I am confused about the intuition behind the definition of a sufficient statistic. The part definition of a sufficient statistic that I am confused about is why the conditional distribution of a sample given the value of the statistic does not depend on the parameter of interest.
I guess this is almost more of a question about conditional distributions maybe. But is there any intuition behind why this makes sense? I understand conditional probability in the sense of events but I am having a hard time understanding why conditioning on the statistic would make this new distribution not depend on the parameter of interest. Does conditioning on a statistic that has all the information about the parameter "remove" that information from the sample? I guess I'm confused about how to conceptually think about conditioning and why it makes sense that the new distribution doesn't depend on the parameter.
The idea for a sufficient statistic is that:
A sufficient statistic can be a function that can give you all the information to compute any estimate of your parameter. Now, roughly speaking, conditional distributions given the parameter is actually a function with no unknown parameters,i.e: $$f(x\mid\theta=2)=2\times\exp(-2x) $$ In this paradeigm $X$ follows the exponential distribution given the parameter value $(=2)$.
So the sufficient statistic has the meaning that for any value of the parameter the conditional distribution of $X\mid T$ is the same. Is this helpful?