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A Bayesian filter has been developed for tracking an extended object in clutter based on two simple axioms: (i) the numbers of received target and clutter measurements in a frame are Poisson distributed (so several measurements may originate from the target) and (ii) target extent is modelled by a spatial probability distribution and each target-related measurement is an independent 'random draw' from this spatial distribution (convolved with a sensor model). Diffuse spatial models of target extent are of particular interest. This model is especially suitable for a particle filter implementation, and examples are presented for a Gaussian mixture model and for a uniform stick target convolved with a Gaussian error. A rather restrictive special case that admits a solution in the form of a multiple hypothesis Kalman filter is also discussed and demonstrated.