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Performing efficient inference on high dimensional discrete Bayesian Networks (BNs) is challenging. When using exact inference methods the space complexity can grow exponentially with the tree-width, thus making computation intractable. This paper presents a general purpose approximate inference algorithm, based on a new region belief approximation method, called Triplet Region Construction (TRC). TRC reduces the cluster space complexity for factorized models from worst-case exponential to polynomial by performing graph factorization and producing clusters of limited size. Unlike previous generations of region-based algorithms, TRC is guaranteed to converge and effectively addresses the region choice problem that bedevils other region-based algorithms used for BN inference. Our experiments demonstrate that it also achieves significantly more accurate results than competing algorithms.