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97is the number of components (this is the same as equation 3.15 of section 3.2).K∑f(Y i |K, θ) = P j Φ(Y i |θ j ) (5.70)j=1Under the assumption that all pixels are independent, the likelihood for thewhole image is given by equation 5.71, where N is the number of data points.f(Y |K, θ) =M-Step⎛⎞N∏ K∑⎝ P j Φ(Y i |θ j ) ⎠ (5.71)i=1 j=1The M-step consists of finding the maximum likelihood estimate of θ conditionalon Q. Think of Q as a matrix with one row for each pixel and one column foreach component. The mixture proportions are easily obtainable from Q as shownin equation 5.72.ˆP j = 1 NN∑ˆQ ij (5.72)Since Q ij is the probability of pixel i being in component j, we know that eachrow of Q must sum to 1. The sum of all the elements of Q will be equal to N.This agrees with the fact that ∑ j P j = 1.We now find estimates for the density parameters. For a Gaussian mixture,i=1we need estimates for the mean µ j and variance σ 2 jfor each component j. For aPoisson mixture, only the mean is needed. Formulas for the mean and varianceestimates are gven in equations 5.73 and 5.74. In essence, these are simply weightedversions of the usual maximum likelihood estimators, with the weights given byQ.ˆµ j =∑ Ni=1 ˆQij Y i∑i ˆQ ij(5.73)