  Numerical integration and emulation are fundamental topics across scientific
fields. We propose novel adaptive quadrature schemes based on an active
learning procedure. We consider an interpolative approach for building a
surrogate posterior density, combining it with Monte Carlo sampling methods and
other quadrature rules. The nodes of the quadrature are sequentially chosen by
maximizing a suitable acquisition function, which takes into account the
current approximation of the posterior and the positions of the nodes. This
maximization does not require additional evaluations of the true posterior. We
introduce two specific schemes based on Gaussian and Nearest Neighbors (NN)
bases. For the Gaussian case, we also provide a novel procedure for fitting the
bandwidth parameter, in order to build a suitable emulator of a density
function. With both techniques, we always obtain a positive estimation of the
marginal likelihood (a.k.a., Bayesian evidence). An equivalent importance
sampling interpretation is also described, which allows the design of extended
schemes. Several theoretical results are provided and discussed. Numerical
results show the advantage of the proposed approach, including a challenging
inference problem in an astronomic dynamical model, with the goal of revealing
the number of planets orbiting a star.
