# Dienekes is a fool

Dienekes has driven his car off a cliff.

He now expects everyone to believe that I got the derivation of the expected value of the f3 statistic for my phylogeny all wrong, but that by pure dumb luck I arrived at a complicated function that takes values from his own analysis as input and just happens to give values close to empirical f3 statistics as output.

Dienekes is too stupid to correctly derive the expected value of the f3 statistic for my phylogeny.

The 16 path combinations are below.

The sum of the 16 weights is equal to 1, as is required.

The sum of the 16 terms gives the expected value of f3(E; S, A).

Path combination 1
E → S path 1: E → M → S
E → A path 1: E → M → C → N → A
Weight: αβαγ
Drift edge overlap: 0
Term: 0

Path combination 2
E → S path 1: E → M → S
E → A path 2: E → M → C → V → G → A
Weight: αβα(1 − γ)
Drift edge overlap: 0
Term: 0

Path combination 3
E → S path 1: E → M → S
E → A path 3: E → N → A
Weight: αβ(1 − α)γ
Drift edge overlap: 0
Term: 0

Path combination 4
E → S path 1: E → M → S
E → A path 4: E → N → C → V → G → A
Weight: αβ(1 − α)(1 − γ)
Drift edge overlap: 0
Term: 0

Path combination 5
E → S path 2: E → M → C → N → S
E → A path 1: E → M → C → N → A
Weight: α(1 − β)αγ
Drift edge overlap: CM + CN
Term: α(1 − β)αγ(CM + CN)

Path combination 6
E → S path 2: E → M → C → N → S
E → A path 2: E → M → C → V → G → A
Weight: α(1 − β)α(1 − γ)
Drift edge overlap: CM
Term: α(1 − β)α(1 − γ)CM

Path combination 7
E → S path 2: E → M → C → N → S
E → A path 3: E → N → A
Weight: α(1 − β)(1 − α)γ
Drift edge overlap: 0
Term: 0

Path combination 8
E → S path 2: E → M → C → N → S
E → A path 4: E → N → C → V → G → A
Weight: α(1 − β)(1 − α)(1 − γ)
Drift edge overlap: −CN
Term: −α(1 − β)(1 − α)(1 − γ)CN

Path combination 9
E → S path 3: E → N → C → M → S
E → A path 1: E → M → C → N → A
Weight: (1 − α)βαγ
Drift edge overlap: −CM − CN
Term: −(1 − α)βαγ(CM + CN)

Path combination 10
E → S path 3: E → N → C → M → S
E → A path 2: E → M → C → V → G → A
Weight: (1 − α)βα(1 − γ)
Drift edge overlap: −CM
Term: −(1 − α)βα(1 − γ)CM

Path combination 11
E → S path 3: E → N → C → M → S
E → A path 3: E → N → A
Weight: (1 − α)β(1 − α)γ
Drift edge overlap: 0
Term: 0

Path combination 12
E → S path 3: E → N → C → M → S
E → A path 4: E → N → C → V → G → A
Weight: (1 − α)β(1 − α)(1 − γ)
Drift edge overlap: CN
Term: (1 − α)β(1 − α)(1 − γ)CN

Path combination 13
E → S path 4: E → N → S
E → A path 1: E → M → C → N → A
Weight: (1 − α)(1 − β)αγ
Drift edge overlap: 0
Term: 0

Path combination 14
E → S path 4: E → N → S
E → A path 2: E → M → C → V → G → A
Weight: (1 − α)(1 − β)α(1 − γ)
Drift edge overlap: 0
Term: 0

Path combination 15
E → S path 4: E → N → S
E → A path 3: E → N → A
Weight: (1 − α)(1 − β)(1 − α)γ
Drift edge overlap: 0
Term: 0

Path combination 16
E → S path 4: E → N → S
E → A path 4: E → N → C → V → G → A
Weight: (1 − α)(1 − β)(1 − α)(1 − γ)
Drift edge overlap: 0
Term: 0