[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-117176-en":3,"doc-seo-117176-105":30,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":4,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":27,"seo_description":14,"update_tm":28,"read_time":29},117176,4398048950312,"Violet","https://ap-avatar.wpscdn.com/avatar/400002538284de19e3c?_k=1778320343897328908",8,"Research & Report","Pareto Smoothed Importance Sampling - Research article","Importance weighting stabilizes Monte Carlo integration when samples come from an auxiliary distribution rather than the target, yet estimates can become highly variable when importance ratios have a heavy right tail. This occurs when target features are poorly matched by the approximating distribution. Pareto smoothed importance sampling stabilizes extreme ratios by fitting a generalized Pareto model to their upper tail. The approach provides stabilized effective sample size, Monte Carlo error estimates, and Pareto k convergence diagnostics, including finite-sample convergence guidance usable for any Monte Carlo estimator.","This is an electronic reprint of the original article.  \nThis reprint may differ from the original in pagination and typographic detail.  \nVehtari, A; Simpson, Daniel; Gelman, Andrew; Yao, Yuling; Gabry, Jonah  \nPareto Smoothed Importance Sampling  \nPublished in:  \nJournal of Machine Learning Research  \nPublished: 01/01/2024  \nDocument Version  \nPublisher's PDF, also known as Version of record  \nPublished under the following license:  \nCC BY  \nPlease cite the original version:  \nVehtari, A. , Simpson, D. , Gelman, A. , Yao, Y. , & Gabry, J. (2024) . Pareto Smoothed Importance Sampling. Journal of Machine Learning Research, 25, Article 72. [https://www.jmlr.org/papers/v25/19-556.html](https://www.jmlr.org/papers/v25/19-556.html)  \nThis material is protected by copyright and other intellectual property rights, and duplication or sale of all or part of any of the repository collections is not permitted, except that material may be duplicated by you foryour research use or educational purposes in electronic or print form. You must obtain permission for anyother use. Electronic or print copies may not be offered, whether for sale or otherwise to anyone who is not an authorised user.  \nPareto Smoothed Importance Sampling  \nAki Vehtari [aki.vehtari@aalto.fi](aki.vehtari@aalto.fi)  \nDepartment of Computer Science Aalto University  \nDaniel Simpson  \nNormal Computing  \nAndrew Gelman  \nDepartments of Statistics and Political Science Columbia University  \n[dan@normalcomputing.ai](dan@normalcomputing.ai)  \n[gelman@stat.columbia.edu](gelman@stat.columbia.edu)  \nYuling Yao [yyao@flatironinstitute.org](yyao@flatironinstitute.org)  \nCenter for Computational Mathematics Flatiron Institute  \nJonah Gabry [jgabry@gmail.com](jgabry@gmail.com)  \nDepartment of Statistics Columbia University  \nEditor: Vikash Mansinghka  \nAbstract  \nImportance weighting is a general way to adjust Monte Carlo integration to account for draws from the wrong distribution, but the resulting estimate can be highly variable when the importance ratios have a heavy right tail. This routinely occurs when there are aspects of the target distribution that are not well captured by the approximating distribution, in which case more stable estimates can be obtained by modifying extreme importance ratios.  \nWe present a new method for stabilizing importance weights using a generalized Pareto distribution 􀀌t to the upper tail of the distribution of the simulated importance ratios. The method, which empirically performs better than existing methods for stabilizing importance sampling estimates, includes stabilized e􀀋ective sample size estimates, Monte Carlo error estimates, and convergence diagnostics. The presented Pareto ^k 􀀌nite sample convergence rate diagnostic is useful for any Monte Carlo estimator.  \nKeywords: importance sampling, Monte Carlo, Bayesian computation, diagnostics  \n1. Introduction  \nImportance sampling is a simple modi􀀌cation to the Monte Carlo method for computing expectations that is useful when there is an auxiliary distribution g (􀀒) that is easier to directly sample from than the target distribution p(􀀒), which may be only known up to a proportionality constant (Hammersley and Handscomb, 1964) . The starting point is the  \n􀀍c2024 Aki Vehtari, Daniel Simpson, Andrew Gelman, Yuling Yao, and Jonah Gabry.  \nLicense: CC-BY 4.0, see [https://creativecommons.org/licenses/by/4.0/. Attribution](https://creativecommons.org/licenses/by/4.0/. Attribution) requirements are provided at  \n[http://jmlr.org/papers/v25/19-556.html](http://jmlr.org/papers/v25/19-556.html).  \nVehtari, Simpson, Gelman, Yao, and Gabry  \nsimple Monte Carlo estimate for the expectation of a function h,  \nIh = Ep(h) = Z h (􀀒)p (􀀒) d􀀒 􀀙 1S  \nwhich requires exact draws 􀀒 s; s = 1; : : : ; S from p (􀀒) .  \nSX h(􀀒 s); s=1  \nThe self-normalized importance  \nsampling estimate for the same expectation is  \nP~~1~~Sss1hr(s􀀒~~ ~~s) ; rs = r (􀀒 s) = pg((􀀒􀀒ss)) ; (1)  \nwhich only requires draws 􀀒 s from a propos","cbCainRQjQjBg4ai","https://ap.wps.com/l/cbCainRQjQjBg4ai","pdf",3823262,1,59,"English","en",105,"# Introduction\n## Importance sampling and importance weights\n# Pareto smoothed importance sampling (PSIS)\n## Stabilizing importance weights using a generalized Pareto tail\n## Diagnostics and error assessment","[{\"question\":\"Why can importance sampling estimates become unstable?\",\"answer\":\"When importance ratios have a heavy right tail, the resulting weighted estimator can be highly variable and even have effectively infinite variance, especially if the proposal distribution poorly captures the target’s characteristics.\"},{\"question\":\"How does Pareto smoothed importance sampling stabilize importance weights?\",\"answer\":\"PSIS fits a generalized Pareto distribution to the upper tail of the simulated importance ratios and modifies extreme ratios, producing more stable importance sampling estimates.\"},{\"question\":\"What diagnostics and uncertainty outputs does PSIS provide?\",\"answer\":\"PSIS includes stabilized effective sample size estimates, Monte Carlo error estimates, and a Pareto k finite-sample convergence rate diagnostic that can be used for any Monte Carlo estimator.\"}]","Pareto Smoothed Importance Sampling - 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