(Zsitek )T W oldZsitek Sample Clauses

(Zsitek )T W oldZsitek. Jaakkola and Haussler (1999) introduce an alternative approach to optimize the logistic regression model by dual optimization. The original maximization of the log-likelihood of a primal problem is replaced with the minimization of the dual form log-likelihood, which guarantees the same optimum. The linear decomposition becomes feasible for the dual optimization. In the following, we use a new response indicator si, taking value 1 if individual i is fully observed and -1 otherwise, to better represent the primal form of the log-likelihood function. The logistic regression model for the response indicator becomes Pr(si = ±1|z) = 1/(1 + exp(−sizT β)), where β ∈ R(p+1)×1 is a vector of nuisance parameters to be esti- mated and zi is the i-th row of Z. The primal problem is to maximize the log-likelihood l(β) = − Σn log(1 + exp(−sizT β)) − λβT β/2. The penalty λβT β/2 is introduced to give a superior generalization performance, especially when p is large. Instead of solving the primal problem, we solve the dual problem which is represented by dual parameter Σ Σ ΣT ψ ∈ Rn×1 as: n n n min J(ψ) = ψ ψ js s j z z j − H(ψ ), (4.1) ψ 2λ