Factorization Theorem Let f θ ( x ) {\displaystyle f_{\theta }(x)} be a statistical model and suppose that f θ ( s ) = h ( s ) f θ ( T ( s ) ) {\displaystyle f_{\theta }(s)=h(s)f_{\theta }(T(s))} then T ( s ) {\displaystyle T(s)} is a sufficient statistic for the model. PROOF s 1 {\displaystyle s_{1}} and s 2 {\displaystyle s_{2}} : T ( s 1 ) = T ( s 2 ) {\displaystyle T(s_{1})=T(s_{2})} f θ ( T ( s 1 ) ) = f θ ( T ( s 2 ) ) {\displaystyle f_{\theta }(T(s_{1}))=f_{\theta }(T(s_{2}))} L ( θ | s 1 ) = h ( s 1 ) f θ ( T ( s 1 ) ) h ( s 2 ) f θ ( T ( s 2 ) ) h ( s 2 ) f θ ( T ( s 2 ) ) = c ( s 1 , s 2 ) L ( θ | s 2 ) {\displaystyle L(\theta |s_{1})={{h(s_{1})f_{\theta }(T(s_{1}))} \over {h(s_{2})f_{\theta }(T(s_{2}))}}h(s_{2})f_{\theta }(T(s_{2}))=c(s_{1},s_{2})L(\theta |s_{2})}