Can you elaborate on what you mean by not taking into account the prior? What is the output of aoftmax if not a probability?

Well, it's a probability of the output given the fixed, fitted model parameters. This is the likelihood. This is not the probability of getting that output, since we don't know what the parameters should really be. See the inverse probability fallacy [0]. This confusion is literally the main reason behind the replication crisis of scientific research.

In ML, you typically have a model f(w;u), and you are trying to fit the parameters w to some data, given some hyperparameters u that represent some kind of prior information. In frequentist techniques, these could be regularization parameters.

In frequentist techniques, the hyperparameters the fitted parameters are assumed fixed, and not random. In contrast, Bayesian methods assume it is random as well. This extra uncertainty is neglected by the predictions of frequentist techniques.

Nonetheless, the uncertainty is real.

[0] https://en.wikipedia.org/wiki/Confusion_of_the_inverse

Thanks!