Ok good to hear they at least wanted to release the preprint at the same time as going public, but why post childish and arrogant memes as damage control? Like this one, compared to the manuscript it's simply a lie:

Eikä myöskään taida olla todellista. Trump puhui hiirillä tehtävistä hormonikokeista, projektien yhteishinta noin 8M$. Kuva Twitteristä <- ilmeisesti tiktokista <- valkoisen talon nettisivuilta.

For example, looking at a cube from one corner to the opposite corner makes it look like a 2d-hexagon. Looking at a 4d-hypercube similarly would make it look like a rhombic dodecahedron (as I'm sure everyone is aware of after the neat 3blue1brown video earlier this week). Etc.

If we instead put polar bears to leaf 4 of tree A, we only need some more brown bears to infer gene flow and its direction to samples 1 and 2. This methodological limitation is in our opinion to blame for the different conclusions we did with another group in summer 2022.

In this instance, to infer gene flow direction using tree S we would have to put polar bears somewhere, say, leaf 4. But then leaf 3 would have to be an unadmixed sister sample to polar bears - no such population exists! Even the 100,000 year old subfossil samples are admixed.

And here's the corresponding results for the two other five-leaf trees A=((((1,2),3),4),5) and Q=(((1,2),3),(4,5)). In our empirical example of the (controversial?) brown bear - polar bear admixture, we found tree A particularly useful. The phylogeny is what it is after all.

Without using the singleton patterns (first four columns), we have almost identical ability to distinguish events as DFOIL has. The only difference is gene flow from one of the populations 1-4 into 5, which DFOIL can't see while we can but not distinguish from some other events.

I came up with eight statistics for tree S. Like DFOIL: patterns come in pairs that cancel out when there's no gene flow, and the statistics attain some consistent sign (-/0/+) under gene flow. First four can be used with ancient DNA because they don't have singleton patterns.

Sometimes, ufortunately, there's not a unique solution. For example, (00--) could signify 2->34 (the population ancestral to 3 and 4), 2->34, 5->1, 1234->1 or 12345->1. These two issues along with DFOIL being restricted to tree S (there's two others) is what we set out to solve.

There's four statistics, check if they're significantly positive or negative or not, and compare the 4-tuple ("signature") to this table (asterisk means the gene flow event was not considered by Pease and Hahn). Signature (-0++) is evidence for gene flow event 4->1 and so on.

For the singleton patterns to cancel each other out, we need more assumptions. If one sample is ancient for example, the symmetry is broken. For gene tree shown here producing pattern BAAAA there exists no companion producing ABAAA, so the statistic is not expected to be zero.

Under ILS alone, all four statistics are expected to be zero: we can view them in pairs that cancel each other out by the symmetries in tree S, by the same logic the D-statistic employs. But in each statistic, one of the pairs (red) is not like the others. The singleton patterns!

DFOIL uses a fifth sample so that the population history is "symmetric" S=(((1,2),(3,4)),5). Instead of one statistic there's four, each taking positive contribution from four allelic patterns and negative from another four. The denominators are again only for inference.

Biallelic patterns arise from mutations in gene trees, so the patterns ABBA and BABA are equally common at least when it comes to ILS - but gene flow is another thing. Here's the collection of unidirectional gene flow events that would make ABBA more likely i.e. D positive.

But by the symmetry between 1 and 2 in the population history, whatever happened to sample 1 could have just as well happened to 2 and vice versa. We don't have to know the prob. of a gene tree to know it's the same with another gene tree where the roles of 1 and 2 are reversed.

Mushroom haul: