What does this equation mean?

The basic tool for estimating what fraction of a genome derives from an archaic population is comparative allele sharing: for a given modern population, count how much more genetic variation it shares with an archaic genome than a population known to lack that admixture does, and scale by how much variation the archaic and comparison genomes share with each other overall. A simplified version of the widely used D -statistic framework expresses this as: $\hat{f} = \frac{\sum_{i} (P_2(i) - P_1(i))\,(P_4(i) - P_1(i))}{\sum_{i} (P_3(i) - P_1(i))\,(P_4(i) - P_1(i))}$. where $P_1$ through $P_4$ are allele frequencies at site i in an outgroup, a candidate-admixed population, a comparison population without the admixture, and the archaic source, respectively. The estimator exploits a simple logic: if population $P_2$ has interbred with the archaic lineage $P_4$ and $P_3$ has not, $P_2$ should share systematically more rare variants with $P_4$ than $P_3$ does, in proportion to the fraction of its genome that traces to that admixture event. This is why admixture claims in the literature are reported as percentages with confidence intervals rather than as a binary yes/no — the estimate is statistical, built from many thousands of sites, not read off a single marker.