an optimal tempering schedule [2][5]. The work also addresses noisy scenarios, where the evaluation of the posterior is itself a random variable, with applications to energy-based models, approximate Bayesian computation, and reinforcement learning [4][5]. A key insight highlighted in the review is that an importance sampling scheme employing a proposal density close to the optimal one can outperform the ideal Monte Carlo technique, which is why these approaches are also known as variance reduction methods [5]. The first part of the survey exhaustively analyzes the use of a single or multiple proposal densities for approximating one or several integrals, discussing optimal weights and optimal sample allocation across different multiple-proposal settings [4][5]. The second part turns to more specialized application frameworks and specific strategies for proposal adaptation and construction [5]. The authors also describe the behavior of different divergences used to adapt the proposal density toward the optimal proposal and provide theoretical and empirical comparisons [2][4]. While the survey centers on minimum-variance optimality, it acknowledges that other notions—such as minimizing Kullback-Leibler or Renyi divergences, maximizing effective sample size, or achieving robustness to model misspecification—are meaningful in practice but lie beyond the scope of the present review [4].