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Using the central observation that margin-based weighted classification error (modeled using Minimum Phone Error (MPE)) corresponds to the derivative with respect to the margin term of margin-based hinge loss (modeled using Maximum Mutual Information (MMI)), this article subsumes and extends marginbased MPE and MMI within a broader framework in which the objective function is an integral of MPE loss over a range of margin values. Applying the Fundamental Theorem of Calculus, this integral is easily evaluated using finite differences of MMI functionals; lattice-based training using the new criterion can then be carried out using differences of MMI gradients. Experimental results comparing the new framework with margin-based MMI, MCE and MPE on the Corpus of Spontaneous Japanese and the MIT OpenCourseWare/MIT-World corpus are presented.