91Ó°ÊÓ

Let X have p.d.f. or p.f. \(f\left( {x;\theta } \right)\) conditional on θ. Let T = r(X) be a statistic. We need to prove that for every possible prior p.d.f. for θ, the posterior p.d.f. of θ given X = x depends on x only through r(x).

By Bayes’ theorem

\(g\left( {\theta |x} \right) = \frac{{f\left( {x;\theta } \right)f\left( \theta \right)}}{{\int {f\left( {x;u} \right)f\left( u \right)du} }}\)

Fisher-Neyman Factorization Theorem

\({X_1},...,{X_n}\) be i.i.d r.v. with pdf \(f\left( {x;\theta } \right)\) and let \(T = r\left( {{X_1},...,{X_n}} \right)\) be a statistic .T is sufficient statistic for \(\theta \)iff

\(f\left( {x;\theta } \right) = u\left( x \right)\nu \left( {r\left( x \right);\theta } \right)\) where \(u\,\,\,{\rm{and}}\,\,\,\nu \) are non-negative functions.

By Fisher-Neyman Factorization Theorem

\(g\left( {\theta |x} \right) = \frac{{u\left( x \right)\nu \left( {r\left( x \right);\theta } \right)f\left( \theta \right)}}{{\int {u\left( x \right)\nu \left( {r\left( x \right);z} \right)f\left( z \right)dz} }}\)

This proves that for every possible prior p.d.f. for θ, the posterior p.d.f. of θ given X = xdepends on xonly through r(x).