Probabilities of trees

[James Francis Lyons-Weiler]

On Thu, 25 Sep 1997, Richard Zander wrote:

> Okay, one last time. After the latest flurry of go-rounds from the
> cladistic adepts, I figure the following:
>
>   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
> antiparsimonious, using ad hoc assumptions about ancestral relationships
> 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.