Adaptive Quadrature Schemes for Bayesian Inference via Active Learning

Luca Martino

Abstract:

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 specic schemes based on Gaussian and Nearest Neighbors bases. For
the Gaussian case, we also provide a novel procedure for tting 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 showthe 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.