Gregor writes:

I have some data on body measurements of breed of domestic animals that are in danger of extinction. Therefore, whole population sizes are small, from 500 to 3000 and I have data up to 200 or 300 (for larger populations). There is also difference in number of males and females and some inbreeding, but this is not relevant for the question. I am aware of standard result from surveys, where one should correct standard error of the mean if sample of data can not be viewed as a small part of population. Are you aware of any pointers for Bayesian setting in such cases? I am interseted in simple independent or unit sampling as well as cluster sampling.

My response: The quick answer (from books such as Lohr’s) is that each term in the variance multiplied by (1-sampling.fraction), where sampling.fraction is (#clusters sampled)/(total #clusters) for the group-level variance, and (#units in sample)/(total #units in population) for the data-level variance.

For the full Bayesian inference, what you do is make inference for the unsampled units and clusters, then avg with the observed data to get inference for the population. See Section 7.4 of BDA (2nd edition) for more. We cover unit sampling as well as cluster sampling.