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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.
 Fall '13
 F.Borrelli
 The Aeneid

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