{"id":149949,"date":"1998-01-01T00:00:00","date_gmt":"1998-01-01T00:00:00","guid":{"rendered":"https:\/\/www.microsoft.com\/en-us\/research\/msr-research-item\/ensemble-learning-in-bayesian-neural-networks\/"},"modified":"2018-10-16T21:49:22","modified_gmt":"2018-10-17T04:49:22","slug":"ensemble-learning-in-bayesian-neural-networks","status":"publish","type":"msr-research-item","link":"https:\/\/www.microsoft.com\/en-us\/research\/publication\/ensemble-learning-in-bayesian-neural-networks\/","title":{"rendered":"Ensemble learning in Bayesian neural networks"},"content":{"rendered":"<div class=\"asset-content\">\n<p>Bayesian treatments of learning in neural networks are typically based either on a local Gaussian approximation to a mode of the posterior weight distribution, or on Markov chain Monte Carlo simulations. A third approach, called `ensemble learning&#8217;, was introduced by Hinton (1993). It aims to approximate the posterior distribution by minimizing the Kullback-Leibler divergence between the true posterior and a parametric approximating distribution. The original derivation of a deterministic algorithm relied on the use of a Gaussian approximating distribution with a diagonal covariance matrix and hence was unable to capture the posterior correlations between parameters. In this chapter we show how the ensemble learning approach can be extended to full-covariance Gaussian distributions while remaining computationally tractable. We also extend the framework to deal with hyperparameters, leading to a simple re-estimation procedure. One of the benefits of our approach is that it yields a strict lower bound on the marginal likelihood, in contrast to other approximate procedures.<\/p>\n<\/div>\n<p><!-- .asset-content --><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Bayesian treatments of learning in neural networks are typically based either on a local Gaussian approximation to a mode of the posterior weight distribution, or on Markov chain Monte Carlo simulations. A third approach, called `ensemble learning&#8217;, was introduced by Hinton (1993). It aims to approximate the posterior distribution by minimizing the Kullback-Leibler divergence between [&hellip;]<\/p>\n","protected":false},"featured_media":0,"template":"","meta":{"msr-url-field":"","msr-podcast-episode":"","msrModifiedDate":"","msrModifiedDateEnabled":false,"ep_exclude_from_search":false,"_classifai_error":"","footnotes":""},"msr-content-type":[3],"msr-research-highlight":[],"research-area":[13556],"msr-publication-type":[193716],"msr-product-type":[],"msr-focus-area":[],"msr-platform":[],"msr-download-source":[],"msr-locale":[268875],"msr-post-option":[],"msr-field-of-study":[],"msr-conference":[],"msr-journal":[],"msr-impact-theme":[],"msr-pillar":[],"class_list":["post-149949","msr-research-item","type-msr-research-item","status-publish","hentry","msr-research-area-artificial-intelligence","msr-locale-en_us"],"msr_publishername":"Springer Verlag","msr_edition":"Generalization in Neural Networks and Machine Learning","msr_affiliation":"","msr_published_date":"1998-01-01","msr_host":"","msr_duration":"","msr_version":"","msr_speaker":"","msr_other_contributors":"","msr_booktitle":"Generalization in Neural Networks and Machine Learning","msr_pages_string":"215\u2013237","msr_chapter":"","msr_isbn":"","msr_journal":"","msr_volume":"","msr_number":"","msr_editors":"","msr_series":"","msr_issue":"","msr_organization":"","msr_how_published":"","msr_notes":"","msr_highlight_text":"","msr_release_tracker_id":"","msr_original_fields_of_study":"","msr_download_urls":"","msr_external_url":"","msr_secondary_video_url":"","msr_longbiography":"","msr_microsoftintellectualproperty":1,"msr_main_download":"211387","msr_publicationurl":"","msr_doi":"","msr_publication_uploader":[{"type":"file","title":"bishop-ensemble-nato-98.pdf","viewUrl":"https:\/\/www.microsoft.com\/en-us\/research\/wp-content\/uploads\/2016\/02\/bishop-ensemble-nato-98.pdf","id":211387,"label_id":0}],"msr_related_uploader":"","msr_attachments":[{"id":211387,"url":"https:\/\/www.microsoft.com\/en-us\/research\/wp-content\/uploads\/2016\/02\/bishop-ensemble-nato-98.pdf"}],"msr-author-ordering":[{"type":"text","value":"D. 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