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\begin{document}
\title{\textbf{The serially-sampled coalescent}}
\date{July 12, 2007}
\author{\textsc{Alexei J. Drummond}}
\maketitle
\section{A simple example}
Consider the situation in which there are 4 individuals sampled, two in the present (A, B) and two sampled $\tau$ time units in the past. Going back in time, the probability that there is no coalescent between A and B before time $\tau$ is:
\begin{equation}
p_{nc} = e^{-\tau/\theta}
\end{equation}
And consequently the probability of coalescence is:
\begin{equation}
p_{c} = 1 - p_{nc}
\end{equation}
If there is a coalescence before time $\tau$ then the tree must be one of the following topologies: ((A,B),(C,D)), (((A,B),C),D), (((A,B),D),C).
Now consider the topology ((A,B),(C,D)). Conditional on coalescence of (A,B) before time $\tau$ it has a probability of $\frac{1}{3}$. However if there is no coalescence before time $\tau$ it has it normal coalescent probability of $\frac{1}{9}$ (being a symmetrical tree shape). This gives a total probability for this tree shape of:
\begin{equation}
p_{((A,B),(C,D))} = \frac{p_c}{3} + \frac{p_{nc}}{9}
\end{equation}
Likewise the probability of topologies (((A,B),C),D) and (((A,B),D),C) can be calculated as:
\begin{equation}
p_{(((A,B),C),D)} = \frac{p_c}{3} + \frac{p_{nc}}{18}
\end{equation}
\begin{equation}
p_{(((A,B),D),C)} = \frac{p_c}{3} + \frac{p_{nc}}{18}
\end{equation}
The probability of the two remaining symmetrical trees are:
\begin{equation}
p_{((A,C),(B,D))} = \frac{p_{nc}}{9}
\end{equation}
\begin{equation}
p_{((A,D),(B,C))} = \frac{p_{nc}}{9}
\end{equation}
The probability of each of the remaining asymmetric trees is:
\begin{equation}
\frac{p_{nc}}{18}
\end{equation}
Taking $\tau/\theta = 0.5$ then $p_{nc} = 0.607$ and $p_c = 0.393$ giving a probability of ((A,B),(C,D)) of:
\begin{equation}
p_{((A,B),(C,D))} = 0.199
\end{equation}
the probability of (((A,B),C),D) is:
\begin{equation}
p_{(((A,B),C),D)} = 0.165
\end{equation}
the probability of ((A,C),(B,D)) is:
\begin{equation}
p_{((A,C),(B,D))} = 0.0674
\end{equation}
and the probability of (((C,D),B),A) is:
\begin{equation}
p_{(((C,D),B),A)} = 0.0337
\end{equation}
Work out the rest :-) Check out examples/testCoalescent.xml to see these results from an MCMC run.
\end{document}
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