Bayes-Optimal Decision Rule Under a Hidden Regime

A hidden regime $R \in \{0, 1\}$ has prior probability $P(R = 1) = \pi$. Conditional on $R = r$, you observe a signal $X \sim N(\mu_r, 1)$ where $\mu_1 = +0.4$ and $\mu_0 = -0.2$. After observing $X$, you choose an action $u \in \{-1, +1\}$ and receive payoff $u \cdot Y$, where $Y \in \{-1, +1\}$ with regime-dependent probabilities $P(Y = +1 \mid R = 1) = 0.56$ and $P(Y = +1 \mid R = 0) = 0.48$. 1. Derive the Bayes-optimal decision rule $u^{*}(x)$ that maximizes $E[uY \mid X = x]$, and show it is a threshold rule in $x$. 2. Express the threshold $x^{*}$ explicitly in terms of $\pi$ and Gaussian density terms. 3. For $\pi = 0.3$, compute the threshold numerically and state whether it is closer to $\mu_0 = -0.2$ or $\mu_1 = +0.4$. Explain why.