A representation theorem for a minimally sufficient statistic by Bahadur The Statement of the Problem

I'd appreciate help in proving the following, unproven theorem from a classic article by Bahadur ([BAH], Theorem 6.3) (the expressions in square brackets are my annotations; they do not appear in the original):


If $T$ is a sufficient statistic for $P$ [with $P$ dominated by a $\sigma$-finite measure], there exists a field $T_0$ of [measurable] subsets of the range of $T$ such that the subfield $T^{-1}\left(\mathbf{T}_0\right)$ is necessary and sufficient [= minimally sufficient] for $P$.


A clarifications of the terms and symbols used in the statement of the problem


Denote by $\overline{\mathbb{R}}$ the extended real line $\mathbb{R}\cup\left\{\pm\infty\right\}$, denote by $\mathfrak{B}$ the standard Borel field on $\mathbb{R}$ and by $\overline{\mathfrak{B}}$ - the standard Borel field on $\overline{\mathbb{R}}$.


Let $M:=\left(X,\mathbf{S}\right)$ be a measurable space and let $P$ be a non-empty set of probability measures on $M$. Set $\mathbf{M}:=\left(M,P\right)$. It is assumed that $P$ is $\sigma$-finitely dominated, i.e. that there's some $\sigma$-finite measure on $M$, $\lambda$, such that $p\ll\lambda$ for all $p\in P$. We write $P\ll\lambda$ to designate this.


If $\mathbf{S}_0$ is a sub-$\sigma$-algebra of $\mathbf{S}$, $p\in P$, and $f,g:X\rightarrow\overline{\mathbb{R}}$, we write

$$
f=g\space\left[\mathbf{S}_0,p\right]
$$

to indicate that there's some $N\in\mathbf{S}_0$, such that $p(N)=0$ and for all $x\in X\backslash N$, $f(x)=g(x)$. Analogous interpretations apply to the statements $f\neq g\space\left[\mathbf{S}_0,p\right]$, $f<g\space\left[\mathbf{S}_0,p\right]$, $f\leq g\space\left[\mathbf{S}_0,p\right]$ etc.


We write

$$
f=g\space\left[\mathbf{S}_0,P\right]
$$

to indicate that $f=g\space\left[\mathbf{S}_0,P\right]$ for all $p\in P$. Analogous interpretations apply to the statements $f\neq g\space\left[\mathbf{S}_0,P\right]$, $f<g\space\left[\mathbf{S}_0,P\right]$, $f\leq g\space\left[\mathbf{S}_0,P\right]$ etc.


Suppose $\mathbf{S}_0$ is a sub-$\sigma$-algebra of $\mathbf{S}$ and $p\in P$. For every $A\in\mathbf{S}_0$, we let $\mathrm{Pr}_{\left(M,p\right)}\left[A\mid\mathbf{S}_0\right]$ denote a fixed, but otherwise arbitrary, determination of the conditional probability of $A$ given $\mathbf{S}_0$, w.r.t. $\left(M,p\right)$. Consequently, $H:X\rightarrow\overline{\mathbb{R}}$ is any other determination of this conditional probability iff

$$
H=\mathrm{Pr}_{\left(M,p\right)}\left[A\mid\mathbf{S}_0\right]\space\left[\mathbf{S}_0,p\right]
$$


If $\mathbf{S}_0$, $\mathbf{S}_1$ are sub-$\sigma$-algebras of $\mathbf{S}$, we write

$$
\mathbf{S}_0\subseteq\mathbf{S}_1\space\left[\mathbf{S}, P\right]
$$

iff for every $A\in\mathbf{S}_0$ there's some $B\in\mathbf{S}_1$ such that $p\left(A\Delta B\right)=0$ for all $p\in P$. We write $\mathbf{S}_0=\mathbf{S}_1\space\left[\mathbf{S}, P\right]$ iff $\mathbf{S}_0\subseteq\mathbf{S}_1\space\left[\mathbf{S}, P\right]$ and $\mathbf{S}_0\supseteq\mathbf{S}_1\space\left[\mathbf{S}, P\right]$.


A sub-$\sigma$-algebra of $\mathbf{S}$, $\mathbf{S}_0$, is sufficient for $P$ iff for every $A\in\mathbf{S}$ there's some $\mathbf{S}_0/\overline{\mathfrak{B}}$-measurable function $\varphi_A:X\rightarrow\overline{\mathbb{R}}$ such that

$$
\varphi_A=\mathrm{Pr}_{\left(M,p\right)}\left[A\mid\mathbf{S}_0\right]\space\left[\mathbf{S}_0,p\right]
$$

for all $p\in P$.


