Suppose $x_i \sim N(10,4)$ - ie, the distribution is known.
There is a noisy signal $s_i \sim N(x_i, \sigma_e^2)$ and I want to estimate $\sigma_e$.
I see some pairs ($s_i, x_i$) but they are not 'random' (there is some selection on $s_i$ - say, I see these values mostly for large values of $s_i$).
What approach would you recommend for estimating $\sigma_e$?
My first thought was to simply take $\hat\sigma_e = \sqrt{\sum(s_i-b_i)^2/N}$ but I think this would be wrong since my data contains mostly larger $s_i$?
I next tried to model the distribution of $x_i | s_i$. The Bayesian posterior distribution is $N((\frac{\sigma_{e}^2}{\sigma_{e}^2+4})10 + (\frac{4}{\sigma_{e}^2+4})s_{i}, \frac{\sigma_{e}^2*4}{\sigma_{e}^2+4})$ where the $4$ and $10$ come from the distribution of $x_i$ above.
But I do not know how to proceed. I believe that the difference between the expected value of $x_i|s_i$ and the actual observed value should be informative of $\sigma_e$ somehow, but I am confused since $\sigma_e$ enters both the expected value term and the variance term above. Trying to find the MLE by differentiating the log likelihood above becomes very complicated. Is there something obvious that I am missing? Or any approach that you would recommend to estimate $\sigma_e$?
Thanks!