to replace ( with ln ( wbxxb ( wbxj ( where the matrix ( - ) ( - and vector rc- reparameterize all such that (recall [ ])we [ bc- ] observe that the random response is assumed to have conditional probability distribution for which the log-odds ratio with respect to class and "referenceclass (in this case is linear the separating boundaries between two pairs of classes are again affine hyperplanes the model ( completely specifies the distribution of ynamelyexp(zy+ ( wbxpc = exp(zk where is an arbitrary constantsay corresponding to the "referenceclass and [ zc ]: note that ( wbxis the ( )-st component of softmax( )where exp(zsoftmax exp(zk is the softmax function and [ zc ]finallywe can write the classifier as softmax argmax + { , - in summarywe have the sequence of mappings transforming the input into the output yx we softmax(zargmax + { , - in example we will revisit the multi-logit model and reinterpret this sequence of mappings as neural network in the context of neural networksw is called weight matrix and is called bias vector the parameters and have to be learned from the training datawhich involves minimization of the supervised version of the cross-entropy training loss ( ) lossf (yi xi ) (yi wbxi )ln (yi wbxi = = using the softmax functionthe cross-entropy loss can be simplified toc lossf ( ) ( wbx)-zy+ ln exp(zk ( = the discussion on training is postponed until where we reinterpret the multilogit model as neural netwhich can be trained using the limited-memory bfgs method (exercise note that in the binary case ( )where there is only one vector to be estimatedexample already established that minimization of the cross-entropy training loss is equivalent to likelihood maximization