A sub-$\sigma$-algebra of $\mathbf{S}$, $\mathbf{S}_0$, is necessary for $P$ iff $\mathbf{S}_0\subseteq\mathbf{S}_1\space\left[\mathbf{S}, P\right]$ whenever $\mathbf{S}_1$ is a sub-$\sigma$-algebra of $\mathbf{S}$ that is sufficient for $P$.


Let $Y$ be a non-empty set. An $\mathbf{S}/Y$-statistic is a function $T:X\rightarrow Y$ that is surjective on $Y$. The $\sigma$-algebra induced by $T$ on $Y$, $\mathbf{T}$, consists of all the subsets $B\subseteq Y$, for which $T^{-1}(B)\in\mathbf{S}$. $T$ is said to be sufficient for $P$, resp. necessary for $P$, iff $T^{-1}\left(\mathbf{T}\right)$ is.

A Useful, Related Result: The Fisher-Neyman Factorization Theorem

The following classic result, known as the Fisher-Neyman Factorization Theorem, is proved in Bahadur's article (Corollary 6.1). It gives a characterization of sufficiency in the case that $P$ is $\sigma$-finitely dominated. Assume the same terms and notations defined in the previous section.


Let there be given a $\sigma$-finite measure $\lambda$ on $\mathbf{S}$ such that $P\ll\lambda$. A necessary and sufficient condition that a statistic $T$ be sufficient for $P$ is that there exist a nonnegative function $h$ on $X$ and a set $\left\{g_p:p\in P\right\}$ of nonnegative functions with domain $Y$ such that


*

*
$h$ is an $\mathbf{S}$-measurable function;



*
For each $p$, $g_p\circ T$ is an $\mathbf{S}$-measurable function;



*
For each $p$, $dp = h\cdot\left(g_p\circ T\right)\space d\lambda$ on $\mathbf{S}$.



References

[BAH] Bahadur, R. R. Sufficiency and Statistical Decision Functions. Ann. Math. Statist. Volume 25, Number 3 (1954), 423-462.

 A: 
Define $\mathbf{S}_0 := T^{-1}\left(\mathbf{T}\right)$. Choose a countable subset of $P$, $P_0=\left\{p_1,p_2,\dots\right\}$, such that every $\mathbf{S}$-$P_0$ null set (i.e. every set $B\in\mathbf{S}$ such that $p(B)=0$ for all $p\in P_0$) is also $\mathbf{S}$-$P$ null and vice-versa. The existence of $P_0$ is guaranteed by [HAL] lemma 7; alternatively [BAH] remark preceding theorem 6.1. Choose any sequence of positive, real constants that sum to $1$ and define

$$\lambda_0:\mathbf{S}\rightarrow\overline{\mathbb{R}},\space\space\lambda_0(B):=\sum_i c_i p_i(B)$$


According to [BAH] theorem 6.1, $\lambda_0$ is a probability measure on $M$ and to each $p\in P$ there exists a non-negative, $\mathbf{S}_0$-measurable function $g_p:X\rightarrow\overline{\mathbb{R}}$ such that

$$g_p=\frac{dp}{d\lambda_0}\space\left[\mathbf{S}_0,\lambda_0\right]$$


According to the factorization theorem ([HAL] lemma 2; also [BAH] lemma 3.2), to each $p\in P$ there's a unique $h_p:Y\rightarrow\overline{\mathbb{R}}$ such that $g_p=h_p\circ T$. According to [BAH] lemma 3.1, $h_p$ is $\mathbf{T}/\overline{\mathfrak{B}}$-measurable.


Define

$$A_p(r):=\left\{x:g_p(x)<r\right\}$$

for all $p\in P$, $r\in\mathbb{R}$ and

$$\mathbf{S}^*:=\sigma\left(A_p(r):p\in P, r\in\mathbb{R}\right)$$

According to [BAH] theorem 6.2, $\mathbf{S}^*$ is necessary and sufficient for $P$.


Define

$$
A'_p(r):=\left\{y:h_p(y)<r\right\}
$$

for all $p\in P$ and $r\in\mathbb{R}$ and

$$
\mathbf{T}_0:=\sigma\left(A'_p(r):p\in P, r\in\mathbb{R}\right)
$$

Then $\mathbf{T}_0\subseteq\mathbf{T}$ and $\mathbf{S}^*=T^{-1}\left(\mathbf{T}_0\right)$.
$\square$

References


*


*
[BAH] Bahadur, R. R. Sufficiency and Statistical Decision Functions. Ann. Math. Statist. Volume 25, Number 3 (1954), 423-462.



*
[HAL] Halmos, Paul R., Savage, L. J. "Application of the Radon-Nikodym Theorem to the Theory of Sufficient Statistics". Annals of Mathematical Statistics, 20, 225-241 (1949) 



