Theorem (Factorization theorem): The statistic $T(X)$ is sufficient for $\theta$ if and only if the model $p(x \mid \theta)$ can be factorized as follows:
\[p(x \mid \theta) = g(t(x), \theta) h(x).\]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 $\theta$.
\[\begin{alignat*}{2} L(\theta \mid \x) &= p(\x \mid \theta) &\hspace{8em}& \href{likelihood.html}{\text{Def. likelihood}} \\ &= g(t, \theta) h(\x) && \text{Factorization theorem; T sufficient} \\ &\propto L(\theta \mid t) && \text{Define $L(\theta \mid t) = g(t, \theta)$} \end{alignat*}\]