I hope you are doing well and safe. I’d like to comment and ask for a variant of the model proposed in the paper Struggles with Survey weighting and Regression Modeling (Gelman, 2007).
Quoting Gelman “However, the implicit weights (9) from hierarchical regression do depend on the data, implicitly, through the hyperparameters in \sum_y and \sum_β, which are estimated from the data. Thus, the appropriate weights could differ for different survey responses.”
As it can be seen in equation (10), the posterior estimates are a weighted average of the cell mean and the overall mean (partial-pooling effect).
My question is related to the fact that sometimes we have the observed means for each of the poststratified cells, instead of individual observations. If we knew the corresponding count of individuals (and within-group standard deviation), can we replicate the shrinkage effect including this information in the priors?