Survey Statistics: perfect collinearity in the sample but not in the population

In 2019, Andrew blogged about collinearity in Bayesian models. In the comments, he pointed to an example from Bayesian Data Analysis, 2nd edition (BDA2). I think it is a useful example to keep in mind when extrapolating from sample to population. Since folks (like me) may only have BDA3 on their shelf, I thought I’d talk thru it. Pretend it is 1980 and we are at the US Census Bureau. We just revamped the occupational coding system, and it’s so much better ! We want 1980-style codes on all our old data that only had 1970-style codes. Let’s trade in our peasant blouses for some shoulder pads. Say we have double-coded training data (n = 10,000) with: O_1980 = occupation coded in the 1980 coding system O_1970 = occupation coded in the 1970 coding system E = education, either high or low I = income, either high or low We want to impute O_1980 for the single-coded full dataset (N = 1,000,000) with only O_1970, E, and I. Consider everyone with the a specific occupation according to the 1970…