Noisy target - problem in understanding distribution

[iamds]

Noisy targets:

" Indeed, we can formally
express any function f as a distribution P (y I x) by choosing P (y I x) to be
zero for all y except y = f (x) . Therefore, there is no loss of generality if we
consider the target to be a distribution rather than a function"

I am not able to understand how is no loss in generality ensured by considering target distribution and not target function ?

[yaser]

Originally Posted by iamds

I am not able to understand how is no loss in generality ensured by considering target distribution and not target function ?

If we can express a function as a distribution, then considering only distributions will not exclude functions, hence there would be no loss in generality. The fact that we can indeed express a function as a distribution is based on using a "delta function" which allows distributions to put all the probability on a single value of , thus making it effectively a function since is uniquely determined by .

[waleed]

Originally Posted by yaser

If we can express a function as a distribution, then considering only distributions will not exclude functions, hence there would be no loss in generality. The fact that we can indeed express a function as a distribution is based on using a "delta function" which allows distributions to put all the probability on a single value of , thus making it effectively a function since is uniquely determined by .

thank you yaser