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There's still something that confuses me about how many of these algorithms work, and how results are presented in the literature.

Let's consider a Maximum Likelihood based algorithm like MrBayes or RAxML: Users set a random number seed which generates the starting tree. For many of our datasets, different seeds which result in different ML results, as the algorithms are initialized with different trees.

I'm not entirely sure how one is supposed to interpret this, especially as my experience with ML methods is that the initial step is irrelevant to the global/local min/max in parameter space----the chains just take longer to converge.

How should one interpret these results? Are users to run 1000s of trees at different parameter values, and then chose the most optimal likelihood value? That seems rather ad hoc, as does bootstrapping, etc.

Is the dataset fundamentally flawed?

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  • $\begingroup$ MrBayes is definitely not doing maximum likelihood, as its name suggests. $\endgroup$ – kmm Sep 21 '16 at 3:03
  • $\begingroup$ @kmm Was trying to make this question relevant to several algorithms, and failed. Thanks for the help! I'll try to edit $\endgroup$ – ShanZhengYang Sep 22 '16 at 18:00
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In short, you have chosen two examples that do not use maximum likelihood as you know it in other contexts. In most statistical contexts, the ML is a single number which can be calculated analytically, so it it always the same for a given data set. This is not the case for either MrBayes or RAxML but for different reasons.

MrBayes

The likelihood criterion in MrBayes is the marginal likelihood of the posterior, given effectively the data conditioned on the priors. This likelihood comes from a stochastic MCMC sampling of the parameter space. If all is well behaved, then the chains and/or runs will converge in the same general location. But then the different possible topologies need to be summarized in some way.

RAxML

RAxML generates essentially random starting trees by random sequence additions to build up trees. The subtrees are then rearranged to find a "best" tree. Again, different starting points may lead to different best trees. But if all goes well, analyses will end up with the same tree. This process is described in this chapter.

In both cases, if you start in a different place, you may end up in a different place. There may be many trees that are, within some criterion, equally likely. If you are familiar with parsimony methods, the analogy is that of multiple equally parsimonious trees.

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  • $\begingroup$ I'm not sure I follow this: "If all is well behaved, then the chains and/or runs will converge in the same general location. But then the different possible topologies need to be summarized in some way." (1) If the chains converge in the same general location, wouldn't the topologies be equivalent? (2) How would one summarize the different topologies---consensus trees? $\endgroup$ – ShanZhengYang Sep 22 '16 at 17:29
  • $\begingroup$ Also, " There may be many trees that are, within some criterion, equally likely. If you are familiar with parsimony methods, the analogy is that of multiple equally parsimonious trees." How does one proceed in the literature then? Do you report all equally likely trees? Use (again) consensus tree methods? $\endgroup$ – ShanZhengYang Sep 22 '16 at 17:31
  • $\begingroup$ Rather than there being a single tree with the highest (maximum) likelihood, there are a family of trees that are all possible, just some more likely than others. The topologies don't have to be equivalent. A consensus tree is the summary of those trees considered most likely or credible (in the Bayesian sense). Yes, you would report the consensus tree (or as many as you want). $\endgroup$ – kmm Sep 22 '16 at 17:32

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