Advances in Importance Sampling

Abstract

Importance sampling (IS) is a Monte Carlo technique for the approximation of intractable distributions and integrals with respect to them. The origin of IS dates from the early 1950s. In the last decades, the rise of the Bayesian paradigm and the increase of the available computational resources have propelled the interest in this theoretically sound methodology. In this paper, we first describe the basic IS algorithm and then revisit the recent advances in this methodology. We pay particular attention to two sophisticated lines. First, we focus on multiple IS (MIS), the case where more than one proposal is available. Second, we describe adaptive IS (AIS), the generic methodology for adapting one or more proposals.

Large Data and (Not Even Very) Complex Ecological Models: When Worlds Collide

“…through the self-normalized importance sampling (SNIS) estimator (Elvira and Martino, 2021). These importance sampling weights can be used to obtain summary statistics/distributions of interest.…”

“…This computational cost is in terms of (1) time per each iteration (due to the number of auxiliary variables and cost to evaluate the likelihood function), and (2) length of MCMC simulations required since poorer mixing is often observed due to increased correlation between the parameters (notably for the random effects, ǫ 1 , and σ 2 ). Alternatively, smaller subsamples provide subposteriors for which it is (relatively) compu-tationally fast to obtain a sample from but where the following importance sampling algorithm may suffer from increased particle depletion due to differences between the subposterior and full posterior (see, Elvira and Martino, 2021 for further discussion). Further, in this case, there is an increased computational cost in the calculation of the importance sampling weight, as this is a function of the remaining data, though this is minimised when using an alternative (biased) weight calculation making use of repeated histories (see Section 4.3, consideration (iii)).…”

“…The realisations of the Markov chain are then reweighted via an importance sampling algorithm (for a review of importance sampling, see for example, Tokdar and Kass, 2010;Elvira and Martino, 2021) to obtain an estimate of the posterior distribution for the full dataset. Multiple sets of subsampled data can be taken and analysed in parallel, independently of each other, and subsequently combined to decrease the Monte Carlo error of the posterior estimates.…”

Policy Learning and Evaluation with Randomized Quasi-Monte Carlo

“…QMC with such low-discrepancy point sets can reach rates of convergence arbitrarily close to O N −1 when f satisfies some regularity conditions (Papageorgiou, 2003). In comparison, common variance-reduction techniques in reinforcement learning (e.g., control variates or importance sampling) only improve the constant factors in front of the O N −1 /2 rate (Glynn and Szechtman, 2002;Elvira and Martino, 2021).…”

Approximating Bayes in the 21st Century

“…In order to address such questions, we bring together in one place, and using a common notational framework, the four main 1 We acknowledge, of course, that there have been many concurrent advances in MCMC and importance sampling designed, in particular, to deal with the problem of scale. We refer the reader to: Green et al (2015), Robert et al (2018) and Dunson and Johndrow (2019) for broad overviews of modern developments in MCMC; to Betancourt (2018) for a review of Hamiltonian Monte Carlo (HMC); to Naesseth et al (2019) for a recent review of sequential Monte Carlo (exploiting importance sampling principles as it does); and to Hoogerheide et al (2009), Tokdar and Kass (2010) and Elvira and Martino (2021) for other advances in importance sampling.…”