My question is a simplification of a statement in this book that i.i.d. Gaussian random variables $X_1, X_2, …, X_n \sim \mathcal{N}(\Theta, 1)$ are conditionally independent of $\Theta$ given their sum $X_1 + … + X_n$.
I started working on the statement with $n=2$, aiming to show that the conditional distribution of $X_1,X_2$ does not involve the parameter $\Theta$. Immediately, I run into the problem of having the region $X_1 + X_2 = c$ having zero area in $\mathbb{R}^2$. Because of this, I cannot talk about
$$
\int f_{X_1,X_2|Z}(x,y|z) dxdy
$$
where $Z=X_1+X_2$ because any neighborhood of any point in $X_1+X_2=z$ for a fixed $z$ will contain points outside of the restricted region. Working with a conditional distribution does not look promising to me.
My question is: is there an alternative way to show conditional independence of $X_1,X_2$ with the mean $\Theta$ given their sum $X_1+X_2$?
Best Answer
Yes, there is! It is enough to prove that $\sum_i X_i=T$ is the sufficient estimator of $\theta$
(you can prove it in many ways, e.g. factorization theorem)
More, as Gaussian belongs to the Exponential family, T is not only sufficient but also complete and minimal.
Now what you are looking to prove is exactly the definition of sufficiency of T