{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,18]],"date-time":"2026-08-18T04:39:16Z","timestamp":1787027956142,"version":"build-2736575974"},"reference-count":37,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2017,2,2]],"date-time":"2017-02-02T00:00:00Z","timestamp":1485993600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100000923","name":"Australian Research Council","doi-asserted-by":"publisher","award":["DP140104402"],"award-info":[{"award-number":["DP140104402"]}],"id":[{"id":"10.13039\/501100000923","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Go8\/DAAD Australia-Germany Joint Research Cooperation Scheme","award":["RG123832"],"award-info":[{"award-number":["RG123832"]}]},{"name":"French Agence Nationale de la Recherche Chair of Excellence","award":["TUCOROM"],"award-info":[{"award-number":["TUCOROM"]}]},{"name":"Institute Pprime, Poitiers, France"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>We compare the application of Bayesian inference and the maximum entropy (MaxEnt) method for the analysis of \ufb02ow networks, such as water, electrical and transport networks. The two methods have the advantage of allowing a probabilistic prediction of \ufb02ow rates and other variables, when there is insuf\ufb01cient information to obtain a deterministic solution, and also allow the effects of uncertainty to be included. Both methods of inference update a prior to a posterior probability density function (pdf) by the inclusion of new information, in the form of data or constraints. The MaxEnt method maximises an entropy function subject to constraints, using the method of Lagrange multipliers,to give the posterior, while the Bayesian method \ufb01nds its posterior by multiplying the prior with likelihood functions incorporating the measured data. In this study, we examine MaxEnt using soft constraints, either included in the prior or as probabilistic constraints, in addition to standard moment constraints. We show that when the prior is Gaussian,both Bayesian inference and the MaxEnt method with soft prior constraints give the same posterior means, but their covariances are different. In the Bayesian method, the interactions between variables are applied through the likelihood function, using second or higher-order cross-terms within the posterior pdf. In contrast, the MaxEnt method incorporates interactions between variables using Lagrange multipliers, avoiding second-order correlation terms in the posterior covariance. The MaxEnt method with soft prior constraints, therefore, has a numerical advantage over Bayesian inference, in that the covariance terms are avoided in its integrations. The second MaxEnt method with soft probabilistic constraints is shown to give posterior means of similar, but not identical, structure to the other two methods, due to its different formulation.<\/jats:p>","DOI":"10.3390\/e19020058","type":"journal-article","created":{"date-parts":[[2017,2,2]],"date-time":"2017-02-02T11:24:06Z","timestamp":1486034646000},"page":"58","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":9,"title":["Comparison Between Bayesian and Maximum Entropy Analyses of Flow Networks\u2020"],"prefix":"10.3390","volume":"19","author":[{"given":"Steven","family":"Waldrip","sequence":"first","affiliation":[{"name":"School of Engineering and Information Technology, The University of New South Wales, Canberra, ACT 2600, Australia"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5781-1259","authenticated-orcid":false,"given":"Robert","family":"Niven","sequence":"additional","affiliation":[{"name":"School of Engineering and Information Technology, The University of New South Wales, Canberra, ACT 2600, Australia"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2017,2,2]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Sivia, D., and Skilling, J. 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