How does one reduce the Maximum Likelihood rule to a pairwise comparison of the probability densities? I am studying an article, where the authors have discussed something about "the ML decision rule reducing to a pairwise comparison of the conditional PDFs" because the means and variances are in ascending order.
To remove confusion, I am attaching a screengrab. As you can see, they use condition (11) to invoke the pairwise comparison of the PDFs in (12). I have checked the reference that they cited, that is, [31], but I did not find anything related to that discussion.
Why are the PDFs equal?
Could anyone explain how this works?
I'm not sure if I'm asking the right questions here, feel free to direct me towards the right questions.


The distributions in (12) are equal because the decision thresholds $\eta_1, \dots, \eta_{M-1}$ are chosen to make them equal.
Consider a simple case, where you have two distributions and you want to assign an observation to one of them.  If your decision rule is, essentially, "assign the observation to whichever distribution is most likely to have generated it", clearly you will choose the one with the highest probability density at the observed value.   The decision threshold will be the value at which the two probability densities are equal; observations that fall below the threshold will be assigned to one distribution, and those that fall above the threshold will be assigned to the other.

In this case, the two distributions are equal at the value $-0.0636$, which becomes the decision threshold.  Equation (12) is simply a mathematical expression of this relationship between the decision thresholds and the distribution values at those thresholds, extended to more than two distributions.  To the left of the threshold, the observation is more likely to have come from the black distribution, so we would assign it to that one, and similarly for an observation to the right of the threshold.  At the threshold, the likelihood that an observation comes from the black distribution is equal to the likelihood that it comes from the blue distribution, so it doesn't matter - mathematically - which one we assign it to.
