{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/holonomic-gradient-method-for-the-probability","title":"Holonomic gradient method for the probability content of a simplex region with a multivariate normal distribution","arxiv_id":"1512.06564","date":"2015-12-21","proceeding":null,"authors":["Tamio Koyama"],"abstract":"We use the holonomic gradient method to evaluate the probability content of a simplex region under a multivariate normal distribution. This probability equals to the integral of the probability density function of the multivariate Gaussian distribution on the simplex region. For this purpose, we generalize the inclusion--exclusion identity which was given for polyhedra, to the faces of a polyhedron. This extended inclusion--exclusion identity enables us to calculate the derivatives of the function associated with the probability content of a polyhedron in general position. We show that these derivatives can be written as integrals of the faces of the polyhedron.","url_abs":"https://arxiv.org/abs/1512.06564v2","url_pdf":"https://arxiv.org/pdf/1512.06564v2.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"holonomic-gradient-method-for-the-probability","repo_url":"https://github.com/tkoyama-may10/simplex","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}