I have seen this identity in some economics journals, but cannot derive it myself. The identity is formulated as follows:
Let's say the correct realization of $\mu$ is either 1/2 or -1/2. And the prior on the correct realization is 1/2. Let $X_t$ represent a Brownian motion. i.e. $X_t = \mu t + Z_t$. \ Can we show then, that $Pr(\mu=1/2|X_t) = 1/[1+e^{X_t}]$?