# Probabilities of trees

James Francis Lyons-Weiler weiler at ERS.UNR.EDU
Thu Sep 25 07:26:57 CDT 1997

```On Thu, 25 Sep 1997, Richard Zander wrote:

> Okay, one last time. After the latest flurry of go-rounds from the
>
>   Maximum likelihood techniques, if you allow all the assumptions,
> introduce a probability model (if you allow that) which allows
> calculation of posterior probabilities (chance that a given tree is the
> correct tree). With a small data set, you can get a probability greater
> than .5 for a particular tree, and so you have a phylogenetic
> hypothesis. With large data sets, however, the tree with greatest
> probability is generally less than .5, lots less. Only some parts of
> that tree may be the same in all trees of the .5 "credible region."
> There is a pool of trees, therefore, that probably include the true
> tree, but the true tree is improbably the same as the tree of maximum
> probability.

Recall that L refers to the likelihood of the data,
given the model, and given the tree.  Most people tend
to think of tree as a parameter, i.e., part of the model;
people are trying to move away from that position in part
because when you treat the tree as a parameter, some funny
things happen.  Max Lik assumes that for any model, there
exists one and only one set of parameter values that will
achieve the maximum likelihood of the data.  This assumption
is not well supported by theory.  So the major points of
likelihood (as applied to phylogenetics) are: it is
consistent (right answer is guaranteed) if and only if
one has the right model and an infinite amount of data.

This has been the by-line of maximum likelihood for some time.
Consistency aside, the method can also give you the right
answer (the true tree) with a finite amount of data
given then wrong model of evolution.  There are other
equally compelling difficulties that have recently emerged,
and the good thing is the following: the maximum likelihood
people are taking them seriously, and are not (for the most
part) overly dismissive of functional critical analysis
of the technique.

>   Maximum parsimony techniques are parsimonious in eliminating trees
> that are grossly unreasonable by the theory of common descent with
> modification. At the (vague) point that the method begins to eliminate
> trees that are not unreasonable phylogenetic hypotheses, it is
> to eliminate all but the tree that assumes that ALL covariance that can
> possibly be ascribed to common ancestry must be so. We get then the tree
> of maximum synapomorphy, or alternatively of minimum convergence.

I would say "apparent convergence".

This
> is only very doubtfully the true tree. There is no probability involved
> since the assumption that all covariance possible with a particular
> polarization must be treated as ancestrally shared is false, because at
> least at the species level, characters are usually simple and commonly
> often separately evolved (judging at least from cladistic homoplasy) if
> not in the group studied then in related groups (the larger the group,
> the smaller the consistency index, which is expected of course).

I'm not sure that a fallacious assumption means that
no probability is involved.  A plausible analogy:
I may take the sample mean as the best estimate of
the population mean, and in doing so I'm making
a probabilistic statement (because the sample mean
is the maxc lik estimate of the population mean);
however, there are distributional assumptions that
may be violated.  So the probability statements are
there, even if I'm not using them correctly.

One type of probability statement that leaps out
of max pars is that evolutionary change (in the
characters) is rare.  That means that evolutionary
events themselves are assigned a low probability
in the gestalt model applied by max pars. This
asumption is perfectly tenable under some conditions
but is not under others.
>
>   The tree of maximum synapomorphy is valuable as a basis for a
> classification emphasizing theoretically possible ancestral
> relationships, but it must be recognized as overly interpreted, and
> hardly a "reconstruction through a discovery process." It can also be
> used as a tree of minimum convergence: overlaid on a phenetic ordination
> it can identify theoretical convergence of taxa.

"theoretically possible" under its own criterion.

James L_W.

```