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The probability for class j with the exception of the last class is

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Pj(Xi) = exp(XiBj)/((sum\[j=1..(k-1)\]exp(Xi*Bj))+1)

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The last class has probability

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1-(sum\[j=1..(k-1)\]Pj(Xi))

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	= 1/((sum\[j=1..(k-1)\]exp(Xi*Bj))+1)

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The (negative) multinomial log-likelihood is thus:

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L = -sum\[i=1..n\]\{

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	sum\[j=1..(k-1)\](Yij \* ln(Pj(Xi)))

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	+(1 - (sum\[j=1..(k-1)\]Yij))

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	\* ln(1 - sum\[j=1..(k-1)\]Pj(Xi))

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	\} + ridge * (B^2)

In order to find the matrix B for which L is minimised, a Quasi-Newton Method is used to search for the optimized values of the m*(k-1) variables. Note that before we use the optimization procedure, we 'squeeze' the matrix B into a m*(k-1) vector. For details of the optimization procedure, please check weka.core.Optimization class.

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