Adaptive Quadrature Schemes for Bayesian Inference via Active Learning

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 maxi...

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Bibliographic Details
Published in:IEEE access 2020, Vol.8, p.208462-208483
Main Authors: Fernandez, Fernando Llorente, Martino, Luca, Elvira, Victor, Delgado, David, Lopez-Santiago, Javier
Format: Article
Language:English
Subjects:
Active learning
Bayes methods
Bayesian analysis
Bayesian quadrature
Density
Dynamic models
emulation
Emulators
Estimation
experimental design
Importance sampling
Interpolation
Kernel
Machine learning
Maximization
Monte Carlo methods
Monte Carlo simulation
Nodes
numerical integration
Numerical models
Optimization
Quadratures
Sampling methods
Statistical inference
Symmetric matrices
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Summary: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 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.
ISSN:2169-3536
2169-3536
DOI:10.1109/ACCESS.2020.3038333