MATHEMATICAL ENGINEERING TECHNICAL REPORTS On the Interpretation of I-Divergence-Based Distribution-Fitting as a Maximum-Likelihood Estimation Problem
نویسندگان
چکیده
We investigate the mis-match in the classical interpretation of certain distribution-fitting problems as maximum-likelihood (ML) estimation problems in the particular case of the I-divergence. The general relation between Bregman divergences and exponential families shown by Banerjee et al. [8] enables to consider distribution-fitting problems based on Bregman divergences as ML problems based on the corresponding exponential family. This interpretation is however only valid if the data is included in the support of the exponential family. This is the case for the I-divergence, which is associated to the Poisson distribution, when applied to real-valued data. We explain more precisely the reason for this mis-match, and derive an asymptotically justifiable alternative to the usual workaround consisting in quantizing the data, by using the Gamma function as a normalization term.
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