= Fisher-Neyman factorization theorem
{c}
{wiki}
= Factorization criterion for sufficiency
{synonym}
= Factorization theorem for sufficient statistics
{synonym}
Assume the full-data <sample space> and the range of the <statistic> $T$ are <standard Borel spaces>, $T$ is a <measurable function>, and the <statistical model> $\{P_\theta\}$ is dominated by one common <sigma-finite measure> $\nu$. Write $f_\theta=dP_\theta/d\nu$ for its densities. Then $T$ is a <sufficient statistic> if and only if there exist nonnegative <measurable functions> $g_\theta$ and $h$, with $h$ independent of $\theta$, such that
$$
f_\theta(x)=g_\theta(T(x))h(x)\quad\nu\text{-almost everywhere for each }\theta.
$$
The <statistical parameter> can enter $g_\theta$ only through $\theta$ and the observed $T(x)$. <Sufficiency (Stat 210a, Fall 2025) by Will Fithian>[Sufficiency, Will Fithian, “Factorization theorem”: common dominating measure and almost-everywhere factorization]{ref}
For discrete data, take <counting measure>; for ordinary continuous data, take <Lebesgue measure>. Independence, identical sampling laws, smooth dependence on $\theta$, and a fixed positive <support of a probability distribution> are not requirements of this theorem. Parameter-dependent support must nevertheless be included in $f_\theta$, usually through an <indicator function>, and must fit the displayed factorization. The <sufficient statistic for a symmetric uniform sample> illustrates exactly this: its <indicator function> depends on the unknown scale through the observed absolute maximum. The <null sets> in the theorem concern versions of densities; the factors are not generally themselves normalized <probability distributions>.
For a discrete <sample space>, the cancellation behind the theorem is particularly transparent. On a group with positive <probability>, factorization gives
$$
P_\theta(X=x\mid T=t)=\frac{h(x)\mathbf1_{\{T(x)=t\}}}{\sum_{z:T(z)=t}h(z)}.
$$
Conversely, multiply a common conditional <probability mass function> by $P_\theta(T=T(x))$ to obtain the required factorization. For continuous data, a sum over a group is not a density integral over the original <Lebesgue measure>: the group may have measure zero. Use the <regular conditional distribution> formulation and the <Fisher-Neyman factorization theorem> instead of that discrete calculation.
Back to article page
