In the literature and textbooks, one often sees supervised learning expressed as a conditional probability, e.g.,
$$\rho(\vec{y}|\vec{x},\vec{\theta})$$
where $\vec{\theta}$ denotes a learned set of network parameters, $\vec{x}$ is an arbitrary input, and $\vec{y}$ is an arbitrary output. If we assume we have already learned $\vec{\theta}$, then, in words, $\rho(\vec{y}|\vec{x},\vec{\theta})$ is the probability that the network will output an arbitrary $\vec{y}$ given an arbitrary input $\vec{x}$.
I am having a hard time reconciling how, after learning $\vec{\theta}$, there is still a probabilistic aspect to it. Post training, a network is, in general, a deterministic function, not a probability. For any specific input $\vec{x}$, a trained network will always produce the same output.
Any insight would be appreciated.