= Binary Brownian drift filter
{title2=$\mathbb E[\alpha\mid\mathcal F_t^X]=a\tanh(aX_t)$}
For $X_t=W_t+\alpha t$ with an independent equiprobable drift $\alpha\in\{-a,a\}$, the <likelihood ratio> between the two drifts is $e^{2aX_t}$. 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 $q_t=\mathbb P(\alpha=a\mid\mathcal F_t^X)=(1+e^{-2aX_t})^{-1}$. Thus the observed drift is $a(2q_t-1)=a\tanh(aX_t)$. The process $\widehat W_t=X_t-\int_0^t a\tanh(aX_s)ds$ 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>.
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