Take the tour ×   $$ S \sim N(\mu, \sigma^2) $$  be a normally
distributed random variable with known $\mu$ and $\sigma^2$.
Suppose, we observe  $$ X = \begin{cases} T & \text{if $S \ge
0$}, \\ -T & \text{if $S<0$},\end{cases} $$ where $T \in
\mathbb{R}$. The probability distribution of $X$ is given by: $$
p(x) = Q\left(\frac{-\mu}{\sigma}\right)\delta(x-
T)+Q\left(\frac{\mu}{\sigma}\right)\delta(x+T) $$  I want to
optimize the value of $T$ such that $X$ conveys as much
information about $S$ as possible.  My Attempt:  a. I tried
minimizing the Kullback–Leibler divergence between the
distribution of $X$ and $S$, but as mentioned here, it is not
possible.  b. I tried to calculate the mutual information
between the two distributions, it turned out to be independent
of $\alpha$.  Is there any other way of formulating this
problem? I feel quite confident that there must be such $T$ for
which $X$ explains $S$ better, e.g., assume $\mu=10000$ then a
value of $T$ near $10000$ will better explain $S$ than say
$T=2$? On