  Much work on social media opinion polarization focuses on a flat
categorization of stances (or orthogonal beliefs) of different communities from
media traces. We extend in this work in two important respects. First, we
detect not only points of disagreement between communities, but also points of
agreement. In other words, we estimate community beliefs in the presence of
overlap. Second, in lieu of flat categorization, we consider hierarchical
belief estimation, where communities might be hierarchically divided. For
example, two opposing parties might disagree on core issues, but within a
party, despite agreement on fundamentals, disagreement might occur on further
details. We call the resulting combined problem a hierarchical overlapping
belief estimation problem. To solve it, this paper develops a new class of
unsupervised Non-negative Matrix Factorization (NMF) algorithms, we call Belief
Structured Matrix Factorization (BSMF). Our proposed unsupervised algorithm
captures both the latent belief intersections and dissimilarities, as well as a
hierarchical structure. We discuss the properties of the algorithm and evaluate
it on both synthetic and real-world datasets. In the synthetic dataset, our
model reduces error by 40%. In real Twitter traces, it improves accuracy by
around 10%. The model also achieves 96.08% self-consistency in a sanity check.
