  Monte Carlo methods represent the "de facto" standard for approximating
complicated integrals involving multidimensional target distributions. In order
to generate random realizations from the target distribution, Monte Carlo
techniques use simpler proposal probability densities to draw candidate
samples. The performance of any such method is strictly related to the
specification of the proposal distribution, such that unfortunate choices
easily wreak havoc on the resulting estimators. In this work, we introduce a
layered (i.e., hierarchical) procedure to generate samples employed within a
Monte Carlo scheme. This approach ensures that an appropriate equivalent
proposal density is always obtained automatically (thus eliminating the risk of
a catastrophic performance), although at the expense of a moderate increase in
the complexity. Furthermore, we provide a general unified importance sampling
(IS) framework, where multiple proposal densities are employed and several IS
schemes are introduced by applying the so-called deterministic mixture
approach. Finally, given these schemes, we also propose a novel class of
adaptive importance samplers using a population of proposals, where the
adaptation is driven by independent parallel or interacting Markov Chain Monte
Carlo (MCMC) chains. The resulting algorithms efficiently combine the benefits
of both IS and MCMC methods.
