Modeling group interactions of heterogenous voters in the US Senate

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Statistical models of interacting systems on discrete spaces can be effective causal models - for example, of yes/no voting - but their discrete sample space can turn normalization into a combinatorially complex endeavour: for example, normalizing the pairwise Ising model on the N dimensional binary (hyper)cube is NP-Complete. This lack of normalization can limit their utility and prevent rigorous comparisons to other models. Pairwise interacting models also suffer from quadratic parameter growth as the dimensionality of the sample space grows, unless interactions are structured in some way: for example, homogeneous interactions between groups (a block structured model). Group-structured pairwise interacting models can be effective causal models as well, and are easily normalizable, but aren’t able to capture individual heterogeneity that we suspect exists in some systems, e.g., political systems where every representative/voter has their own ideology (that there is individual heterogeneity is part of our prior). We describe results in exactly normalizing group-interacting pairwise Ising models with heterogeneous individual (linear and local) preferences within polynomial time complexity N^k, where N is the number of individuals and k is the number of groups. We discuss generalizations of this approach to effective low rank approximations of interacting systems, as well as potential applications to social systems, namely the US Senate.