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In this problem, we study the output representation and decision rule performed by a multilayer perceptron. In theory for an M-class classification problem in which the union of the M distinct classes forms the entire input space, we need a total of M outputs to represent all possible classification decisions, as depicted in Fig. P4.7. In this figure, the vector xj denotes the jth prototype (i.e., unique sample) of an m-dimensional random vector x to be classified by a multilayer perceptron The kth of M possible classes to which x can belong is denoted by ℓk Let ykj be the kth output of the network produced in response to the prototype xj, as shown by

After a multilayer perceptron is trained, what should the optimum decision rule be for classifying the M outputs of the network?
To address this problem, consider the use of a multilayer perceptron embodying a logistic function for its hidden neurons and operating under the following assumptions:
• The size of the training sample is sufficiently large to make a reasonably accurate estimate of the probability of correct classification.
• The back-propagation algorithm used to train the multilayer perceptron does not get stuck in a local minimum.
Specifically, develop mathematical arguments for the property that the M outputs of the multilayer perceptron provide estimates of the a posteriori class probabilities.