We develop an extension to Poisson factorization, to model Multinomial data using a moment parametrization. Our construction is an alternative to the canonical construction of generalized linear models. This is achieved by defining K component Poisson Factorization models and constraining the sum of observation tensors across components. A family of fully conjugate tensor decomposition models for binary, ordinal or multinomial data is devised as a result, which can be used as a generic building block in hierarchical models for arrays of such data. We give parameter estimation and approximate inference procedures based on Expectation Maximization and variational inference. The flexibility of the resulting model on binary and ordinal matrix factorizations is illustrated. Empirical evaluation is performed for movie recommendation on ordinal ratings matrix, and for knowledge graph completion on binary tensors. The model is tested for both prediction and producing ranked lists.
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