Pseudo likelihood estimation in network tomography

Network monitoring and diagnosis are key to improving network performance. The difficulties of performance monitoring lie in today´s fast growing Internet, accompanied by increasingly heterogeneous and unregulated structures. Moreover, these tasks become even harder since one cannot rely on the collaboration of individual routers and servers to directly measure network traffic. Even though the aggregatory nature of possible network measurements gives rise to inverse problems, existing methods for solving inverse problems are usually computationally intractable or statistically inefficient. In this paper, a pseudo likelihood approach is proposed to solve a group of network tomography problems. The basic idea of pseudo likelihood is to form simple subproblems and construct a product of marginal likelihood of subproblems by the ignoring their dependences. As a result, it keeps a good balance between the computational complexity and the statistical efficiency of the parameter estimation. Some statistical properties of the pseudo likelihood estimator, such as consistency and asymptotic normality, are established. A pseudo expectation-maximization (EM) algorithm is developed to maximize the pseudo log-likelihood function. Two examples with simulated or real data are used to illustrate the pseudo likelihood proposal: (1) internal link delay distribution inference through multicast end-to-end measurements; (2) origin-destination matrix estimation through link traffic counts.