Should I use priors to balance unbalanced population sizes?

Question

Should I use priors to balance unbalanced population sizes?

So e.g. if some variable shows negatively skewed distribution of population sizes, then should I infer that it probably "should be that way" or should I speculate about correcting it towards e.g. uniformly or normally distributed?

Given that I expect that such variables should be inferred based on "uniform distribution" or "normal distribution".

An example, suppose that we have the variable "education level". If there are more low education samples than higher ones (positive skew), then should I select a prior that corrects this more towards uniform, since the assumption would be that:

we ought to infer the correlations in the data based on "if there were same amounts of different education classes"? (since having more of low education might mess the data by introducing too much variation)

Answer 1

Forget about methods for a moment and start by clarifying what it is you are even doing. What is your goal and what is the function of the "balancing" you propose?

Let's start with a bit of context on "balancing" samples. Sometimes, we undertake statistical problems where we have good a priori knowledge of the distribution of a characteristic in the population of interest (e.g., from some past census or something like this) and we can then adjust our inference if our sample has a distribution of that characteristic that is different from the population. For example, if I know a priori that my population of interest has essentially 50/50 males and females, but my sample has a lot more males, then I could adjust my inference accordingly (downweighting males and upweighting females) to make an inference about the average characteristics in the population. In this case the "balancing" that occurs happens because I am making an inference about a population average which is known a priori to consist of a particular proportion of people of different types, and those proportions differ from my sample proportions. (See related questions here and here for some theory on this.)

Now, presumably you are making an inference about something, so what is the something, and what are you "balancing" for? If you are making an inference about a larger population then what (if any) knowledge do you have of the distribution of education in that larger population? Do you have some good reason to believe that education classes in the population of interest have equal size? (Highly unlikely I imagine.) Alternatively, if you have no a priori information on the distribution of education in your population then isn't your sample the best estimator of this, and so do you really want to be "balancing" that sample?

In either case, the "balancing" that occurs in these situations has nothing to do with a Bayesian prior --- rather, it is something that comes out of what it is you are making an inference about. Once you sort this out, you will be in a better position to create an appropriate estimator (Bayesian or classical).