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