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I know that every class has the same covariance matrix $\Sigma$ in linear discriminant analysis (LDA), and in quadratic discriminant analysis (QDA) they are different. When using gaussian mixture model (GMM) in supervised classification, we fit a Gaussian with its own mean and variance to each class in data. So what is the difference between QDA and GMM?

Is my understanding of GMM wrong? Maybe I should fit more than one Gaussian to each class in order to model subgroups in it. But I am not sure if this is true or not.

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    $\begingroup$ Please don't forget, when asking a question, to expand your abbreviations. $\endgroup$ – ttnphns Mar 1 '16 at 8:03
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    $\begingroup$ A discriminant analysis is not only a classification but also a dimensionality reduction. In DA, classifiers are the discriminants extracted, not the input variables. $\endgroup$ – ttnphns Mar 1 '16 at 8:06
  • $\begingroup$ So, if we consider just the classification part, are QDA and GMM the same? $\endgroup$ – groove Mar 1 '16 at 9:04
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If you're given class labels $c$ and fit a generative model $p(x, c) = p(c) p(x|c)$, and use the conditional distribution $p(c|x)$ for classification, then yes you're essentially performing QDA (the decision boundary will be quadratic in $x$). Under this generative model, the marginal distribution of the data $x$ is exactly the GMM density (say you have $K$ classes):

$$p(x) = \sum_{k \in \{1,...,K\}} p(c=k) p(x|c=k) = \sum_{k=1}^K \pi_k \mathcal{N}({x};{\mu}_k, {\Sigma}_k)$$

"Gaussian mixture" typically refers to the marginal distribution above, which is a distribution over $x$ alone, as we often don't have access to the class labels $c$.

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