Suppose I have two Gaussian distributions $N(\mu_0, \sigma), N(\mu_1, \sigma)$ with identical variance, where $\mu_0, \sigma$ are known and $\mu_1$ is unknown. Suppose I realize a sample $x$ from one of these two distributions.
What can I say about the probability that $x$ is not from $N(\mu_0, \sigma)$? A reasonable thing to do would be to compute a p-value. That is, assume that $x$ does come from $N(\mu_0, \sigma)$ and then compute $P(x \leq t)$ under $N(\mu_0, \sigma)$. If this probability is sufficiently small then I can reject the thought that $x$ is sampled from $N(\mu_0, \sigma)$. Is this the only thing I can do?
What can I say about the probability that $x$ is sampled from $N(\mu_1, \sigma)$? My first thought is nothing, since I don't know $\mu_1$. For all I know $\mu_1 = \mu_0$. But what if I assume that $\mu_1 > \mu_0$. What can I say about the probability that $x$ is sampled from a distribution with greater mean than $\mu_0$? If I knew both means then I could compute probabilities as above for both and choose the distribution with greater probability as the one more likely to have generated $x$, but I don't know $\mu_1$.
To give some intuition about what I'm thinking about here. I could imagine sampling many times from both $N(\mu_0, \sigma), N(\mu_1, \sigma)$, and running a t-test to determine if $\mu_0 < \mu_1$. But I don't have many samples, I know $\mu_0, \sigma$, and I have a single sample $x$, and I want to know what I can say about it likely coming from a distribution with greater mean than $\mu_0$.