This post from John Cook about the hierarchy of U.S. census divisions (nation, region, division, state, county, census tract, block group, census block, household, family, person) reminded me of something that I’ve discussed before but have never really worked out, which is the construction of families of hierarchical models for taxonomic data structures.

A few ideas come into play here:

1. Taxonomic structures come up a lot in social science. Cook gives the example of geography. Two other examples are occupation and religion, each of which can be categorized in many layers. Check out the Census Bureau’s industry and occupation classifications or the many categories of religion, religious denomination, etc.

2. Hierarchical modeling. If you have a national survey of, say, 2000 people classified by county, you won’t want to just do a simple “8 schools“-style hierarchical model, because if you do that you’ll pool the small counties pretty much all the way to the national average. You’ll want to include county-level predictors. One way of doing this is to include indicators for state, division, and region: that is, a hierarchical model over the taxonomy. This would be a simple example of the “unfolding flower” structure that expands to include more model structure as more data come in, thus avoiding the problem that the usual forms of weak priors are very strong as the dimensionality of the model increases while the data remain fixed (see Section 3 of my Bayesian model-building by pure thought paper from 1996).

I think there’s a connection here to quantum mechanics, in the way that, when a system is heated, new energy levels appear.

3. Taxonomies tend to have fractal structure. Some nodes have lots of branches and lots of depth, others just stop. For example, in the taxonomy of religious denominations, Christians can be divided into Catholics, Protestant, and others, then Protestants can be divided into mainline and evangelical, each which can be further subdivided, whereas some of the others, such as Mormons, might not be divided at all. Similarly, some index entries in a book will have lots of sub-entries and can go on for multiple columns, within which some sub-entries have many sub-sub-entries, etc., and then other entries are just one line. You get the sort of long-tailed distribution that’s characteristic of self-similar processes. Mandelbrot wrote about this back in 1955! This might be his first publication of any kind about fractals, a topic that he productively chewed on for several decades more.

My colleagues and I have worked with some taxonomic models for voting based on geography:
– Section 5 of our 2002 article on the mathematics and statistics of voting power,
– Our recent unpublished paper, How democracies polarize: A multilevel perspective.
These are sort of like spatial models or network models, but structured specifically based on a hierarchical taxonomy. Both papers have some fun math.

I think there are general principles here, or at least something more to be said on the topic.