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In artificial neural network we can use softmax activation function in the output layer to get a probability distribution of multi classes. Do any machine learning classification algorithms other than neural networks can also be used for such multi class prediction?

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There may be a lot of models that uses softmax function. The simpler one is the multinomial logistic regression.

Given an input vector, this model assigns a score to each class : $$ score(X_i,class_k)=\beta_k \cdot X_i$$

These score are then converted to probabilities using a softmax layer

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