{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T21:40:44Z","timestamp":1760218844908,"version":"build-2065373602"},"reference-count":32,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2014,8,4]],"date-time":"2014-08-04T00:00:00Z","timestamp":1407110400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>A general approach to Bayesian learning revisits some classical results, which study which functionals on a prior distribution are expected to increase, in a preposterior sense. The results are applied to information functionals of the Shannon type and to a class of functionals based on expected distance. A close connection is made between the latter and a metric embedding theory due to Schoenberg and others. For the Shannon type, there is a connection to majorization theory for distributions. A computational method is described to solve generalized optimal experimental design problems arising from the learning framework based on a version of the well-known approximate Bayesian computation (ABC) method for carrying out the Bayesian analysis based on Monte Carlo simulation. Some simple examples are given.<\/jats:p>","DOI":"10.3390\/e16084353","type":"journal-article","created":{"date-parts":[[2014,8,4]],"date-time":"2014-08-04T08:06:18Z","timestamp":1407139578000},"page":"4353-4374","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":9,"title":["Learning Functions and Approximate Bayesian Computation Design: ABCD"],"prefix":"10.3390","volume":"16","author":[{"given":"Markus","family":"Hainy","sequence":"first","affiliation":[{"name":"Department of Applied Statistics, Johannes Kepler University, 4040 Linz, Austria"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Werner","family":"M\u00fcller","sequence":"additional","affiliation":[{"name":"Department of Applied Statistics, Johannes Kepler University, 4040 Linz, Austria"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Henry","family":"P. 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