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Forgive me if this question has been asked here before, because it is something which should be very easy to find, but I can't seem to find an answer no matter where I search.

The question is simply where to learn the mathematics that goes into things like popular ancestry tests, and also more academic things like determining ancestry components of historical groups (e.g. usage of the terms Ancient North Eurasian, Eastern European Hunter-Gatherers and all that).

It is clear that if someone gets an ancestry test saying say 32% Scandinavian, then of course that doesn't mean 32% of their base pairs have a convenient "Scandinavian" label attached to them, rather there is some statistical inference going on behind these percentages, and I would like to understand that.

Suppose I have the raw data of my own fully sequenced genome, and also a database of the genomes of many individuals from various populations (of course grouping them into populations is already something that involves some assumptions that I would like to learn more about). Where would I learn how to analyze that myself to produce something like the results of an ancestry test? Is there a textbook someone could recommend that gets into the actual algorithms used?

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I'll give here a simple, non-technical answer because I'm assuming you don't need to actually perform an analysis of ancestry.

So, detecting ancestry is a non-trivial task. Given your genome sequence, you would need to compare some "informative" regions of the genome with the homologous sequences of some population (say, of a database with other genomes). These informative regions are usually some parts of the genome that vary across individuals (variation is used because differences are informative: some populations vary on particular sites, distinct from other populations). At the core, this is a question on how to compare "character strings" (DNA is composed of 4 characters, namely, A, T, C, and G). But these strings exist in a complicated structure: a human genome is partitioned in 23 pairs of chromosomes, within each individual. However, the question of ancestry is not about individual sequences really, but about population-level changes in DNA composition. So, in fact, you need to consider population-level factors: size of the populations to consider, the rates of recombination (DNA exchange across chromosomic pairs), mutation rates, and even population structure (people move geographically!).

Given these (and, many other) factors, people build models of "coalescence": given a sequence of interest, how likely is that it shares some ancestor with another query sequence? So, the models try to relate these two sequences (say, the one you are interested), with a query (say, a "consensus" Scandinavian sequence), and then make a model of a 3rd sequence (the ancestor!). This process is repeated to test many hypothesis, so you end up with many probabilities. On top of this, you can estimate the ancestry for any given part of your genome, and this is what most companies do (say, 23 and me).

In summary, you are correct that a "% Scandinavian" does not mean sequence similarity per se. It implies an estimation of common ancestry. This estimation comes from models of shared ancestry. If you're unsatisfied with this simple answer, and want a more technical answer, I recommend reading this paper. An intermediate-level explanation is found here.

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