= Sufficient statistic
{title2=$\mathcal L_\theta(X\mid T)=K(T,\cdot)$}
{wiki}
A <statistic> $T=T(X)$ is sufficient for a <statistical model> $\{P_\theta:\theta\in\Theta\}$ when the <conditional distribution> of the full data $X$ given $T$ has a version independent of the unknown <statistical parameter> $\theta$. Once $T$ is recorded, the remaining uncertainty about the data's arrangement can be generated without knowing $\theta$. <Sufficiency (Stat 210a, Fall 2025) by Will Fithian>[Sufficiency, Will Fithian, “Sufficiency”: definition]{ref}
For data and <statistic> taking values in <standard Borel spaces>, express this precisely with one <Markov kernel> $K$, common to every <statistical parameter> value:
$$
P_\theta(X\in B\mid T)=K(T,B)\quad P_\theta\text{-almost surely},\qquad \theta\in\Theta,
$$
for each measurable event $B$. Thus $K(t,\cdot)$ is a <regular conditional distribution> of $X$ given $T=t$. Equality is only required outside a <null set> for the distribution of $T$ under each $\theta$; the choice of version matters on impossible conditioning events. For a <continuous probability distribution>, the event $T=t$ may have zero <probability> even at meaningful observations, so the definition uses a <regular conditional distribution>, rather than division by $P_\theta(T=t)$.
A <sufficient statistic> is a property of a specified <statistical model>, rather than of a formula for summarizing data in isolation. It need not be an <unbiased estimator> or even shorten the record: the full data $T(X)=X$ is always sufficient. An <estimator> instead targets a quantity such as a <mean>. An <ancillary statistic> has a parameter-independent marginal <probability distribution>, whereas a <sufficient statistic> has a parameter-independent conditional full-data law. A <minimal sufficient statistic> permits the greatest reduction among <sufficient statistics>. A <complete statistic> satisfies a separate condition about zero <expectations>; a <complete sufficient statistic> must satisfy both conditions.
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