{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,10]],"date-time":"2026-10-10T13:03:17Z","timestamp":1791637397609,"version":"4.3.4"},"reference-count":10,"publisher":"Oxford University Press (OUP)","issue":"4","license":[{"start":{"date-parts":[[1995,11,1]],"date-time":"1995-11-01T00:00:00Z","timestamp":815184000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/academic.oup.com\/journals\/pages\/open_access\/funder_policies\/chorus\/standard_publication_model"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[1995,11,1]]},"abstract":"<jats:title>SUMMARY<\/jats:title>\n               <jats:p>It is shown that the usual Laplace approximation is not a valid asymptotic approximation when the dimension of the integral is comparable with the limiting parameter n. The formal Laplace expansion for multidimensional integrals is given and used to construct asymptotic approximations for high dimensional integrals. One example is considered in which the dimension of the integral is O(n \u00a01\/2) and the relative error of the unmodified Laplace approximation is O(1). Nevertheless, it is possible to construct a valid asymptotic expansion by regrouping terms in the formal expansion according to asymptotic order in n.<\/jats:p>","DOI":"10.1111\/j.2517-6161.1995.tb02060.x","type":"journal-article","created":{"date-parts":[[2018,12,7]],"date-time":"2018-12-07T06:16:45Z","timestamp":1544163405000},"page":"749-760","source":"Crossref","is-referenced-by-count":152,"title":["Laplace Approximation of High Dimensional Integrals"],"prefix":"10.1093","volume":"57","author":[{"given":"Zhenming","family":"Shun","sequence":"first","affiliation":[{"name":"University of Chicago \u00a0 USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Peter","family":"McCullagh","sequence":"additional","affiliation":[{"name":"University of Chicago \u00a0 USA"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"286","published-online":{"date-parts":[[2018,12,5]]},"reference":[{"key":"2024090921171194100_rssb02060-cit-0001","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4899-3424-6","volume-title":"Asymptotic Techniques for Use in Statistics.","author":"Barndorff-Nielsen","year":"1989"},{"key":"2024090921171194100_rssb02060-cit-0002","volume-title":"Asymptotic Expansions of Integrals.","author":"Bleistein","year":"1986"},{"key":"2024090921171194100_rssb02060-cit-0003","doi-asserted-by":"crossref","first-page":"9","DOI":"10.1080\/01621459.1993.10594284","article-title":"Approximate inference in generalized linear mixed models","volume":"88","author":"Breslow","year":"1993","journal-title":"J. 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