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Adaptive quadrature schemes for Bayesian inference via active learning

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Document pages: 43 pages

Abstract: Numerical integration and emulation are fundamental topics across scientificfields. We propose novel adaptive quadrature schemes based on an activelearning procedure. We consider an interpolative approach for building asurrogate posterior density, combining it with Monte Carlo sampling methods andother quadrature rules. The nodes of the quadrature are sequentially chosen bymaximizing a suitable acquisition function, which takes into account thecurrent approximation of the posterior and the positions of the nodes. Thismaximization does not require additional evaluations of the true posterior. Weintroduce two specific schemes based on Gaussian and Nearest Neighbors (NN)bases. For the Gaussian case, we also provide a novel procedure for fitting thebandwidth parameter, in order to build a suitable emulator of a densityfunction. With both techniques, we always obtain a positive estimation of themarginal likelihood (a.k.a., Bayesian evidence). An equivalent importancesampling interpretation is also described, which allows the design of extendedschemes. Several theoretical results are provided and discussed. Numericalresults show the advantage of the proposed approach, including a challenginginference problem in an astronomic dynamical model, with the goal of revealingthe number of planets orbiting a star.

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