Origin-destination (OD) network tomography problem is the estimation of OD traffic counts from measurable traffic counts at router interfaces. In this paper the problem is formulated as a linear inverse problem with additive noise and is resolved using Bayesian inversion approach. Both OD traffic counts and noise are modelled as Gaussian random functions, and are represented by Karhunen-Lo?ve expansion, respectively. The posterior random function of OD traffic counts given the link counts is also represented as the Karhunen-Lo?ve expansion. With the singular system of routing matrix, we thus can found the optimal estimator of OD traffic counts analytically.