Determining optimal binary decision rule threshold from observations with unknown priors?
My intuition is that it would be difficult to derive the right decision threshold you expect to find:

$$\tau = \frac{1}{2}\left(\mu_0 + \mu_1\right) - \frac{\sigma^2}{\lVert\mu_0 - \mu_1\rVert^2} \log \frac{\pi}{1 - \pi}\left(\mu_0 - \mu_1\right)$$

From the global statistics you are considering (sample mean: $\pi \mu_0 + (1 - \pi) \mu_1$ ; standard deviation: more complex expression but I doubt it would involve a log).

I would approach the problem this way:


If the assumption that $\sigma$ is small can be made

I'm mentioning that, because keep in mind that the decision threshold is affected by $\pi$ only if $\sigma$ is sufficiently high to allow both classes to overlap. If the $\mu$s are distant by more than a few $\sigma$, class prior probabilities have nothing to say in the decision process!


Run k-means on your observations ($\sigma$ is small and is shared by both classes, so k-means is in this case EM for the mixture model). If you just want to binarize these observations and no other data, you can stop here.
If you have new observations to binarize, and you know they are generated by the same process, you can use the class centroids found by k-means on your training data as estimates of $\mu$, and use the middle as a decision threshold. 

If no assumption about $\sigma$ can be made


Run the EM algorithm (with pooled, diagonal covariance) on your training data. Use the inferred "soft class membership" variables to binarize your observations.
Compute the decision threshold $\tau$ from the parameters given by EM to binarize new data generated by the same process.