The study presents a suite of methods for analyzing population mixture in human history, implemented in the ADMIXTOOLS software package. These methods allow for formal tests of admixture and inference of mixture proportions and dates. The researchers analyzed 934 individuals from 53 diverse populations using a custom SNP array with 629,433 sites, revealing a significant signal of admixture into northern Europe, with one ancestral population related to present-day Basques and Sardinians and another related to northeast Asian and American populations. This suggests a history of admixture between Neolithic migrants and the indigenous Mesolithic population of Europe. The study discusses various methods for detecting population mixture, including local ancestry-based methods and global ancestry-based methods. The latter, such as principal component analysis and STRUCTURE/ADMIXTURE, are powerful tools for detecting population substructure but do not provide formal tests for admixture. The researchers propose an alternative approach based on phylogenetic tree-based models, which allows for explicit inferences about historical population relationships. The study introduces five methods: the three-population test, D-statistics, F4-ratio estimation, admixture graph fitting, and rolloff. These methods are designed to formally test for population mixture and allow researchers to build models of population relationships that fit genetic data. The three-population test is a formal test of admixture and can provide clear evidence of admixture, even if the gene flow events occurred hundreds of generations ago. D-statistics are also a formal test for admixture and can provide information about the directionality of gene flow. F4-ratio estimation allows inference of the mixing proportions of an admixture event, even without accurate surrogates for the ancestral populations. Admixture graph fitting allows one to build a model of population relationships for an arbitrarily large number of populations simultaneously. Rolloff is an approach for estimating the date of admixture by modeling the decay of admixture linkage disequilibrium in the target population. The study also discusses the effect of ascertainment processes on f- and D-statistics, showing that these methods are robust to various ascertainment strategies. Simulations demonstrate that these methods reliably detect mixture events and provide accurate estimates of mixture proportions, even for widely varied demographic histories. The study concludes that these methods provide a robust framework for analyzing population mixture in human history.The study presents a suite of methods for analyzing population mixture in human history, implemented in the ADMIXTOOLS software package. These methods allow for formal tests of admixture and inference of mixture proportions and dates. The researchers analyzed 934 individuals from 53 diverse populations using a custom SNP array with 629,433 sites, revealing a significant signal of admixture into northern Europe, with one ancestral population related to present-day Basques and Sardinians and another related to northeast Asian and American populations. This suggests a history of admixture between Neolithic migrants and the indigenous Mesolithic population of Europe. The study discusses various methods for detecting population mixture, including local ancestry-based methods and global ancestry-based methods. The latter, such as principal component analysis and STRUCTURE/ADMIXTURE, are powerful tools for detecting population substructure but do not provide formal tests for admixture. The researchers propose an alternative approach based on phylogenetic tree-based models, which allows for explicit inferences about historical population relationships. The study introduces five methods: the three-population test, D-statistics, F4-ratio estimation, admixture graph fitting, and rolloff. These methods are designed to formally test for population mixture and allow researchers to build models of population relationships that fit genetic data. The three-population test is a formal test of admixture and can provide clear evidence of admixture, even if the gene flow events occurred hundreds of generations ago. D-statistics are also a formal test for admixture and can provide information about the directionality of gene flow. F4-ratio estimation allows inference of the mixing proportions of an admixture event, even without accurate surrogates for the ancestral populations. Admixture graph fitting allows one to build a model of population relationships for an arbitrarily large number of populations simultaneously. Rolloff is an approach for estimating the date of admixture by modeling the decay of admixture linkage disequilibrium in the target population. The study also discusses the effect of ascertainment processes on f- and D-statistics, showing that these methods are robust to various ascertainment strategies. Simulations demonstrate that these methods reliably detect mixture events and provide accurate estimates of mixture proportions, even for widely varied demographic histories. The study concludes that these methods provide a robust framework for analyzing population mixture in human history.