n which is known to be nonnegative and monotonically

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Unformatted text preview: e) Optimality condition for ML estimation. Let ℓθ (x1 , . . . , xK ) be the log-likelihood function for K IID samples, x1 , . . . , xk , from the distribution or density pθ . Assuming log pθ is differentiable in θ, show that 1K (1/K )∇θ ℓθ (x1 , . . . , xK ) = c (xi ) − E c (x). θ K i=1 (The subscript under E means the expectation under the distribution or density pθ .) Intepretation. The ML estimate of θ is characterized by the empirical mean of c(x) being equal to the expected value of c(x), under the density or distribution pθ . (We assume here that the maximizer of ℓ is characterized by the gradient vanishing.) 6.4 Maximum likelihood prediction of team ability. A set of n teams compete in a tournament. We model each team’s ability by a number aj ∈ [0, 1], j = 1, . . . , n. When teams j and k play each other, the probability that team j wins is equal to prob(aj − ak + v > 0), where v ∼ N (0, σ 2 ). You are given the outcome of m past games. These are organized as ( j (i) , k (i) , y (i) )...
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This note was uploaded on 09/10/2013 for the course C 231 taught by Professor F.borrelli during the Fall '13 term at Berkeley.

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