Common-reference factorization criterion for sufficiency

Suppose $T$ is sufficient, and let $K$ be its common kernel. Fix $\theta$. Disintegrating $P_{\theta }$ and $P_0$ with this same kernel gives $P_{\theta }(A)=\int K(y,A)P_{\theta }^T(dy)$ and $P_0(A)=\int K(y,A)P_0^T(dy)$ for every $A$. Since $P_{\theta }\ll P_0$, also $P_{\theta }^T\ll P_0^T$; let $r_{\theta }=d{P}_{\theta }^T\mathrm{/}d{P}_0^T$. Change of measure [F5] gives $P_{\theta }(A)=\int K(y,A)r_{\theta }(y)P_0^T(dy)$. The disintegration identity under $P_0$ extends to the nonnegative weight $r_{\theta }$ by [F9], making this ${\int }_A{r}_{\theta }(T)d{P}_0$. Uniqueness in [F4] gives $d{P}_{\theta }\mathrm{/}d{P}_0=r_{\theta }(T)$ $P_0$-almost surely.

This proves that the derivative has a $\sigma (T)$-measurable version. Modify that version on the $\sigma (T)$-measurable null set where it is negative or infinite to obtain a finite nonnegative version. By [F6, F7] this version has a Borel factor $g_{\theta }(T)$ on the standard Borel target; reset target values where $g_{\theta }$ is negative or infinite to zero, which preserves the identity $P_0$-almost surely.

Conversely suppose $d{P}_{\theta }\mathrm{/}d{P}_0=g_{\theta }(T)$ $P_0$-almost surely for every $\theta$. By [F3], a regular conditional distribution under $P_0$ factors through $T$ to a kernel $K_0$ on $\mathcal{Y}$. Change of measure [F5] shows $P_{\theta }^T(C)={\int }_C{g}_{\theta }(y)P_0^T(dy)$. For every $A$ and Borel $C$, $P_{\theta }(A\cap T^{-1}C)=\int 1_A{1}_C(T)g_{\theta }(T)d{P}_0={\int }_C{g}_{\theta }(y)K_0(y,A)P_0^T(dy)={\int }_C{K}_0(y,A)P_{\theta }^T(dy)$. The weighted conditional identity follows from [F9]. Thus the same $K_0$ is a conditional law for every parameter, and [F1] gives sufficiency.