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I know that the difference between generative and discriminative classifiers is that the generative ones directly model the distribution of the observed data while the discriminative ones do not.

But then I come across the term discriminative training in which there are Minimum Classification Error(MCE) and Maximum Mutual Information(MMI) estimation. These two methods seems to estimate the parameters for the generative classifiers. It confuses me where the discriminative comes from.

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Generative training is to $\hat M$(learned model) close to the overall joint distribution $P^{*}(Y,X)$ while discriminative training is to get $\hat P(Y|X)$ to be close to $P^{*}(Y|X)$. And the discriminative training is usually performed in the context of undirected models.

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