= Brownian endpoint sufficiency for a constant drift
{c}
For $Y_s=\lambda s+W_s$ and fixed $t>0$, subtracting $(s/t)Y_t$ leaves the <Brownian bridge> $W_s-(s/t)W_t$. It is independent of $(\lambda,Y_t)$, since it is independent of $\lambda$ and has zero <covariance> with $W_t$ in their jointly <Gaussian> law. The observed path is determined by this bridge and its endpoint, so the bridge gives no further information about $\lambda$. Hence conditioning on the full observation <filtration> at $t$ gives the same posterior as conditioning on $Y_t$ alone.
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