notes.html

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layout: notes
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<section>
<h1> COMPSCI 369 final notes</h1>
</section>

<section>
<h2> Assignment 3 </h2>

<p>The coalescent likelihood should be:</p>

$$ p(T|N_e) = \prod_{k=2}^{n} \left( \frac{1}{N_e} \exp\left(\color{red}{-}\frac{k (k-1) t_k}{ 2N_e}\right) \right)  $$

<p>where $t_k$ is the time duration that the tree $T$ has $k$ lineages.</p>

<p>You may find that you need to calculation this target distribution in log space to avoid numerical underflow:</p>

$$ \ln p(T|N_e) = \sum_{k=2}^{n}  \left(-\ln(N_e) -\frac{k (k-1) t_k}{ 2N_e}\right)   $$

<p> The acceptance probability ratio becomes $\alpha = \min(1, exp(\ln p(N_e'|T)-\ln p(N_e|T)))$</p>

<p> Remember $p(N_e|T) \propto p(T|N_e)p(N_e)$.
</section>

<section>
<h2> MCMC algorithm</h2>

<p> Pseudocode for one dimensional problem on positive real parameter $\theta$: </p>

<ul style="list-style-type:none;">
<li> Set $i = 0, \theta = 1$, chain = array[$MAX\_STEPS$];
<li> While ($i < MAX\_STEPS$) {
  <ul style="list-style-type:none;">
  <li> Generate a uniform random variate $U \sim \mbox{Uniform}\left(-10,10 \right)$; 
  <li> Set $\theta' = \theta + U$
  <li> if ($\theta' > 0$) {
  <ul style="list-style-type:none;">
    <li> Set $\alpha = \min(1, \pi(\theta')/\pi(\theta)$);
    <li> Generate a unit uniform random variate $V \sim \mbox{Uniform}\left(0,1 \right)$;
    <li> if ($\alpha > V$) then set $\theta = \theta'$;
  </ul>
  <li> }
  <li> Set chain[i] = $\theta$;
  <li> Set $i = i + 1$;
  </ul>
  <li> }
</ul>

</section>

<section>
<h2> MCMC in tree space simulation</h2>

<a href="https://walterxie.github.io/TreeSpace/"> Tree Space simulation </a>

</section>



<section>
<h2> What is not examinable</h2>

<ul> 
<li>Three guest lectures will not be examined
<li> Continuous-time Markov chains (CTMCs) are examinable except for
   <ol>
   <li> slide 2/7; Don't need to know the differential form for CTMCs probability distribution
   <li> Slide 2/20; Don't need to know the JC distance correction with site rate heterogeneity
   </ol>
<li> Bayesian phylogenetics is examinable except for
   <ol>
   <li> slide 10; Don't need to be able to reproduce the stochastic trajectory formalism at the top of this slide; 
   <b>do need</b> to know that the Metropolis-Hastings acceptance probability achieves the target distribution as the equilibrium.
   </ol>
</ul>
</section>