In this paper, we formalize and study the properties of a new kind of iterative estimation algorithm which has recently appeared in the literature in e.g. [1, 2, 3]. We refer to this algorithm as the ”iterative pseudo log-MAP function maximization (IPLFM) algorithm. We give a definition of the pseudo log-MAP function (PLF) in terms of regions of a factor graph and prove some of its properties. In particular, we provide a correspondence between the zeroth, first and second order behaviors of the PLF and the (minimum) Bethe free energy associated to the considered factor graph. Based on these properties, we prove some results pertaining to the fixed points and the local convergence of the IPLFMalgorithm. In particular, we relate the fixed points of the IPLFM-algorithm to the stationary points of the Bethe free energy and to the fixed points of the EM algorithm. Moreover, we provide necessary and sufficient conditions for the local convergence of the IPLFM algorithm.