Suppose there are two random variables, $X_1$ and $X_2$, and we're trying to infer $\theta$.
If $X_1$ and $X_2$ are conditionally independent, then is $f(\theta|X_1)$ a sufficient statistic for $X_1$?
I.e., will this hold:
$f(\theta|X_1,X_2) = f(\theta|X_2,f(\theta|X_1))$
This seems to be the case for the Kalman filter (yesterday's posterior is sufficient statistic for all prior information). Is this a general rule?
I'm sure this is well-studied but I can't find the right terminology.