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QDA (quadratic discriminant analysis) assumes that the K different classes are generated by K different multivariate Gaussians, each with potentially different mean vector and covariance matrix.

If these assumptions hold, will QDA give the same classification results as using EM (expectation maximization) clustering with Gaussian likelihoods?

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See my answer to a related question https://stats.stackexchange.com/a/253235/71228

Yes, both methods use essentially the same assumptions and are capable of learning the exact same kind of underlying data distribution (mixture of Gaussians); the difference is the use case.

Your question is somewhat ill-posed, since you want to compare a supervised learning method (classification given class labels) to an unsupervised learning method (clustering with no class labels).

If you're talking about classification performance given class labels, and assuming somehow you could identify the class labels with clusters learned from EM (and more importantly you somehow know how many true clusters there are!), I would guess QDA would give slightly better results, since no label information is used in clustering the data with GMM+EM.

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