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1. Suppose we have n independent paired comparisons between K competitors with pa- rameters. The likelihood of the parameters under the Bradley-Terry model can be written K ·În (,,,)** + j i=1 ji L(A) = IIII where, in the usual notation, wij denotes the number of times that competitor i beats competitor j. (a) Without performing any simplification to the above expression for L(A), write down the corresponding log likelihood, l(A). (b) It can be shown that the partial derivative of the log likelihood with respect to X, is Σjti Wij Wij + W ji Σ di + Xj Xi j‡i 1 Show that expression (1) is equal to the following expression X; \/\/}. -Σ di + j‡i Wij ΣWij j#i X(u) = Wji d₂ + dj (c) Set the expression in (2) equal to 0 and hence derive an update equation for A₁. This is an alternative to the update equation in Zermelo's algorithm. Wi Σω; j‡i (d) Show that when A = 1K (a vector of K ones) and we have synchronous updates (that is, all À parameters updated at the same time), the update equation from this alternative algorithm can be written as (1) (2) which corresponds to the ratio of wins to losses for competitor i.

1. Suppose we have n independent paired comparisons between K competitors with pa- rameters. The likelihood of the parameters under the Bradley-Terry model can be written K ·În (,,,)** + j i=1 ji L(A) = IIII where, in the usual notation, wij denotes the number of times that competitor i beats competitor j. (a) Without performing any simplification to the above expression for L(A), write down the corresponding log likelihood, l(A). (b) It can be shown that the partial derivative of the log likelihood with respect to X, is Σjti Wij Wij + W ji Σ di + Xj Xi j‡i 1 Show that expression (1) is equal to the following expression X; \/\/}. -Σ di + j‡i Wij ΣWij j#i X(u) = Wji d₂ + dj (c) Set the expression in (2) equal to 0 and hence derive an update equation for A₁. This is an alternative to the update equation in Zermelo's algorithm. Wi Σω; j‡i (d) Show that when A = 1K (a vector of K ones) and we have synchronous updates (that is, all À parameters updated at the same time), the update equation from this alternative algorithm can be written as (1) (2) which corresponds to the ratio of wins to losses for competitor i.

MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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1. Suppose we have n independent paired comparisons between K competitors with pa- rameters. The likelihood of the parameters under the Bradley-Terry model can be written K Wij Xi L(A) = IIII ( + ++ +)* i=1 ji where, in the usual notation, wij denotes the number of times that competitor i beats competitor j. (a) Without performing any simplification to the above expression for L(A), write down the corresponding log likelihood, l(A). (b) It can be shown that the partial derivative of the log likelihood with respect to X₂ is Σjti Wij Wij + W ji Xi di + X j Show that expression (1) is equal to the following expression 1 X; {{{~ + + ₂ } Σ Wij j j‡i Σ j‡i X(u) Σ j‡i = (c) Set the expression in (2) equal to 0 and hence derive an update equation for λ₂. This is an alternative to the update equation in Zermelo's algorithm. Wji Xi + Aj = (d) Show that when A 1K (a vector of K ones) and we have synchronous updates (that is, all À parameters updated at the same time), the update equation from this alternative algorithm can be written as Wi Σω; j‡i (1) which corresponds to the ratio of wins to losses for competitor i.
Transcribed Image Text:1. Suppose we have n independent paired comparisons between K competitors with pa- rameters. The likelihood of the parameters under the Bradley-Terry model can be written K Wij Xi L(A) = IIII ( + ++ +)* i=1 ji where, in the usual notation, wij denotes the number of times that competitor i beats competitor j. (a) Without performing any simplification to the above expression for L(A), write down the corresponding log likelihood, l(A). (b) It can be shown that the partial derivative of the log likelihood with respect to X₂ is Σjti Wij Wij + W ji Xi di + X j Show that expression (1) is equal to the following expression 1 X; {{{~ + + ₂ } Σ Wij j j‡i Σ j‡i X(u) Σ j‡i = (c) Set the expression in (2) equal to 0 and hence derive an update equation for λ₂. This is an alternative to the update equation in Zermelo's algorithm. Wji Xi + Aj = (d) Show that when A 1K (a vector of K ones) and we have synchronous updates (that is, all À parameters updated at the same time), the update equation from this alternative algorithm can be written as Wi Σω; j‡i (1) which corresponds to the ratio of wins to losses for competitor i.
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