Say a variable $x$ follows uninformative prior on $\left(-\infty,\infty\right)$ and if $x<x^{*}$, there is a signal generated with probability $p$; with probability $1-p$, no signal is ever generated. If $x\geq x^{*}$, no signal is generated as well.
I understand that conditional on observing the signal, the posterior becomes uninformative on $\left(-\infty,x^{*}\right)$. However, what is the posterior conditional on not observing the signal? In particular, is this still uninformative distribution on $\left(-\infty,\infty\right)$?