Set-up. I am modelling a normally distributed stochastic variable $\theta\sim N\left(\mu_\theta, \sigma_\theta^2\right)$, and calculating the conditional expectation of this based on a number of signals $S_i=\theta+\epsilon_i$ with $\epsilon_i\sim N\left(0, \sigma_i^2\right)$ so that $S_i\sim N\left(\mu_\theta, \sigma^2_\theta+\sigma_i^2\right)$. The error terms $\epsilon_i$ are independent from each other.
Question: How do you calculate the conditional expectation of a normally distributed variable based on several correlated normally distributed variables?
Here, how do you you calculate the conditional mean in, for instance, the case with two signals, $E[\theta|S_1=s_1, S_2=s_2] $, and is there a general formula going forward with more signals?
My attempt. I realize that if there is only one signal, i may use that $E[Y|X=x]=\mu_Y+\sigma_{YX}/\sigma^2_X(x-\mu_X)$ which yields:
$$ E[\theta|S_1=s_1]=\mu_\theta+\frac{\sigma_\theta^2}{\sigma^2_\theta+\sigma_1^2}\left(s_1 - \mu_\theta\right) = \left(1-\frac{\sigma_\theta^2}{\sigma^2_\theta+\sigma_1^2}\right)\mu_\theta+\left(\frac{\sigma_\theta^2}{\sigma^2_\theta+\sigma_1^2}\right)s_1 $$
I have tried simply applying the same formula once more, to get
$$E[\theta|S_1=s_1, S_2=s_2]=E[\theta|S_1=s_1]+\frac{\sigma^2_\theta}{\sigma_\theta^2+\sigma_2^2}\left(s_2-\mu_\theta\right) $$
Is this approach generally correct? A paper i am working based on presents the two-variable case in a different manner as below, but does not offer much explanation, or how general the result is. Nor do they proceed beyond two signals.
$$ E[\theta|S_1=s_1, S_2=s_2]= \left(1-\frac{V(\theta|S_1=s_1)}{V(\theta|S_1=s_1)+\sigma_2^2} \right)E[\theta|S_1=s_1]+\frac{V(\theta|S_1=s_1)}{V(\theta|S_1=s_1)+\sigma_2^2}s_2 $$
Any aid is greatly appreciated.