Hi,
I have had a few replies off-list which have made me try and clarify
what I mean. I think the distinction needs to be made between two
types of probability: the probability that an outcome is 0 or 1 Pr(y|
\theta) and the probability density of \theta, Pr(\theta | \gamma),
indexed by parameter(s) \gamma. It seems to me that in order to make
the problem identifiable an informative prior (or an assumption) is
required for the root state. It seems to me that the strong prior
Pr(\theta=0.5|\gamma) =1 is used, and then justified because int
Pr(y=0| \theta)Pr(theta)dtheta=int Pr(y=1| \theta)Pr(theta)dtheta=0.5.
A non-informative prior distribution for \theta could be U(0,1). This
also induces a prior on y of the same form, int Pr(y=0|
\theta)Pr(theta)dtheta=int Pr(y=1| \theta)Pr(theta)dtheta=0.5, but
that is not the main motivation for choosing U(0,1).
For example, is this not worrying:
library(ape)
n<-100
tree<-rcoal(n) # random tree
y<-rbinom(n, 1, 0.2) # random data unconnected to the tree
m1<-ace(y, tree, type = "d", model="SYM")
m2<-ace(y, tree, type = "d", model="ARD")
anova(m1, m2) # asymmetric evolutionary transition rates strongly
supported
y<-rbinom(n, 1, 0.5) # random data unconnected to the tree but p=0.5
m1<-ace(y, tree, type = "d", model="SYM")
m2<-ace(y, tree, type = "d", model="ARD")
anova(m1, m2) # asymmetric evolutionary transition not supported
Cheers,
Jarrod
Quoting Jarrod Hadfield <j.hadfi...@ed.ac.uk> on Thu, 16 Aug 2012
12:30:30 +0100:
Hi,
I have been helping someone with some analyses and came across some
routines to estimate asymmetric transition rates between discrete
characters. This surprised me because its fairly straightforward to
prove that asymmetric transition rates cannot be identified using
data collected on the tips of a phylogeny. However when I run these
routines (e.g. ace) likelihood ratio tests often suggest strong
support for asymmetric rates. This seems to arise because there is
an implicit (and unjustified) prior point mass on the ancestral
state *probability*. If anyone could point me to reference that
states this that would be great? I really don't want to be saying we
have support for processes that we need a fossil record, not just a
phylogeny, to understand.
Cheers,
Jarrod
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