# Probabilities and trees

Richard Zander bryo at COMMTECH.NET
Sun Sep 21 11:56:42 CDT 1997

```Regarding probability that the shortest tree is the true tree, one
might judge that the probability of a character state being evolved
twice is half that of one. Given that (admittedly doubtful) assumption,
then a three taxon data set, with two taxa sharing one advanced state,
gives three trees, one the shortest tree and two being one step longer.
The shortest tree should have a .5 probability of being the correct tree
(assuming it must be one of the three), since the two one-step longer
trees are each half as probable as the shortest tree, and .5 + .25 + .25
= 1.
On the other hand, maybe probabilities for shortest trees being the
true tree are not calculable in this or any other fashion (I think this
is both T. DiB. and J. L.-W.'s perception). Maximum parsimony requires
that *all* shared character states be interpreted when possible as due
homoplasy, what is the chance that one, or two, of the synapomorphies
are falsely interpreted, that in fact we have an extra branch based on
an extra step? If there is no cladistic homoplasy in the shortest tree,
we still get a fairly similar tree, but if there is some cladistic
homoplasy, the extra step or two would imply having to consider vastly
different topologies.
Maximum parsimony is impressive in discarding a relatively large
number of unreasonable hypotheses (long trees), but when approaching the
shortest tree the method becomes overly precise, due to a mathematical
freak: taking the method to its limit often results in only one tree!
Voila, it must be the true tree or something resembling the true tree!
Now, the coldest day of the year is not the Winter Solstice, and the
true tree is not necessarily or even likely to be the shortest tree.
There may not be a decent, discernable hypothesis in phylogenetic
analysis by maximum parsimony unless Bremer support is greater than,
what? "several"?
I think J. Lyons-Weiler said the same thing as this, but in different,
more mathematically based, accurate, and harder-to-understand words.
It's difficult to keep track of the welter of ideas in this thread. Some
post-modernists say all ideas are pastiches of those of other people,
and all publications are "cut-ups" and rearrangements of previous ones.
This must happen often in listservering.

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Richard H. Zander, Buffalo Museum of Science
1020 Humboldt Pkwy, Buffalo, NY 14211 USA bryo at commtech.net
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