  Monte Carlo sampling methods are the standard procedure for approximating
complicated integrals of multidimensional posterior distributions in Bayesian
inference. In this work, we focus on the class of Layered Adaptive Importance
Sampling (LAIS) scheme, which is a family of adaptive importance samplers where
Markov chain Monte Carlo algorithms are employed to drive an underlying
multiple importance sampling scheme. The modular nature of LAIS allows for
different possible implementations, yielding a variety of different performance
and computational costs. In this work, we propose different enhancements of the
classical LAIS setting in order to increase the efficiency and reduce the
computational cost, of both upper and lower layers. The different variants
address computational challenges arising in real-world applications, for
instance with highly concentrated posterior distributions. Furthermore, we
introduce different strategies for designing cheaper schemes, for instance,
recycling samples generated in the upper layer and using them in the final
estimators in the lower layer. Different numerical experiments, considering
several challenging scenarios, show the benefits of the proposed schemes
comparing with benchmark methods presented in the literature.
