{"id":396312,"date":"2025-10-24T16:25:19","date_gmt":"2025-10-24T22:25:19","guid":{"rendered":"http:\/\/xianblog.wordpress.com\/?p=61206"},"modified":"2025-10-24T16:25:19","modified_gmt":"2025-10-24T22:25:19","slug":"approximating-evidence-via-bounded-harmonic-means-and-hpd-regions-with-known-volumes","status":"publish","type":"post","link":"https:\/\/www.r-bloggers.com\/2025\/10\/approximating-evidence-via-bounded-harmonic-means-and-hpd-regions-with-known-volumes\/","title":{"rendered":"Approximating evidence via bounded harmonic means (and HPD regions with known volumes)"},"content":{"rendered":"\r\n\r\n
\r\n[This article was first published on R \u2013 Xi'an's Og<\/a><\/strong>, and kindly contributed to R-bloggers<\/a>]. (You can report issue about the content on this page here<\/a>)\r\n
Want to share your content on R-bloggers? click here<\/a> if you have a blog, or here<\/a> if you don't.\r\n<\/div>\n

\"\"<\/p>\n

Following a suggestion by Christian Hennig<\/a> at JSM 2024<\/a>, I started working with my PhD student Dana Naderi on a detailed assessment of the method we proposed in 2009 with Darren Wraith for evidence approximation<\/a>. (The method was briefly mentioned in a Physical Review paper<\/a> and also briefly illustrated in our 2010 San Antonio<\/a> survey of evidence approximation methods<\/a> with Jean-Michel Marin.) Well,\u00a0 it took longer than expected but we eventually completed our paper<\/a> on the approximation of evidence by bounded harmonic means, exploiting the general identity<\/a> of Alan Gelfand and Dipak Dey (1994). Following our 2009 idea, the free function in Gelfand & Dey representation is chosen as a Uniform distribution on an HPD region, since this insures boundedness (and hence finite variance) for the resulting estimator. This followed a revival of the method, renamed THAMES, by Metodiev et al. in 2023<\/a>, where the authors approximate the HPD region with an ellipsoid derived from a Normal distribution centred at the highest of the HPD points, whose covariance matrix is estimated from the posterior sample. With the drawback that this ellipsoid may as well include low probability regions. Our approach (ECMLE, standing for elliptical coverings for marginal likelihood estimation<\/em>) is aiming at staying within the actual and targeted HPD region by creating non-overlapping ellipsoids from simulations from the posterior and by using a Uniform density on that collection as the reverse importance function. The resulting estimator is unbiased, since the volume of the set is known (when based on a second, independent, sample of simulations from the posterior, a requirement that was not clearly explicated in our earlier survey<\/a>). In the meanwhile, that is, while we were close to conclude our paper, Metodiev et al.<\/a> produced a modified version of THAMES, where they truncate the original ellipsoid to intersect with the HPD region of interest, as discussed in an earlier \u2018Og entry<\/a> (with a reply from the authors<\/a>). While their paper is more focussed on mixture inference and include other aspects on Bayesian inference for mixture, we rewrote ours to include a comparison between the methods, which proves satisfactory, especially in larger dimensions.<\/p>\n

\"\"<\/p>\n\n

\r\n
To leave a comment<\/strong> for the author, please follow the link and comment on their blog: R \u2013 Xi'an's Og<\/a><\/strong>.<\/div>\r\n
\r\nR-bloggers.com<\/a> offers daily e-mail updates<\/a><\/strong> about R<\/a> news and tutorials about learning R<\/a> and many other topics. Click here if you're looking to post or find an R\/data-science job<\/a>.\r\n\r\n
Want to share your content on R-bloggers? click here<\/a> if you have a blog, or here<\/a> if you don't.\r\n<\/div>","protected":false},"excerpt":{"rendered":"
Following a suggestion by Christian Hennig at JSM 2024, I started working with my PhD student Dana Naderi on a detailed assessment of the method we proposed in 2009 with Darren Wraith for evidence approximation. (The method was briefly mentioned in a Physical Review paper and also briefly illustrated in our 2010 San …<\/div>\n
<\/div>\n
<\/div>\n","protected":false},"author":56,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":[],"categories":[4],"tags":[],"aioseo_notices":[],"jetpack-related-posts":[],"amp_enabled":true,"_links":{"self":[{"href":"https:\/\/www.r-bloggers.com\/wp-json\/wp\/v2\/posts\/396312"}],"collection":[{"href":"https:\/\/www.r-bloggers.com\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.r-bloggers.com\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.r-bloggers.com\/wp-json\/wp\/v2\/users\/56"}],"replies":[{"embeddable":true,"href":"https:\/\/www.r-bloggers.com\/wp-json\/wp\/v2\/comments?post=396312"}],"version-history":[{"count":4,"href":"https:\/\/www.r-bloggers.com\/wp-json\/wp\/v2\/posts\/396312\/revisions"}],"predecessor-version":[{"id":400755,"href":"https:\/\/www.r-bloggers.com\/wp-json\/wp\/v2\/posts\/396312\/revisions\/400755"}],"wp:attachment":[{"href":"https:\/\/www.r-bloggers.com\/wp-json\/wp\/v2\/media?parent=396312"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.r-bloggers.com\/wp-json\/wp\/v2\/categories?post=396312"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.r-bloggers.com\/wp-json\/wp\/v2\/tags?post=396312"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}