**BACKGROUND:** The [softmax function](https://en.wikipedia.org/wiki/Softmax_function) is the most common choice for an activation function for the last dense layer of a multiclass neural network classifier. The outputs of the softmax function have mathematical properties of probabilities and are--in practice--presumed to be (conditional) probabilities of the classes given the features:

1. First, the softmax output for each class is between $0$ and $1$.
2. Second, the outputs of all the classes sum to $1$.

**PROBLEM:** However, just because they have mathematical properties of probabilities does not automatically mean that the softmax outputs are in fact probabilities. In fact, there are other functions that also have these mathematical properties, which are also occasionally used as activation functions.

**QUESTION:** *"Do softmax outputs represent probabilities in the usual sense?"* In other words, do they really reflect chances or likelihoods? (I use likelihood in the colloquial sense here.)