bv_cvxbook_extra_exercises

# find the maximum likelihood estimate xml and plot it

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Unformatted text preview: , i = 1, . . . , m, meaning that game i was played between teams j (i) and k (i) ; y (i) = 1 means that team j (i) won, while y (i) = −1 means that team k (i) won. (We assume there are no ties.) 53 (a) Formulate the problem of ﬁnding the maximum likelihood estimate of team abilities, a ∈ Rn , ˆ given the outcomes, as a convex optimization problem. You will ﬁnd the game incidence matrix A ∈ Rm×n , deﬁned as y (i) l = j (i) Ail = − y (i) l = k (i) 0 otherwise, useful. The prior constraints ai ∈ [0, 1] should be included in the problem formulation. Also, we ˆ note that if a constant is added to all team abilities, there is no change in the probabilities of game outcomes. This means that a is determined only up to a constant, like a potential. But ˆ this doesn’t aﬀect the ML estimation problem, or any subsequent predictions made using the estimated parameters. (b) Find a for the team data given in team_data.m, in the matrix train. (This matrix gives the ˆ outcomes for a tournament in which each team plays each other team once.) You may ﬁnd the CVX function log_n...
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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 University of California, Berkeley.

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