Noisy Targets as deterministic target function

[davidrcoelho]

Hi everyone,

As I understood, noisy target is when we observe that for a same value of x we get different values of y.

Then, we model this function as a distribution P(y|x) (instead of a deterministic one)

But later on in the book, Yaser says:

"This view suggests that a deterministic target function can be considered a special case of a noisy target, just with zero noise. Indeed, we can formally express any function f as a distribution P(y|x) by choosing P(y|x) to be zero for all y except y = f(x)".

My question is:

How we can consider a value of y is equal to f(x), since the function f is a distribution.
Suppose we have two different values of y for the same input x, what of these two y's is different to f(x)?