Next Article in Journal
On the Binary Input Gaussian Wiretap Channel with/without Output Quantization
Previous Article in Journal
The Second Law: From Carnot to Thomson-Clausius, to the Theory of Exergy, and to the Entropy-Growth Potential Principle
Article Menu
Issue 2 (February) cover image

Export Article

Open AccessArticle
Entropy 2017, 19(2), 58;

Comparison Between Bayesian and Maximum Entropy Analyses of Flow Networks†

School of Engineering and Information Technology, The University of New South Wales, Canberra, ACT 2600, Australia
Author to whom correspondence should be addressed.
Received: 22 December 2016 / Accepted: 22 January 2017 / Published: 2 February 2017
View Full-Text   |   Download PDF [289 KB, uploaded 16 February 2017]


We compare the application of Bayesian inference and the maximum entropy (MaxEnt) method for the analysis of flow networks, such as water, electrical and transport networks. The two methods have the advantage of allowing a probabilistic prediction of flow rates and other variables, when there is insufficient 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 finds 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. View Full-Text
Keywords: maximum entropy analysis; Bayesian inference; probability; flows; networks maximum entropy analysis; Bayesian inference; probability; flows; networks
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).

Share & Cite This Article

MDPI and ACS Style

Waldrip, S.H.; Niven, R.K. Comparison Between Bayesian and Maximum Entropy Analyses of Flow Networks†. Entropy 2017, 19, 58.

Show more citation formats Show less citations formats

Note that from the first issue of 2016, MDPI journals use article numbers instead of page numbers. See further details here.

Related Articles

Article Metrics

Article Access Statistics



[Return to top]
Entropy EISSN 1099-4300 Published by MDPI AG, Basel, Switzerland RSS E-Mail Table of Contents Alert
Back to Top