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Optical burst switching (OBS) is envisioned as the paradigm for a high-bandwidth Internet. Losses due to contention is a serious problem in OBS networks. Since the core nodes have limited capabilities, loss rate cannot be determined at the nodes. Thus, estimating the losses on a path purely based on end-to-end measurements and thereafter develop proactive measures for loss reduction is an attractive option. For the first time, we apply a tomographic technique that can estimate the losses at the core nodes from end-to-end measurement between the edge nodes. We use passive unicast tomography to minimize the network overhead. We model the problem of estimating loss rate at the nodes from path-level measurements as a maximum likelihood problem and solve it using the expectation-maximization algorithm. In simulations, we use a multiple source, multiple destination embedding on the NSFNET topology and observe losses on multiple paths to infer losses at the core nodes. The estimated losses are found to match closely with actual losses measured.