Bayes-Optimal Decision Rule for a Noisy Signal

You observe a noisy signal $S \in \{-1, +1\}$ about an ETF's true value move $V \in \{-1, +1\}$. The prior on $V$ is uniform: $P(V = +1) = P(V = -1) = 0.5$. The signal is informative but imperfect: $P(S = V) = 0.65$. After observing $S$, you must choose an action $a \in \{-1, 0, +1\}$, corresponding to sell, do nothing, or buy. The payoffs are: - If $a \neq 0$ and $a = V$: you earn $+1$ - If $a \neq 0$ and $a \neq V$: you earn $-1$ - If $a = 0$: you earn $0$ 1. Derive the Bayes-optimal decision rule -- that is, the action $a(S)$ that maximizes expected payoff given the observed signal. 2. Compute the overall expected payoff of this optimal strategy.