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I am currently studying the online material of Stanford CS229 and I came across the likelihood function for discriminative(for example, regression) and generative algorithms(for example, naive bayes): +Discriminative:

Discriminative likelihood function

+Generative:

Generative likelihood function

In both cases, m is the number of training examples and all training examples are independent of each other. What I am wondering is why in discriminative likelihood function, the formula is the product of conditional probability of y given x and in generative likelihood function, the formula is the product of joint probability of x and y? Is there some reasoning behind this choice?

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  • $\begingroup$ It might be a simple mistake, but "i.e." implies equality. Discriminative models are not synonymous with regression, and generative models are not synonymous with Naive Bayes. $\endgroup$ – Emre Jun 8 '17 at 15:57
  • $\begingroup$ Oh yeah I am sorry for the mistake as english is not my first language. I'm just trying to show an example for each discriminative and generative model. $\endgroup$ – CuriousAlpaca Jun 8 '17 at 18:14
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    $\begingroup$ That's okay, I assumed you meant "e.g.". To answer your question, generative models allow you to generate samples from the joint distribution. Discriminative models don't learn the distribution of X so reducing complexity if it is not desired (don't model what you don't need). Generative models, on the other hand, allow you to marginalize over missing data. $\endgroup$ – Emre Jun 8 '17 at 18:31
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Discriminative model learn to classify an x into class y: the conditional probability distribution p(y|x).

Generative models learn the joint probability distribution p(x,y).It can be transformed into p(y|x) by a Bayes rule.

You get different likelihood functions because the way those models classify data is different.

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