In Gelman's 8-school example (Bayesian Data Analysis, 3rd edition, Ch 5.5) there are eight parallel experiments in 8 schools testing the effect of coaching. Each experiment yields an estimate for the effectiveness of coaching and the associated standard error.
The authors then build a hierarchical model for the 8 data points of coaching effect as follows:
$$ y_i \sim N(\theta_i, se_i) \\ \theta_i \sim N(\mu, \tau) $$
Question In this model, they assume that $se_i$ is known. I do not understand this assumption -- if we feel that we have to model $\theta_i$, why don't we do the same for $se_i$?
I've checked the Rubin's original paper introducing the 8 school example, and there too the author says that (p 382):
the assumption of normality and known standard error is made routinely when we summarize a study by an estimated effect and its standard error, and we will not question its use here.
To summarize, why don't we model $se_i$? Why do we treat it as known?