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Unformatted text preview: 2.3. ESTIMATION 27 2.3.3 The EMalgorithm for mixture distributions In Section 2.2, we assumed that all distribution parameters were known. We will now relax this requirement, to allow more realistic modelling. We assume that the distribution type for each model class is known, but that the distributions parameters, = { 1 , . . . , K } , are unknown. The relative fre quencies are also unknown. Thus, we need to estimate Y = { , } , using data y = { y 1 , . . . , y n } . Here, notation indicates that each model, k = 1 , . . . , K , has its own unique parameter set k . However, in general this is not absolutely necessary, and there may be common elements in different k . A natural example is when different models have the same variance or covariance structure. When each data point is sampled from a randomly selected model class, the re sulting density is a mixture, p ( y  Y ) = n productdisplay i = 1 K summationdisplay k = 1 p ( y i  x i = k , k ) k , which can be very difficult to handle analytically. However, if the true class x i for each pixel was known, the likelihood would have the much nicer form p ( y , x  Y ) = n productdisplay i = 1 p ( y i  x i , x i ) x i . The EMalgorithm is a general method for calculating maximum likelihood estima tors by augmenting (extending) the observed data with unknown quantities, which can often yield more easily handled likelihood expressions. Definition 2.3 . ( The EMalgorithm ) Augment the data set y with the random variables for unknown quantities, x . The joint, or complete, variable ( y , x ) is an extended version of y . Let L ( Y  y , x ) = p ( y , x  Y ) be the joint likelihood for ( y , x ) , Given an initial parameter estimate Y (0) , iterate the following steps. 1. Estep: Evaluate Q ( Y , Y ( t ) ) = E (log p ( y , x  Y )  y , Y ( t ) ) , i.e. with the expec tation taken over the conditional (or posterior) distribution for x given the...
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 Winter '08
 EDMONDS

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