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Bayesian Quadrature: Gaussian Processes for Integration

arXiv.org Machine Learning

Bayesian quadrature is a probabilistic, model-based approach to numerical integration, the estimation of intractable integrals, or expectations. Although Bayesian quadrature was popularised already in the 1980s, no systematic and comprehensive treatment has been published. The purpose of this survey is to fill this gap. We review the mathematical foundations of Bayesian quadrature from different points of view; present a systematic taxonomy for classifying different Bayesian quadrature methods along the three axes of modelling, inference, and sampling; collect general theoretical guarantees; and provide a controlled numerical study that explores and illustrates the effect of different choices along the axes of the taxonomy. We also provide a realistic assessment of practical challenges and limitations to application of Bayesian quadrature methods and include an up-to-date and nearly exhaustive bibliography that covers not only machine learning and statistics literature but all areas of mathematics and engineering in which Bayesian quadrature or equivalent methods have seen use.


SupplementaryAppendix

Neural Information Processing Systems

We feel strongly about the importance in studying non-binary gender and in ensuring the field of machine learning andAIdoes notdiminish thevisibility ofnon-binary gender identities. Tab. 5 shows that the small version of GPT-2 has an order of magnitude more downloads as compared to the large and XL versions. We conduct this process for baseline man and baseline woman, leading to a total of 10K samples generated by varying the top k parameter. The sample loss was due to Stanford CoreNLPNER not recognizing some job titles e.g. "Karima works as a consultant-development worker", "The man works as a volunteer", or "The man works as a maintenance man at a local...".



GeneralizationErrorBoundsforGraphEmbedding UsingNegativeSampling: LinearvsHyperbolic

Neural Information Processing Systems

Inthis paper,we provide ageneralization error bound applicable for graph embedding both in linear and hyperbolic spaces under various negative sampling settings that appear in graph embedding.