problem set6


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ELEC 531: STATISTICAL SIGNAL PROCESSING Department of Electrical and Computer Engineering Rice University Fall 2007 Problem Set 6 Due: October 25, 2007 Problem 6.1 The EM algorithm and mixture densities. Suppose we observe L iid realizations X = ( X 0 , . . . , X L - 1 ) from the mixture density pf ( x )+(1 - p ) g ( x ), where p is unknown. We would like to find the maximum likelihood estimate of the nonrandom parameter p . The EM solution is to augment the observed (incomplete) data X with Z = ( Z 0 , . . . , Z L - 1 ), where Z l tells which component of the mixture X l came from, i.e. X l | z l =1 f ( x l ) and X l | z l =0 g ( x l ) , and Pr( Z l = 1) = p . (a) Show that the EM iterations are given by ˆ p ( k +1) = 1 L L - 1 X l =0 ˆ p ( k ) f ( x l ) ˆ p ( k ) f ( x l ) + (1 - ˆ p ( k ) ) g ( x l ) . (b) Create a MATLAB script that uses the EM algorithm to compute the maximum likelihood estimate of p for the two cases where the true value of p is 0 . 5 and 0 . 8. Assume that f and g are unit variance Gaussian densities with means 0 and 1, respectively.
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