Comparison Between Bayesian and Maximum Entropy Analysis of Flow Networks

Both the maximum entropy (MaxEnt) and Bayesian methods update a prior to a posterior probability density function (pdf) by the inclusion of new information in the form of constraints or data respectively. To find the posterior, the MaxEnt method maximizes an entropy function subject to constraints, using the method of Lagrange multipliers, whereas the Bayesian method finds its posterior by multiplying the prior with likelihood functions, in which the measured data are substituted into the appropriate terms. The purpose of this work is to develop a Bayesian method to analyze flow networks and compare it to the MaxEnt method. Flow networks include, among others, water and electrical distribution networks and transport networks. The purpose of using probabilistic methods to model these networks is to predict the flow rates (and other variables) when there is not enough information to model them deterministically, and also to incorporate the effects of uncertainty. After developing the Bayesian method, we show that the Bayesian and Maxent methods obtain the same posterior means but, when the prior is a normal distribution, their covariances are different. The Bayesian method incorporates interactions between variables through the likelihood function. It achieves this through second order or higher cross-terms within the posterior pdf. The MaxEnt method however, incorporates interactions between variables using Lagrange multipliers, avoiding second order correlation terms in the posterior covariance. Therefore, the mean value inferences made by the MaxEnt and Bayesian methods are similar, but the MaxEnt method has a numerical advantage in its integrations, as the correlation terms can be avoided.