I'm working on some game theory models that involve modeling players' beliefs, and am not sure how to figure out this probability problem.
Suppose I choose some number $x$ between 0 and 1 inclusive (i.e., $0 \leq x \leq 1)$. You do not see my choice, but instead get a signal $S$ normally distributed around my choice. That is, $S$ is a draw from the normal distribution:
$$Z \sim N(x, \sigma^2)$$
You know $S$ (a draw from $Z$), $\sigma^2$, and that $0 \leq x \leq 1$. What are your beliefs about $x$? That is, what do you believe $P(x < y \mid S)$ for any $y$?
I think you might need to have some prior distribution over $x$ in order to solve this. If you do, I think the best prior to use is the uniform distribution $U(0,1)$, since the choice could be anywhere on that interval.
Thanks for any help!
Note: This question is similar to my previous one about two overlapping distributions, except that in this case there are infinitely many possible distributions, not just two.