Daniel Barker wrote:
> On ancestral state reconstructions. I've recently started using Ziheng
> Yang's terminology - of referring to reconstructions, derived from ML
> transition rates and equilibrium frequencies, as 'empirical Bayes'
> reconstructions. I believe this to be the most useful way to describe
> these methods.
>
> My question. Were does empirical Bayes stand, w.r.t. the Likelihood
> Principle?
>
> Empirical Bayes seems beyond the scope of the Likelihood Principle, or in
> violation of it. The biological hypotheses (here, of ancestral state) are
> not expressed as parameters of the model, so the relative support is not
> judged by a likelihood ratio. On the other hand, by only using the data at
> hand, empirical Bayes does comply with 'the irrelevance of outcomes not
> actually observed'.
>
> If anyone can provide or point to some thoughts on this, I'd be very
> grateful - for ancestral states or in general.
It has long been recognized -- since Anthony Edwards's 1970 paper and
Elizabeth Thompson's brilliant 1975 thesis and book --
that the interior node reconstructions are not, strictly speaking, MLEs.
They are maximum posterior probability estimates instead. The root
ancestral states can be considered MLEs if that state is one of the
parameters of the model. If instead (as often done for DNA and
protein data) the root ancestral state is considered to be drawn from
the equilibrium distribution of the stochastic process, then they too are
MPP estimates.
I would only call them empirical Bayes estimators if one made an ML
estimate of some stuff (the whole tree topology, which sounds like an
extreme case) and then, assuming the correctness of that estimate,
the ancestral states are inferred by being Bayesian. In that case
probably the only thing that would be taken to have a prior on it
would be the ancestral state. Then you could take the MPP as
the modal estimate from the posterior.
So, if I have understood his point correctly, Ziheng is formally correct,
but it is a case where one is not being very Bayesian. For my money,
you are a Bayesian not just if you use a prior, but if you are willing to
use a controversial prior. And in this case the prior is pretty
uncontroversial.
Joe
----
Joe Felsenstein j...@gs.washington.edu
Department of Genome Sciences and Department of Biology,
University of Washington, Box 355065, Seattle, WA 98195-5065 USA
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