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For some classification algorithms, assuming independence of data helps reduce the number of parameters to estimate. Why then not just to apply a method like PCA (or ICA) to the original features to get a set of independent features, apply these easy classification algorithms and project back the results on the original feature space?

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Methods like PCA do not create independent features. They create orthogonal features.

Independence implies orthogonality, but the reverse is not true.

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    $\begingroup$ Yes; independence of two features (X,Y) means the joint distribution f(X,Y) can be factorized into a product of its marginals f(X)f(Y) while orthogonal means E(XY)=0; x and x^2 are orthogonal but dependent $\endgroup$ – Jessica Nov 23 '14 at 2:18

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