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The multi-Bernoulli random finite set (MB-RFS) filter is a recent model for efficiently performing multi-target tracking in video by representing the state as a multi-modal distribution, incorporating data association and target detection into the model itself rather than having them as inputs from external subsystems that can be prone to failure. However, the MB-RFS is based on the non-Bayesian concept of random finite sets and its original derivation does not make it explicit what independence assumptions are being used. We show that the MB-RFS can in fact be reformulated as a purely Bayesian model, define the model and its independence assumptions explicitly and derive simpler update equations that are shown to be identical to the original RFS-based formulas. This equivalence may have implications for further theoretical research aimed at uncovering connections between random finite sets and `classical' Bayesian probability. In addition, a flaw in the original derivation of the MB-RFS is corrected and is shown to greatly improve the performance of the MB-RFS on two publicly available datasets: the VS-PETS 2003 soccer video and an ice hockey video.