People do display distinct genetic clustering based on ancestral geography. If someone has your DNA, they can tell with high certainty where your ancestors came from.

Of course, a substantial number of white Americans would be surprised by discovering they had ancestors from Africa in the recent past. (Most African-Americans, on the other hand, wouldn't be too surprised to find out many of them have European ancestry.)

>People do display distinct genetic clustering based on ancestral geography.

For sure! But I suspect the most meaningful coarse clusters might end up being some large number of divisions of sub-saharan africa and then everyone else. Definitely not the "White", "Black", "Asian" categories people in the states take for granted. And even ignoring the amazing diversity of Africa (compared with the rest of the world), did the Irish become more genetically white when they became culturally accepted as white? Did Latin Americans of European descent become less genetically white when people further North stopped viewing them that way? Why do some races trump others (i.e., why is Barack Obama considered black)? The boundaries of these categories seem very arbitrary and not derived by any rigorous means at all.

I think i have seen this before, it came out basically pretty well aligned with the generally accepted racial groups, except there were i believe 2 groups for sub saharan instead of one group of 'blacks'. Has been a few years since i saw the study so dont quote me on it.

Find the citation please. I'm not even sure what you mean by generally accepted racial groups. In the UK, unlike the US, you would probably distinguish between Indian & Pakistani vs. all other Asians. You might also distinguish Irish as a separate ethnicity and probably would not think of Arabs or Turks as white. This stuff is very culturally dependent and I don't know which set of "racial groups" would be considered canonical.

One of the main confounds seems to be "large immigrant group". For example, even ethnically-100%-Spanish Latinos in the U.S. are considered different from "Anglos", in significant part because it's a large group. That sometimes leads to significant differences between countries. I'd say that for the past 50 years or so, Greek-Americans have been considered part of the undifferentiated majority "white" population. But Greek-Australians are still considered different in Australia, not really properly white, or at least white with an "ethnic" asterisk.

Despite a lying Senator, checking the Native American box is a pretty simple exercise. Are you enrolled with a federally (or state in some cases) recognized tribe? Yes, check box. No, pick something else.

"I wonder what our 'racial' categories would look like if people treated these groupings as a statistical problem, rather than some sort of given natural phenomenon."

The outcome would no doubt have a lot in common with the ugly and ill-fitting suit made for Gulliver by the tailors of Laputa, who used quadrants, rules, and compasses rather than a simple tape measure.