Define:
$y= \theta + \varepsilon + a,$
where $a$ is a choice variable in a behavioral economic model, with equilibrium solution $a^e$, and $\theta$ and $\varepsilon$ are independently distributed random variables with distributions:
$\theta \sim \mathcal{N}(\bar{\theta},\sigma_{\theta}^2)$,
and
$\varepsilon \sim \mathcal{N}(0,\sigma_{\varepsilon}^2)$.
A paper I am reading, on page 173 at bottom states:
$E[E[\theta|y]]=\bar{\theta} + \phi E[\theta + \varepsilon + a - a^e - \bar{\theta}]$,
where
$\phi=\sigma_{\theta}^2/(\sigma_{\theta}^2+\sigma_{\varepsilon}^2)$.
It refers to this result as a "well known" signal extraction result.
Googling the latter I found that for $y= a + b$, with $a$ and $b$ i.i.d standard normal (i.e. mean zero) then:
$E[a|y] = \frac{\sigma_{a}^2}{\sigma_{y}^2}y$.
This differs from the case above in that the latter is not standard normal with mean zero and the definition of $y$ includes a constant.
Hence I am having difficulty deriving the result in the paper. Grateful if someone could explain the steps. Thanks!