I observe a variable $\hat{x}$. I know that my observation is noisy due to some Gaussian measurement noise $\mathcal{N}(0, \sigma)$.
My goal is to compute the probability $p(x > 0\mid\hat{x})$ that the true underlying $x \in \mathbb{R}$ is positive. The a priori probability of any $x$ is unknown, i.e. the prior distribution is assumed to be flat. Also assume I know $\sigma$ precisely. What is $p(x > 0\mid\hat{x})$?
We can find the posterior distribution of $x$:
$$p(x|\hat x)\propto p(\hat x|x)p(x)\propto\exp\left[-\dfrac{(x-\hat x)}{2\sigma^2}\right]\Rightarrow x\sim\mathcal N(\hat x, \sigma^2)$$
Since $x|\hat x$ is normal, we can easily calculate $p(x>0|\hat x)$ using the standard normal CDF, $\Phi$:
$$p(x>0|\hat x)=1-\Phi\left(\dfrac{0-\hat x}{\sigma}\right)=\Phi\left(\dfrac{\hat x}{\sigma}\right)$$
Notice that, in this case, this happens to be the same as one minus the p-value for the frequentist Z-test for $x>0$.
