# How can supervised learning be viewed as a conditional probability of the labels given the inputs?

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.

• I agree, personally. I tend to describe it as a transformation operation from X to Y via Theta. Once the network is trained, the ultimate transformation is effectively deterministic unless you've created some type of random feedback layer somewhere. The transformation returns a probability or likelihood (as you certainly and clearly know) – David Hoelzer Jan 25 at 0:29