I am trying to understand the probabilistic concepts behind classification using neural networks, for the goal of incorporating prior information over the target class distributions.
I am failing to assign the terms for conditional probability (bayes equation):
Considering an artificial neural network (ANN) for classification, the softmax output of such network is the probability of a Class given Data and some learned weights = $P(C|D,w)$
Through regularization and weight initialization we can set priors on the weight distribution $P(w)$, and using prior maps on the classes, we can also use probabilities over the classes $P(C)$.
How are these terms coherent with Bayes theorem? Is the output of the neural network a likelihood over classes, or is it a posterior over classes, in which case what would be the likelihood? How do we include priors for classes and weights?
So far all I can write is that given a prior over classes, the likelihood must look like this $L(D|C)$, although the likelihood of seeing Data D given a class C does not make much sense to me if we are predicting classes. The posterior would then be $L(D|C)P(C)$ ~ $P(C|D)$. And I do not know where the weights fit into all of this.
How does the correct bayesian formulation look like for classification with priors over classes?