Fisher–Neyman factorization theorem (1 Viewer)

[He-Mann]

I understand the proof except for one part of the forward implication (only if) (https://en.wikipedia.org/wiki/Sufficient_statistic#Another_proof).

Specifically,



I tried interpreting this through the Law of Total Probability,



But I'm not sure if it's the same simple reasoning as what wiki did.

 

[InteGrand]

I understand the proof except for one part of the forward implication (only if) (https://en.wikipedia.org/wiki/Sufficient_statistic#Another_proof).

Specifically,



I tried interpreting this through the Law of Total Probability,



But I'm not sure if it's the same simple reasoning as what wiki did.

I think what it's saying is that since t is a function T of X, the probability Pr(X = x, t = y) is equal to Pr(X = x) if y = T(x) and 0 otherwise.

This is because if y = T(x), then the event {X = x, t = y} (i.e. {X = x, t = T(x)}) is equivalent to the event {X = x} (because X equals x and t equals T(x) if and only if X equals x (since t = T(X))). Also, if y ≠ T(x), then Pr(X = x, t = y) is 0, since the event is then impossible (since if X = x, t can't equal y, as t = T(x) ≠ y).

Edit: I mixed up the intended notation a bit here, the "t" I used here is used as a random variable, but the Wikipedia article uses it as the value taken on by the random variable (which I denoted by "y" above).