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Given data where the class is categorical (finite and discrete), there are multiple ways to come up with a classifier. One could use multinomial logistic regression, or support vector clustering (implemented, for example, as SVC in scikit-learn). What is the fundamental difference between these two algorithms? When is one better than the other? Are they fundamentally trying to minimize a different type of error?

An entirely different approach is generative algorithms, and in particular naive Bayes. When is that appropriate as opposed to the discriminative alogirthms above?

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Your question really requires more of an answer than can really fit in a space like this. Luckily there is a good book on the topic and an electronic copy can be downloaded for free. See The Elements of Statistical Learning.

The book has sections/chapters on each of the topics you ask about.

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