: Michael A. Charleston <m.a.charleston at mail.utexas.edu> wrote:
: >I'm new to this, so forgive me for any clangers I may make.
: >You have a phylogeny problem on n taxa,
: [and a space of possible trees]
: >The objective function on this space is something like the parsimony
: >length or the log-likeligood.
: >So you can imagine "hills" and "troughs" in this graph, where the "height"
: >is proportional to the goodness - to be maximised - of the trees.
: ...
: >Anyway - the situation is this: you have a great many trees in one big
: >hill
: ...
: >Somewhere far removed from this
: >broad hill is a single spike, so there are many fewer starting trees (and
: >climbs) that would lead you to the top of this spike, but the objective
: >function value is higher at the top of this spike than it is at the top of
: >the broad hill.
: >
: >My question is this: which tree - or trees - do you consider as "better"?
: The spike, no contest.
: >My view is that there should (ideally) be some commonality between the
: >trees at the top of the hill and at the top of the spike, which is being
: >obscured by out choice of tree perturbation (and therefore tree
: >adjacency), and that by choosing more appropriate criteria for "closeness"
: >of two trees we may figure out what's going on. This may be very naive of
: >me: I would appreciate the views of those reading this.
: >
: >Knowing some properties of the "slopes" of this kind of graph would
: >clearly be useful in determining when this situation will arise, if ever,
: >but that's not the question. If it *does* arise, what do you do?
I always consider this situation to be that of a series of chemical
reactions. Consider an upside down Gibbs Free energy plot (which would
correspond to your spikes). Eventually an equilibrium will be reached
where the largest spike will be favored (assuming that all activation
energies are attainable), and the degree to which this spike will be
favored will be a function of the relative values of the different peaks.
Granted that with enough steps in the reaction between one peak and
another, the critical question becomes how long will it take to reach
this equilibrium? Also, all activation energies may not be readily
attainable. It would seem to me that the liklihood of attaining each
successive increment of an activation energy will be less likely than
the previous (possibly), indicating that the scale would be non-linear
(exponential, I think).
This does not really answer your question, I don't think, but I think
it provides some insight into finding a plausable solution.
asim