Intuitively, it is very clear that the entire data itself is a sufficient statistic for a parameter of interest. Formally, if random variable $X$ represents data, $S(X)$ is a sufficient statistic when $P_\theta(X|S)$ does not depend on $\theta$. But what is $P_\theta(X|X)?$
2026-03-25 11:34:15.1774438455
How can I formally reason that the data itself is a sufficient statistic?
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A statistic $S$ is sufficient for the random observable $X$ if the conditional distribution of $X$ given $S=s$ is independent of $\theta$ for every $s$. By this definition, $S:=X$ is sufficient because given $S=s$, the conditional distribution of $X$ is a unit mass at $s$, and this conditional distribution contains no mention of $\theta$.