  Importance sampling (IS) is a powerful Monte Carlo (MC) methodology for
approximating integrals, for instance in the context of Bayesian inference. In
IS, the samples are simulated from the so-called proposal distribution, and the
choice of this proposal is key for achieving a high performance. In adaptive IS
(AIS) methods, a set of proposals is iteratively improved. AIS is a relevant
and timely methodology although many limitations remain yet to be overcome,
e.g., the curse of dimensionality in high-dimensional and multi-modal problems.
Moreover, the Hamiltonian Monte Carlo (HMC) algorithm has become increasingly
popular in machine learning and statistics. HMC has several appealing features
such as its exploratory behavior, especially in high-dimensional targets, when
other methods suffer. In this paper, we introduce the novel Hamiltonian
adaptive importance sampling (HAIS) method. HAIS implements a two-step adaptive
process with parallel HMC chains that cooperate at each iteration. The proposed
HAIS efficiently adapts a population of proposals, extracting the advantages of
HMC. HAIS can be understood as a particular instance of the generic layered AIS
family with an additional resampling step. HAIS achieves a significant
performance improvement in high-dimensional problems w.r.t. state-of-the-art
algorithms. We discuss the statistical properties of HAIS and show its high
performance in two challenging examples.
