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Cell["\<\
The generating function for the IM model and the probability of incongruence\
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We first obtain the generating function under the IM model for a pair of \
unphased diploids, one sampled from each population.    We expand this so \
that the GF is a sum of terms, each corresponding to a particular path \
realized by the coalescent process under the IM model. 

We then use substitution to map the branch type labels, \[Omega][{}], onto \
the unrooted versions.  I.e. we cannot distinguish an {a,a,b} branch from a \
{b} branch type.\
\>", "Text",
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Cell["\<\
These are all the branch types observed among all the paths in the IM model, \
for both the rooted and unrooted genealogies. \
\>", "Text",
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Cell["\<\
For ease, we instead store the terms corresponding to individual paths as \
elements of a list rather than elements of a series. Each element of the list \
corresponds to a unique sequence of coalesence, migration, and population \
split events.  \
\>", "Text",
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Here, we distinguish the branch types present in the unrooted topologies. \
There are two internal branch types, one of which is incongruent with the \
species tree.  There are two distinguishable external branch types identified \
by the subpopulation from which the subtending leaf node was sampled.  \
\>", "Text",
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Cell["\<\
For each of the paths in the gfList, we extract the branch type variables \
that are present, check for the presence of an incongruent branch type, then we
take the subset of the paths in the gfList that are indeed incongruent.  \
\>", "Text",
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Cell["\<\
We can view individual paths, with terms in the denominator denoting the \
sequence of states through which the population passes.  As an example, \
consider path number 9 (below)

(2 M + \[Delta] + \[Lambda]a + \[Lambda]b + 2 \[Omega][{a}] + 2 \
\[Omega][{b}])  corresponds to the events that can occur and the branch types \
that are present for the initial configuration of the sampled lineages under \
the IM model.  This is shared among all paths. We can break this down as \
follows:  two lineages can migrate (2M), the population split could occur (\
\[Delta]), or coalescence could occur in either the A population (\[Lambda]a) \
or the B population (\[Lambda]b). During this first interval of the \
coalescent process there are two of each external branch type present:  2 \
\[Omega][{a}]  and 2 \[Omega][{b}].

We can parse the terms in the denominator to determine the sequence of events \
in this path and the subsequent configurations of the coalescent process. For \
path number 9, the coalescent history is as follows:

migration from A to B pastward  --> (M + \[Delta] + 3 \[Lambda]b + 2 \
\[Omega][{a}] + 2 \[Omega][{b}]) 
coalescence of an \[OpenCurlyQuote]a\[CloseCurlyQuote] and \[OpenCurlyQuote]b\
\[CloseCurlyQuote] lineage, generating an incongruent branch type --> (M + \
\[Delta] + \[Lambda]b + \[Omega][{a}] + \[Omega][{b}] + \[Omega][{a, b}])
coalescence of  the \[OpenCurlyQuote]b\[CloseCurlyQuote] and \
\[OpenCurlyQuote]ab\[CloseCurlyQuote] lineage, which results in two \
\[OpenCurlyQuote]a\[CloseCurlyQuote] branch types when the topology is \
unrooted --> (M + \[Delta] + 2 \[Omega][{a}])
a second migration from A to B pastward -->  (\[Delta] + \[Lambda]b + 2 \
\[Omega][{a}]) 
the split, moving all lineages to the ancestral AB population --> \
(\[Lambda]ab + 2 \[Omega][{a}])
And finally, coalescence of the remaining \
\[OpenCurlyQuote]a\[CloseCurlyQuote] type lineages. 
\
\>", "Text",
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Cell["\<\
At this point, the discrete event was treated as a competing exponential \
process, similar to coalescence or migration.  We apply the following inverse \
Laplace transform to each path to obtain the expression as a function of a \
discrete event occurring at time Td.\
\>", "Text",
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Cell["\<\
We can obtain the probability of each path by setting the \[Omega] \
branch-length variables to zero, and summing, we obtain the total probability \
of observing an incongruent genealogy under the IM model. Note that this \
probability is a function of the size of populations A and B, but  not the \
ancestral population size. I.e. \[Lambda]ab does not factor into the \
expression.

Setting \[Lambda]a and \[Lambda]b to 1 for simplicity,  we obtain the \
probability of incongruence as a function of the migration rate M and split \
time Td, and we plot this probability as a heatmap as well as for the \
transect where Td = 1.  \
\>", "Text",
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Cell["The GF conditioned on incongruent genealogies ", "Section",
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Cell["\<\
To obtain the generating function (here as a list of terms) for incongruent \
genealogies, we simply retain only the paths that are incongruent, and divide \
by the total probability of incongruence as derived above.

Setting \[Omega][] terms to zero gives us the probability for each \
incongruent path   \
\>", "Text",
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Cell["\<\
To obtain the marginal expected length of external \[OpenCurlyQuote]a\
\[CloseCurlyQuote] branches (respectively, \[OpenCurlyQuote]b\
\[CloseCurlyQuote] branches) for each path,  we take the derivative with \
respect to \[Omega][{a}] (respectively, \[Omega][{b}]), multiply by -1, and \
set all the \[Omega][] variables to 0.\
\>", "Text",
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Here, we consider the difference in the expected external branch lengths for \
each path.

For some paths, the expected \[OpenCurlyQuote]a\[CloseCurlyQuote] and \
\[OpenCurlyQuote]b\[CloseCurlyQuote] branch lengths are equal (i.e. the \
difference =  0.0).  Some paths do have a difference in branch lengths, but \
they are paired with an equally probable symmetric path, thereby cancelling \
each other out (e.g. the path pairs (1 & 2) or (5&6). However, there are \
three paths (indexed as 7,8, and 9 below) that are asymmetrical and produce a \
measurable difference in the expected external branch lengths.\
\>", "Text",
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Cell["\<\
We can verify this for the full symbolic expression  by checking that the \
difference in the total expected external branch lengths is 0 if paths 7,8, \
and 9 are removed.\
\>", "Text",
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Cell["\<\
Here, assuming parameters {\[Lambda]b -> 1, \[Lambda]a -> 1, Td -> 1, \
\[Lambda]ab -> 1} while allowing M to vary, we plot:

i) the probability of incongruence
ii) the expected total (marginal) length of external branch types \
\[OpenCurlyQuote]a\[CloseCurlyQuote] and \[OpenCurlyQuote]b\[CloseCurlyQuote] \
given an incongruent genealogy \
\>", "Text",
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We can also look at individual incongruent paths.  Here, we consider the \
incongruent asymmetric path number 9 above. We plot 

(i) the probability of observing the incongruent path number 9 among any \
paths
(ii) the probability of observing the asymmetric path number 9 among \
incongruent paths (i.e. for the GF conditioned on incongruence)
(iii) The expected length of the external branch types conditioned on \
observing path number 9\
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We can try to get a broader view of where A_i = (a - b) / (a + b) is \
maximised by its relationship with M and T

Very high values of Ai are obtained when T is large and M is very small. \
However, by calculating the probability of incongruence for such values, we \
can also see that they will lead to incongruent topologies being very rare. \
In other words, Ai values close to 1 are unlikely to ever be inferred from \
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Here we define some functions that will help us evaluate and plot the \
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The GF conditioned on observing at least one HetAB type mutation\
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Here, we obtain the generating function for the process conditioned on the \
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Define two GFs: (i) the original which gives the probability of coalescence \
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Now subtract GF_ii from GF_i to obtain the probability of coalescence before \
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We obtain the expected branch lengths for this conditioned GF as before:\
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We define a few functions to make evaluating these expressions easier\
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Finally, we define our measure of asymmetry conditioned on observing at least \
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