Why is it true that if $X \sim f_{\theta}(x) $ (let's assume for simplicty that theta is one dimensional) is some random variable and $T(X)$ a sufficient statistic then $I_{X}(\theta)$ (Fisher information ) is equal to $I_{T(X)}(\theta) $?
It is said that it can be derived from factorization theorem ($f_{\theta}(x) = g_{\theta}(T(x)) \cdot h(x) $) but I can't see how exactly since we don't know the distribution of $T$.