# [Taxacom] Markov chain Monte Carlo

Richard Zander Richard.Zander at mobot.org
Fri Apr 20 10:54:38 CDT 2012

```Peter Hovenkamp
Carl Rothfels

Thanks for clarifying Bayesian MCMC analysis. Statistics is a basis for
modern science, yet there remain mysteries.

Let me get this straight. MCMC analysis searches a sample of the full
population and finds maximal support values for nodes, i.e. it does an
optimality search. Because it is Bayesian, the optimality values for the
nodes are the same as in the unsearched subset of the population,
assuming that all peaks are searched by an efficient algorithm, like 3
or 4 random-sampling lines going at once.

A tree is then presented with compatible nodes and their posterior
probabilities.

Question: Is the tree selected as final that one with the joint
probability (product of all support values) the highest?

When there are equally probable trees, I assume a consensus tree (with
multifurcations) is generated.

So...
Question: Are the support values in Bayesian analysis the probability of
optimality of a clade or the probability of being right? When these are
not the same, there is a problem.

Question: Is the following correct? Say, the posterior probability is
the chance of a clade being correct. The chance that all clades are
correct is then the product of all support values. The probability of
all clades being correct may be low, but the probability of most clades
being right is high. Using binomial calculator, I see the probability of
all nodes of a 40 node tree each at 0.95 probability is 0.13 joint
probability but the probability of at least 36 or more of them being
right (90% of nodes) is 0.95. So molecular trees are mostly right.

That's great. IF the joint probability refers to actually chance of
being correct, not chance of being optimal, in which case maximum
likelihood (Williams) or Bayes Factors (Raftery) may make problems.

Richard

* * * * * * * * * * * *
Richard H. Zander
Missouri Botanical Garden, PO Box 299, St. Louis, MO 63166-0299 USA
Web sites: http://www.mobot.org/plantscience/resbot/ and