During my semester, my class and I were subjected to an interesting exercise that challenged my entire understanding of sufficient statistics (It turns out that the median of this exercise was 0). Here the exercise
But even with the correction that was given, I still don't understand. The first question defines exactly $T_1(Y)$ as the MAP rule, doesn't it? I don't understand how it can't be a sufficient statistic.
Can someone help me ? Thanks !
Definition: Let $X_1, ... ,X_n$ be a random sample from the distribution $p_X(x;\theta )$. A statistic $S = f(X_1 ... , X_n)$ is defined to be a sufficient statistic if and only if the conditional distribution of $X_1, ... ,X_n$ given $S = s$ does not depend on $\theta$ for any value $s$ of $S$.
In other words: given $S$, $(X_1, ... ,X_n)$ contains no additional information about $\theta$.
Transcribing to our current problem: we sample $Y$ from the distribution $p_{Y}(y; h)$. A statistic $T=f(Y)$ is sufficient iff $p_{Y|T}(y;t)$ does not depend on $h$ for any value $t$ of $T$.
Note that $T_1 = -1$ if $y=-1$ or $y=-2$.
If $h=-1$: $$ p_{Y|T_1}(-1;-1) = \frac{0.3}{0.6} = \frac{1}{2} $$
However, if $h=1$: $$ p_{Y|T_1}(-1;-1) = \frac{0.21}{0.4} \neq \frac{1}{2} $$
Thus, $p_{Y|T_1}(y;t)$ depends on the value of $h$.
In other words: given $T_1$, $Y$ still has additional information about $h$.