Optimality in importance sampling: a gentle survey

Monte Carlo (MC) methods are powerful tools for numerical inference and optimization widely employed in statistics, signal processing and machine learning Liu (2004); Robert and Casella (2004). They are mainly used for computing approximately the solution of definite integrals, and by extension, of differential equations (for this reason, MC schemes can be considered stochastic quadrature rules). Although exact analytical solutions to integrals are always desirable, such unicorns are rarely available, specially in real-world systems. Many applications inevitably require the approximation of intractable integrals. Specifically, Bayesian methods need the computation of expectations with respect to posterior probability density function (pdf) which, generally, are analytically intractable Gelman et al. (2013). The MC methods can be divided in four main families: direct methods (based on transformations or random variables), accept-reject techniques, Markov chain Monte Carlo (MCMC) algorithms, and importance sampling (IS) schemes Luengo et al. (2020); Martino et al. (2018). The last two families are the most popular for the facility and universality of their possible application Liang et al. (2010); Liu (2004); Robert and Casella (2004). All the MC methods require the choice of a suitable proposal density that is crucial for their performance Luengo et al. (2020); Robert and Casella (2004).