Let $W_t$ denote the standard Wiener process (zero mean).
Let $B$ be a uniform binary random variable taking values $\pm 1$ independent of the stochastic process $\{W_t\}$.
We define $$Y_t = B + W_t ,\quad 0\leq t\leq 1$$
Suppose I observe the sample path $Y_t,\ 0\leq t\leq 1$.
How do I calculate the conditional distribution of $B$ given the sample path $Y_t,\ 0\leq t\leq 1$?