Let's say I have a random effect intercept. For example:
lme4::lmer(yield ~ 1 + (1|Batch))
How is that different than just ordinary regression using regularization or a prior?
arm::bayesglm(yield ~ 1 + Batch,...)
They seem roughly equivalent to me.
Unified view on shrinkage: what is the relation (if any) between Stein's paradox, ridge regression, and random effects in mixed models? asks something similar. The accepted answer claims that they are equivalent.
However, I am not convinced that it is correct, otherwise what is the need for a package
lme4, which is typically more computationally expensive to fit?
My intuition is that regularization is equivalent to a mixed model with a diagonal covariance matrix for the random effects. Whereas something like
lme4 supports more general covariance structures.