Binary Brownian drift filter ($\mathbb{E}[\alpha \mid {\mathcal{F}}_t^X]=a\tanh (a{X}_t)$)

For $X_t=W_t+\alpha t$ with an independent equiprobable drift $\alpha \in \{-a,a\}$, the likelihood ratio between the two drifts is $e^{2a{X}_t}$. Multiplying the normal distribution densities of successive increments proves this ratio for every observation partition; continuity then gives the full observed-path posterior. The Bayes' theorem gives $q_t=\mathbb{P}(\alpha =a\mid {\mathcal{F}}_t^X)=(1+e^{-2a{X}_t}{)}^{-1}$. Thus the observed drift is $a(2q_t-1)=a\tanh (a{X}_t)$. The process ${\hat{W}}_t=X_t-{\int }_0^{t}a\tanh (a{X}_s)ds$ is an observed-filtration martingale with quadratic variation $t$, so the Lévy characterization of Brownian motion makes it a Brownian motion. The observed state is a scaled diffusion with hyperbolic tangent drift.