Maximum A Posteriori (MAP) inference in state observation models typically covers decoding either marginal MAP state estimates or the joint MAP state sequence estimate. This paper addresses a novel yet fundamental MAP inference method denoted as predecessor decoding. This method recursively decodes the most probable predecessors of a chosen initial state using only the marginal distributions from a forward filtering pass. We elaborate on the motivations, abstract relations and analogues, and in particular, the differences between marginal MAP, joint MAP, and MAP predecessors. We conclude by comparing recent results, where predecessor decoding has been utilized for Gaussian mixture models.