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In Bishop's Machine Learning Book, page 478, Exercise 10.14, he mentions the Expected values of certain parameters given a Gaussian-Wishart Distribution are those given. When trying to prove them, I come to pretty much the same expressions, but I'm having troubles with Eq. 10.64 In the right side of the equation, the term that depends on Beta_k. I don't seem to be able to get where that comes from. Any good suggestion? Thanks |
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Glancing at that term it seems to come from the expectation of the norm of m_k. You can rewrite the quadratic (x-m)^T W (x-m) as x^T W x - 2x^TWm + m^T W m. The expectation of first two terms is just how they look like, but for the last term, once you factor in the covariance of m you have a term that looks like tr(WS), where S is the covariance matrix for m. See this in eq. 296 of the matrix cookbook. The scikits.learn people are reviewing my implementation of DP mixtures of gaussians (they already made it 60 times faster than my original one). You can find the branch that implements it here https://github.com/alextp/scikit-learn/tree/variational-infinite-gmm , and I already included with it a document deriving everything here https://github.com/alextp/scikit-learn/raw/variational-infinite-gmm/doc/modules/dp-derivation.rst . It's in restructured text (a python doc format), but all the equations are in latex.
(Apr 25 '11 at 06:13)
Alexandre Passos ♦
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