Dirichlet compound negative multinomial mixture models and applications

In this paper, we consider an alternative parametrization of Dirichlet Compound Negative Multinomial (DCNM) using rising polynomials. The new parametrization gets rid of Gamma functions and allows us to derive the Exact Fisher Information Matrix, which brings significant improvements to model performance due to feature correlation consideration. Second, we propose to improve the computation efficiency by approximating the DCNM model as a member of the exponential family of distributions, called EDCNM. The novel EDCNM model brings several advantages as compared to the DCNM model, such as a closed-form solution for maximum likelihood estimation, higher efficiency due to computational time reduction for sparse datasets, etc. Third, we implement Agglomerative Hierarchical clustering, where Kullback-Leibler divergence is derived and used to measure the distance between two EDCNM probability distributions. Finally, we integrate the Minimum Message Length criterion in our algorithm to estimate the optimal number of components of the mixture model. The merits of our proposed models are validated via challenging real-world applications in Natural Language Processing and Image/Video Recognition. Results reveal that the exponential approximation of the DCNM model has reduced significantly the computational complexity in high-dimensional feature spaces.