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
(1/K )∇θ ℓθ (x1 , . . . , xK ) =
c (xi ) − E c (x).
(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) )...
View Full Document
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.
- Fall '13
- The Aeneid