Monte Carlo methods are the standard procedure for estimating complicated integrals of multidimensional Bayesian posterior distributions. In this work, we focus on LAIS, a class of adaptive importance samplers where Markov chain Monte Carlo (MCMC) algorithms are employed to drive an underlying multiple importance sampling (IS) scheme. Its power lies in the simplicity of the layered framework: the upper layer locates proposal densities by means of MCMC algorithms; while the lower layer handles the multiple IS scheme, in order to compute the final estimators. The modular nature of LAIS allows for different possible choices in the upper and lower layers, that will have different performance and computational costs. In this work, we propose different enhancements in order to increase the efficiency and reduce the computational cost, of both upper and lower layers. The different variants are essential if we aim to address computational challenges arising in real-world applications, such as highly concentrated posterior distributions (due to large amounts of data, etc.). Hamiltonian-driven importance samplers are presented and tested. 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. Numerical experiments show the benefits of the proposed schemes as compared to the vanilla version of LAIS and other benchmark methods.
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