Posterior Return Probability with Binary Regime Indicators

A binary latent regime $Z \in \{0, 1\}$ drives the sign of tomorrow's FX carry return $R \in \{+1, -1\}$. Your prior is $P(Z = 1) = \theta$. Conditional on the regime, the return probabilities are: $P(R = +1 \mid Z = 1) = a, \quad P(R = +1 \mid Z = 0) = b$ where $0 < b < a < 1$ (regime 1 is the "good" regime for positive returns). You also observe two binary indicators $X_1, X_2 \in \{0, 1\}$ that are conditionally independent given $Z$, with: $P(X_i = 1 \mid Z = 1) = p, \quad P(X_i = 1 \mid Z = 0) = q$ where $0 < q < p < 1$ (each indicator is more likely to fire in regime 1). You observe that neither indicator fired: $X_1 + X_2 = 0$. Compute $P(R = +1 \mid X_1 + X_2 = 0)$ in closed form.