Im reading Chapter6 of Casella Berger's statistical inference that talks about sufficiency principle. I've been confused a lot by the definition of sufficient statistics, here it is:
Basically, what I cant understand lean on those he claimed:" Experimenter2 knows P(X=y|$T$(X)=$T$(x)), a probability distribution on A$_{T(x)}$={y: $T$(y)=$T$(x)}, because this can be computed from the model without knowledge of the true value of θ. Thus, Experimenter2 can use this distribution and a randomization device," My questions:
does the Experimenter2 necessarily know about A$_{T(x)}$={y: $T$(y)=$T$(x)}?
What does P(X=y|$T$(X)=$T$(x)) mean? I've captured the author's example following his definition,
I guess $\frac{p(x|θ)}{q(T(x)|θ)}$ might be an example of P(X=y|$T$(X)=$T$(y)), is that correct?
- For the a randomization device, did the author mean simulate X by Y based on the distribution of Y and the criterion of $T$(y)=$T$(x) while this simluation cant produce more information about θ than X has?
I appreciate all answers in advance.
I have figured out answers to all three questions:
For the first one: Experimenter2 knows T(y) = T(x) while may not necessarily know y. However, Experimenter2 can get a set of all y by enumeration leaning on statistic T(x) = T(Y).
For the second: $\frac{p(x|θ)}{q(T(x)|θ)}$ is indeed an example of P(X=x|T(X)=T (y)), not P(X=y|T(X)=T (y)). The difference here is "X=x" is data achieved by Experiment1 while "X=y", the y is simluated real value of random variable Y based on T(y) = T(x) and T(x) is constant for one experiment. P(X=y|$T$(X) = $T$(x)) represents the probability of random variable X of Experimenter1 achieving simulated real value y based on random variale T(X) = real value T(x).
For the third: Yes.
I think I've tried to make them all clear. Everyone debugging above is welcome. Thanks.