Practical Hilbert space approximate Bayesian Gaussian processes for probabilistic programming

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Journal Title
Journal ISSN
Volume Title
A1 Alkuperäisartikkeli tieteellisessä aikakauslehdessä
Date
2023-02
Major/Subject
Mcode
Degree programme
Language
en
Pages
28
1-28
Series
STATISTICS AND COMPUTING, Volume 33, issue 1
Abstract
Gaussian processes are powerful non-parametric probabilistic models for stochastic functions. However, the direct implementation entails a complexity that is computationally intractable when the number of observations is large, especially when estimated with fully Bayesian methods such as Markov chain Monte Carlo. In this paper, we focus on a low-rank approximate Bayesian Gaussian processes, based on a basis function approximation via Laplace eigenfunctions for stationary covariance functions. The main contribution of this paper is a detailed analysis of the performance, and practical recommendations for how to select the number of basis functions and the boundary factor. Intuitive visualizations and recommendations, make it easier for users to improve approximation accuracy and computational performance. We also propose diagnostics for checking that the number of basis functions and the boundary factor are adequate given the data. The approach is simple and exhibits an attractive computational complexity dueto its linear structure, and it is easy to implement in probabilistic programming frameworks. Several illustrative examples of the performance and applicability of the method in the probabilistic programming language Stan are presented together with the underlying Stan model code.
Description
Funding Information: We thank Academy of Finland (Grants 298742, 308640, and 313122), Instituto de Salud Carlos III, Spain (Grant CD21/00186 - Sara Borrell Postdoctoral Fellowship) and co-funded by the European Union, Finnish Center for Artificial Intelligence, and Technology Industries of Finland Centennial Foundation (Grant 70007503; Artificial Intelligence for Research and Development), and Conselleria de Innovación, Universidades, Ciencia y Sociedad Digital, Generalitat Valenciana, Spain (Grant AICO/2020/285) for partial support of this research. We also acknowledge the computational resources provided by the Aalto Science-IT project. Publisher Copyright: © 2022, The Author(s).
Keywords
Bayesian statistics, Gaussian process, Hilbert space methods, Low-rank Gaussian process, Sparse Gaussian process, Stan
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Citation
Riutort-Mayol , G , Bürkner , P C , Andersen , M R , Solin , A & Vehtari , A 2023 , ' Practical Hilbert space approximate Bayesian Gaussian processes for probabilistic programming ' , STATISTICS AND COMPUTING , vol. 33 , no. 1 , 17 , pp. 1-28 . https://doi.org/10.1007/s11222-022-10167-2