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Probabilistic Integration: A Role in Statistical Computation?
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TL;DR: In this article, the authors examined the case for probabilistic numerical methods in routine statistical computation and established the rates of posterior contraction for these methods, and showed that these methods can in principle enjoy the best of both worlds, leveraging the sampling efficiency of Monte Carlo methods whilst providing a principled route to assess the impact of numerical error on scientific conclusions.
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Abstract: A research frontier has emerged in scientific computation, wherein numerical error is regarded as a source of epistemic uncertainty that can be modelled. This raises several statistical challenges, including the design of statistical methods that enable the coherent propagation of probabilities through a (possibly deterministic) computational work-flow. This paper examines the case for probabilistic numerical methods in routine statistical computation. Our focus is on numerical integration, where a probabilistic integrator is equipped with a full distribution over its output that reflects the presence of an unknown numerical error. Our main technical contribution is to establish, for the first time, rates of posterior contraction for these methods. These show that probabilistic integrators can in principle enjoy the "best of both worlds", leveraging the sampling efficiency of Monte Carlo methods whilst providing a principled route to assess the impact of numerical error on scientific conclusions. Several substantial applications are provided for illustration and critical evaluation, including examples from statistical modelling, computer graphics and a computer model for an oil reservoir.
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Citations
Methods of Numerical Integration
Luc Bauwens,Michel Lubrano,Jean-François Richard +2 more
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TL;DR: Methods of numerical integration will lead you to always think more and more, and this book will be always right for you.
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TL;DR: Scattered data approximation is available in our book collection an online access to it is set as public so you can get it instantly.Thank you for reading scattered data approximation.
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•Posted Content
Variational Fourier features for Gaussian processes
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TL;DR: A new class of positive definite and compactly supported radial functions which consist of a univariate polynomial within their support is constructed, it is proved that they are of minimal degree and unique up to a constant factor.
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TL;DR: A new unifying view, including all existing proper probabilistic sparse approximations for Gaussian process regression, relies on expressing the effective prior which the methods are using, and highlights the relationship between existing methods.