Summary: THE GAPS IN ROGERS' PROOF
OF IDENTIFIABILITY OF THE
ELIZABETH S. ALLMAN, CŽECILE ANŽE, AND JOHN A. RHODES
This note explains the gaps in the published proof of Rogers [Rog01] that the
GTR++I model is identifiable. Since that paper has been widely cited and ac-
cepted as correct, our goal is to clearly indicate where the argument is flawed, and
illustrate, through some examples, the nature of the logical gaps.
We emphasize that we do not prove that the gaps in the published argument
cannot be bridged. Indeed, it seems most likely that the GTR++I model is
identifiable, at least for generic parameters, and it is possible a correct proof might
follow the rough outline of [Rog01]. However, we have not been able to complete the
argument Rogers attempts to make. Moreover, our own proof of the identifiability of
the GTR+ model (manuscript in preparation) follows a different line of argument.
1. Gaps in the published proof
There are two gaps in Rogers' argument which we have identified. In this section
we indicate the locations and nature of these flaws, and in subsequent ones we
elaborate on them individually.
The first gap in the argument occurs roughly at the break from page 717 to
page 718 of the article. To explain the gap, we first outline Rogers' work leading