Factorization theorem

Theorem (Factorization theorem): The statistic $T (X)$ is sufficient for $θ$ if and only if the model $p (x | θ)$ can be factorized as follows:

p (x | θ) = g (t (x), θ) h (x) .

Proof: CB 6.2.6.

Corollary: The likelihood based on a sufficient statistic is equivalent to the likelihood based on the entire data.

Proof: Suppose $T$ is a sufficient statistic for $θ$ .

\begin{aligned} L (θ | x) & = p (x | θ) & Def. likelihood \\ = g (t, θ) h (x) & Factorization theorem; T sufficient \\ \propto L (θ | t) & Define L (θ | t) = g (t, θ) \end{aligned}