Bayesian Inference on Network Tra c Claudia Tebaldi & Mike West

We study Bayesian models and methods for analysing network tra c counts in problems of inference about the tra c intensity between directed pairs of origins and destinations in networks. This is a class of problems very recently discussed by Vardi in a 1996 JASA article, and of interest in both communication and transportation network studies. The current paper develops the theoretical framework of variants of the origin-destination ow problem, and introduces Bayesian approaches to analysis and inference. In the rst, the so-called xed routing problem, tra c or messages pass between nodes in a network, with each message originating at a speci c source node, and ultimately moving through the network to a predetermined destination node. All nodes are candidate origin and destination points. The framework assumes no travel time complications, considering only the number of messages passing between pairs of nodes in a speci ed time interval. The route count, or route ow, problem is to infer the set of actual number of messages passed between each directed origin-destination pair in the time interval, based on the observed counts owing between all directed pairs of adjacent nodes. Based on some development of the theoretical structure of the problem and assumptions about prior distributional forms, we develop posterior distributions for inference on actual origin-destination counts and associated ow rates. This involves iterative simulation methods, or Markov chain Monte Carlo (MCMC), that combine Metropolis-Hastings steps within an overall Gibbs sampling framework. We discuss issues of convergence and related practical matters, and illustrate the approach in a network previously studied in Vardi's 1996 article. We explore both methodological and applied aspects much further in a concrete problem of a road network in North Carolina, studied in transportation ow assessment contexts by civil engineers. This investigation generates critical insight into limitations of statistical analysis, and particularly of non-Bayesian approaches, due to inherent structural features of the problem. A truly Bayesian approach, imposing partial stochastic constraints through informed prior distributions, o ers a way of resolving these problems, and is consistent with prevailing trends in updating tra c ow intensities in this eld. Following this, we explore a second version of the problem that introduces elements of uncertainty about routes taken by individual messages in terms of Markov selection of outgoing links for messages at any given node. For speci ed route choice probabilities, we introduce the concept of a super-network, namely a xed routing problem in which the stochastic problem may be embedded. This leads to solution of the stochastic version of the problem using the methods developed for the original formulation of the xed routing problem. This is also illustrated. Finally, we discuss various related issues and model extensions, including inference on stochastic route choice selection probabilities, questions of missing data and partially observed link counts, and relationships with current research on road tra c network problems in which travel times within links are non-negligible and may be estimated from additional data.