Ratios

[Richard Zander]

Somewhere along the line someone remarked that a hypothesis of maximum
likelihood that was many times better than any one other hypothesis was much
more acceptable than a hypothesis of maximum likelihood that was only a bit
better than the second most likely. I nodded and said to myself, "Well, that
sounds right," and remained uneasy until I actually thought about it.

It depends. Consider a die with 1000 sides (a football sewn of 1000
crescents). Paint one of the sides red and roll the die. The chance of
getting a red side (touching the table) is 1/1000. Paint 250 of the sides
red, and the chance of rolling a red side down is now 250 times the chance
of any one of the unpainted sides. The chance that you actually roll a red
side, however, is only 1/4, and you would lose money if you bet on this in
the long run.

Suppose you think you have a die of 1000 sides with one side red. You gain
additional information that lets you know that there are actually 250 sides
red. The increase in probability is phenomenal. The actually probability of
getting a red side down with that die, however, is still 1/4.

Even if a phylogenetic analysis found a hypothesis much more likely than any
other hypothesis, the probabilistic sum of all the alternative hypotheses
cannot be ignored unless you have additional information. Just saying that
hypothesis of little probability can be ignored doesn't make them go away.
That additional information must explain why the model is NOT that of
rolling a die with as many sides as there are phylogenetic arrangements that
explain the data.

I make these comments because of what may be called "the forward curve of
mystification." Somehow, phylogeneticists manage to come up with yet more
abstruse methods of evolutionary analysis, and just when you figure out why
one method is billed as far more reliable than it actually is, yet another
pops up.

We've dealt with strict cladistics (e.g., see M. Wilkinson et al. Syst.
Biol. 52: 127-130. 2003), and Yang has demonstrated that likelihood ratios
can't be used for comparing the reliabilities of different topologies
because they are already optimized. Bayes Factors are the new rage, watch
for them. These are so mathematically complex they make philosophical
gimmicks like "scientists prefer simplicity" come across as gradeschool
jokes. To the extent they are used to demonstrate reliability solely though
an increase in probability without reference to posterior probability, they
are inappropriate.

______________________
Richard H. Zander
Bryology Group
Missouri Botanical Garden
PO Box 299
St. Louis, MO 63166-0299
richard.zander at mobot.org <mailto:richard.zander at mobot.org>
Voice: 314-577-5180
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