Motivated from an application in economics, I would be interested in a simplified solution for $C$, where
\begin{equation*}
C = E\bigg[exp \bigg\{-\frac{1}{2} \frac{(E[\theta|S]-a)^2}{b}\bigg\}\bigg]
\end{equation*}
and $a$ and $b$ are constants.
However, the distribution of $S$ is a normal mixture and thus I could not find any simplified solution for $C$. Can anybody prove me wrong?
The specific setting is as follows: $S$ is a mixture of two normal distributions with equal means and $\eta_1+\eta_2=1$. Specifically, \begin{equation*} \begin{split} f(S)=\eta_1f_{1\mathcal{N}}(S;\hat{\theta},\sigma_\theta^2+\sigma_{\epsilon_1}^{2})+\eta_2f_{2\mathcal{N}}(S;\hat{\theta},\sigma_\theta^2+\sigma_{\epsilon_2}^{2}). \end{split} \end{equation*}
$\theta$ is $\theta\sim\mathcal{N}(\hat{\theta},\sigma_\theta^2)$. The joint distribution of $S$ and $\theta$ is given by \begin{equation*} \begin{split} f(S,\theta)=\eta_1f_{1\mathcal{N}}(S,\theta;\mu_1,\Sigma_1)+\eta_2f_{2\mathcal{N}}(S,\theta;\mu_2,\Sigma_2) , \end{split} \end{equation*} with \begin{equation*} \boldsymbol{\mu_1}=\boldsymbol{\mu_2}= \begin{pmatrix} \hat{\theta}\\ \hat{\theta} \end{pmatrix} \quad \boldsymbol{\Sigma_1}= \begin{pmatrix} \sigma_\theta^2&\sigma_\theta^2\\ \sigma_\theta^2&\sigma_\theta^2+\sigma_{\epsilon_1}^{2} \end{pmatrix}\quad \boldsymbol{\Sigma_2}= \begin{pmatrix} \sigma_\theta^2&\sigma_\theta^2\\ \sigma_\theta^2&\sigma_\theta^2+\sigma_{\epsilon_2}^{2} \end{pmatrix}\quad \end{equation*}
One can calculate the values of $E_1[\theta|S]$, $ Var_1[\theta|S]$ based on $f_{1\mathcal{N}}(\mu_1,\Sigma_1)$ and $ E_2[\theta|S]$ and $ Var_2[\theta|S]$ based on $f_{2\mathcal{N}}(\mu_2,\Sigma_2)$. It is
\begin{equation*} E_1[\theta|S]=\hat{\theta}+\frac{\sigma_\theta^2}{\sigma_\theta^2+\sigma_{\epsilon_1}^{2}}(S-\hat{\theta}) \quad and \quad Var_1[\theta|S]=\sigma_\theta^2-\frac{\sigma_\theta^4}{\sigma_\theta^2+\sigma_{\epsilon_1}^{2}} \end{equation*}
\begin{equation*} E_2[\theta|S]=\hat{\theta}+\frac{\sigma_\theta^2}{\sigma_\theta^2+\sigma_{\epsilon_2}^{2}}(S-\hat{\theta}) \quad and \quad Var_2[\theta|S]=\sigma_\theta^2-\frac{\sigma_\theta^4}{\sigma_\theta^2+\sigma_{\epsilon_2}^{2}} \end{equation*}
and thus it is
\begin{equation} E[\theta|S]=\dfrac{\eta_1f_{1\mathcal{N}}(S)}{f(S)}E_1[\theta|S]+\dfrac{\eta_2f_{2\mathcal{N}}(S)}{f(S)}E_2[\theta|S]. \end{equation} I doubt that the distribution of $E[\theta|S]$ is well specified.