{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,7]],"date-time":"2026-08-07T08:50:13Z","timestamp":1786092613563,"version":"3.56.0"},"reference-count":21,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2018,8,9]],"date-time":"2018-08-09T00:00:00Z","timestamp":1533772800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>The refinement axiom for entropy has been provocative in providing foundations of information theory, recognised as thoughtworthy in the writings of both Shannon and Jaynes. A resolution to their concerns has been provided recently by the discovery that the entropy measure of a probability distribution has a dual measure, a complementary companion designated as \u201cextropy\u201d. We report here the main results that identify this fact, specifying the dual equations and exhibiting some of their structure. The duality extends beyond a simple assessment of entropy, to the formulation of relative entropy and the Kullback symmetric distance between two forecasting distributions. This is defined by the sum of a pair of directed divergences. Examining the defining equation, we notice that this symmetric measure can be generated by two other explicable pairs of functions as well, neither of which is a Bregman divergence. The Kullback information complex is constituted by the symmetric measure of entropy\/extropy along with one of each of these three function pairs. It is intimately related to the total logarithmic score of two distinct forecasting distributions for a quantity under consideration, this being a complete proper score. The information complex is isomorphic to the expectations that the two forecasting distributions assess for their achieved scores, each for its own score and for the score achieved by the other. Analysis of the scoring problem exposes a Pareto optimal exchange of the forecasters\u2019 scores that both are willing to engage. Both would support its evaluation for assessing the relative quality of the information they provide regarding the observation of an unknown quantity of interest. We present our results without proofs, as these appear in source articles that are referenced. The focus here is on their content, unhindered. The mathematical syntax of probability we employ relies upon the operational subjective constructions of Bruno de Finetti.<\/jats:p>","DOI":"10.3390\/e20080593","type":"journal-article","created":{"date-parts":[[2018,8,9]],"date-time":"2018-08-09T10:36:31Z","timestamp":1533810991000},"page":"593","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":20,"title":["The Duality of Entropy\/Extropy, and Completion of the Kullback Information Complex"],"prefix":"10.3390","volume":"20","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-5480-1351","authenticated-orcid":false,"given":"Frank","family":"Lad","sequence":"first","affiliation":[{"name":"Department of Mathematics and Statistics, University of Canterbury, 8140 Christchurch, New Zealand"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0657-3833","authenticated-orcid":false,"given":"Giuseppe","family":"Sanfilippo","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Computer Science, University of Palermo, 90123 Palermo, Italy"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9326-8652","authenticated-orcid":false,"given":"Gianna","family":"Agr\u00f2","sequence":"additional","affiliation":[{"name":"Department of Economics, Business, and Statistics, University of Palermo, 90128 Palermo, Italy"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2018,8,9]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"40","DOI":"10.1214\/14-STS430","article-title":"Extropy: Complementary dual of entropy","volume":"30","author":"Lad","year":"2015","journal-title":"Stat. Sci."},{"key":"ref_2","unstructured":"Lad, F., and Sanfilippo, G. (2018). Scoring alternative forecast distributions: Completing the Kullback symmetric distance complex. Glob. Local Econ. Rev., Available online: http:\/\/arxiv.org\/abs\/1806.11178."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"13","DOI":"10.1063\/1.4759585","article-title":"Completing the logarithmic scoring rule for assessing probability distributions","volume":"1490","author":"Lad","year":"2012","journal-title":"AIP Conf. Proc."},{"key":"ref_4","unstructured":"De Finetti, B. (1970). Teoria Delle Probabilit\u00e0 (English Version: Theory of Probability), Einaudi. 2 Volumes. Reprinted in Wiley: Chichester, UK, 1974  (1975)."},{"key":"ref_5","unstructured":"De Finetti, B. (1972). Probability, Induction, and Statistics, Wiley."},{"key":"ref_6","unstructured":"Lad, F. (1996). 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Lett."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/20\/8\/593\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T15:17:40Z","timestamp":1760195860000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/20\/8\/593"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,8,9]]},"references-count":21,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2018,8]]}},"alternative-id":["e20080593"],"URL":"https:\/\/doi.org\/10.3390\/e20080593","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2018,8,9]]}}}