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Unformatted text preview: st N ∈ {1, . . . , n} × {1, . . . , n} of pairs of conditionally independent variables: (i, j ) ∈ N means Xi and Xj are conditionally independent. The problem is to compute the maximum likelihood estimate of the mean a and the covariance matrix S , subject to the constraint that Xi and Xj are conditionally independent for (i, j ) ∈ N . 6.3 Maximum likelihood estimation for exponential family. A probability distribution or density on a set D, parametrized by θ ∈ Rn , is called an exponential family if it has the form pθ (x) = a(θ) exp(θT c(x)), 52 for x ∈ D, where c : D → Rn , and a(θ) is a normalizing function. Here we intepret pθ (x) as a density function when D is a continuous set, and a probability distribution when D is discrete. Thus we have a (θ ) = D exp(θT c(x)) dx when pθ is a density, and −1 −1 T a (θ ) = exp(θ c(x)) x∈D when pθ represents a distribution. We consider only values of θ for which the integral or sum above is finite. Many families of distributions have this form, for appropriate choice of th...
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