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The EM Algorithm for Gaussian Mixtures

The EM Algorithm for Gaussian MixturesProbabilistic Learning: Theory and Algorithms, CS 274 AFinite Mixture ModelsWe are given a data setD={x1, .. , xN}wherexiis ad-dimensional vector measurement. Assume thatthe points are generated in an IID fashion from an underlying densityp(x). We further assume thatp(x)isdefined as a finite mixture model withKcomponents:p(x| ) =K k=1 kpk(x|zk, k)where: Thepk(x|zk, k)aremixture components,1 k K. Each is a density or distribution defined overp(x), with parameters k. z= (z1, .. , zK)is a vector ofKbinary indicator variables that are mutually exclusive and exhaustive( , one and only one of thezk s is equal to 1, and the others are 0).zis aK-ary random variablerepresenting the identity of the mixture component that generatedx. It is convenient for mixturemodels to representzas a vector ofKindicator variables. The k=p(zk)are the mixture weights, representing the probability that a randomly selectedxwasgenerated by componentk, where Kk=1 k= complete set of parameters for a mixture model withKcomponents is ={ 1.}

the covariance matrix of the whole data set for each of the initial K covariance matrices) or could be chosen via some heuristic method (such as by using the k-means algorithm to cluster the data first and then defining weights based on k-means memberships).

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