Abstract
The equivalence of multinomial maximum likelihood and the isotonic projection problem: $$\inf \left\{ {\sum\limits_{i = 1}^n {p_i \ln (p_i /r_i )|p \in K,a convex} {\mathbf{ }}subset of the probability vectors{\mathbf{ }}p} \right\}$$ can be established using Fenchel's Duality Theorem and subgradient and complementary slackness relationships of convex analysis, all taking place over the real numbers.
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