Derivation — Proof of the Factorization Theorem

The previous lesson stated the Neyman-Fisher factorization theorem — $T(X)$ is sufficient for $\theta$ if and only if $p(x;\theta) = g(T(x),\theta)h(x)$ — and used it as a computational black box to find sufficient statistics for five different families without ever proving the "if and only if." This note supplies the missing proof in the discrete case, where it reduces to two short conditional-probability computations, one in each direction. It closes with a remark on what changes (and what doesn't) when the theorem is extended to continuous families.

• reproduce the proof that factorization implies sufficiency, by computing the conditional pmf of $X$ given $T(X)=t$ directly;
• reproduce the proof that sufficiency implies factorization, by writing $p(x;\theta)$ as a product of the marginal pmf of $T$ and a conditional term;
• identify exactly where each direction of the proof uses the definitions of sufficiency and factorization;
• state what additional machinery the continuous (dominated-family) version of the theorem requires, without needing to execute it in full generality.

1. Setup
2. Factorization implies sufficiency
3. Sufficiency implies factorization
4. Extension to continuous families