Phylogenetic inference

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Phylogenetic inference
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Phylogenetic inference

• The question is how do we find the best tree?
• Many methods using many techniques, many software
  packages
• New improved methods and tests appear in the
  literature constantly
• WHY SO MANY?
• For molecular data, the trend is towards using
  methods based on explicit models based on
  realistic assumptions

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Classification of phylogenetic methods
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Discrete data and Distances
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Algorithms versus optimality criteria

• Phylogenetic inference is an estimation procedure
  (best estimate)
• Only have information about the contemporary
  molecules (and organisms)
• How do we choose a tree from the set of all
  possible trees?
• Two basic approaches
• Algorithmic just follow a sequence of steps
• Optimality criterion how do we compare trees?

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Algorithmic methods

• Combine tree inference and the definition of a
  preferred tree into a single statement
• Include UPGMA and all forms of pair-group cluster
  analysis, and neighbor joining
• Computationally fast because they go straight to
  the final solution
• The task of finding an optimal tree can not be
  separated from that of evaluating a specific tree

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Optimality criteria

• Two logical steps
• Define an optimality criterion (objective
  function for evaluating trees)
• Find the tree(s) with the best value for the
  objective function (may use algorithms)
• Evolutionary assumptions made in the first step
  are decoupled from the computations involved in
  the second step
• Price for logical clarity is that these methods
  can be very slow

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Use of algorithms

• Different use in the two approaches
• In purely algorithmic methods, the algorithm
  defines the tree selection criterion and is
  fundamental
• In criterion-based methods, algorithms are merely
  tools used in evaluating and searching for
  optimal trees
• Reliability of the tree?

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Optimality criteria

• Parsimony select the tree that minimizes the
  total tree length (number of steps or character
  transformations required to explain a given set
  of data)
• Methods based on models of evolutionary change
  assumptions are made explicit.
• Is parsimonys model-free nature an advantage
  or a disadvantage?
• What are the assumptions for parsimony
  (consistency?)

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Optimality methods (cont.)

• Maximum likelihood evaluates the probability
  that a proposed model of evolution and the
  hypothesized history could give rise to the
  observed data (attempts to estimate the actual
  amount of change or history)
• Usually more consistent estimates have lower
  variance than other methods robust to violations
  of assumptions

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Optimality criteria (cont.)

• Pairwise distance methods also minimize the
  effect of multiple hits when using appropriate
  model to estimate the true evolutionary distance
  between two sequences (less desirable than full
  ML)
• Additive and ultrametric distances can be fitted
  to a tree such that all pairwise distances are
  equal to the sum of the branches along the path
  connecting them in the tree

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TREES AND DISTANCES

• Measures of sequence similarity are used to
  estimate evolutionary changes that occurred
  between 2 sequences
• These measures quantify the evolutionary distance
  between the 2 sequences
• Trees can represent these distances.
• This has motivated a range of tree building
  techniques to convert pairwise distances into
  evolutionary trees.

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A distance measure can be used to build
phylogenies IF it satisfies some basic
requirements...

• It must be a metric
• It must be additive

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1. Metric Distances

• Let d (a,b) be the distance between two sequences
  a and b
• A distance d is a metric if it satisfies 4
  conditions
• d (a,b) 0 (non-negativity)
• d (a,b) d (b,a) (symmetry)
• d (a,c) d (a,b) d (b,c) (triangle
  inequality)
• d (a,b) 0 if and only if ab
  (distinctness)

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Triangle inequality d (a,c) lt d (a,b) d (b,c)
The distance between any pair of sequences must
be no greater than that between those sequences
and a third sequence
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Of these 4 conditions...

• non-negativity
• symmetry
• triangle inequality
• distinctness

• 1, 2 and 4 are generally true for most measures
  of sequence dissimilarity calculated directly
  from sequences.
• Indirect measures of sequence dissimilarity such
  as DNADNA hybridization and immunological
  distance often fail condition 2 (symmetry)

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A distance is ultrametric if it satisfies the
additional three-point condition

• d (a,b) maximum d (a,c), d (b,c)

• This implies that 2 of the 3 pairwise distances
  between 3 taxa are equal, and at least as large
  as the third, defining an isosceles triangle.

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Ultrametric Distances have the very useful
property of implyinga constant rate of evolution

• In fact we have a test for the molecular clock
  called the Relative Rate Test which is simply
  a test of how far the pairwise distances between
  3 sequences depart from Ultrametricity.
• If distances between sequences are Ultrametric
  then the most similar sequences are also the most
  closely related

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2. Additive Distances

• Being Metric or Ultrametric is necessary but not
  sufficient to ensure a measure of change is valid
  and fits a tree exactly.
• It must also satisfy the four-point condition
• d(a,b) d(c,d) maximum d(a,c) d(b,d),
  d(a,d) d(b,c)
• This is equivalent to requiring that of the three
  sums d(a,b) d(c,d), d(a,c) d(b,d) and d(a,d)
  d(b,c) the two largest are equal.

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d (a,b) d (c,d), d (a,c) d (b,d) d (a,d) d
(b,c)
of the three sums...
Ae ef fB Ce ef fD
Ae eC Bf fD
Ae ef fD Ce ef fB
...the two largest are equal.
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An additive distance measure defines a tree...

• Sequence d is equidistant from all other
  sequences
• Sequence c is equidistant from a and b

• For any 2 sequences the value in the distance
  matrix corresponds to the sum of the branch
  lengths along the path between the 2 sequences on
  the tree.
• The presented tree is ultrametric (draw the
  isosceles triangle)

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When distances are not ultrametric but only
metric they can still be represented by a tree..
An additive tree
Also represents additive distances exactly...
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• Notice that sequences b and c are the most
  similar (3), but ARE NOT the most closely related
• Similarity and evolutionary relationship will
  only coincide exactly if the distances are
  ultrametric
• While this tree is additive, it is not ultrametric

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• Observed distances are obtained directly from the
  sequences themselves and patristic distances
  from a tree
• For additive and ultrametric distances, the
  observed and tree distances match exactly

For real data this is rarely the case, indicating
that observed distances cannot be completely
accurately represented by a tree.
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The discrepancy between observed and tree
distances can be used as an indicator of how well
observed distances fit a tree like
representation.
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Distance methods

• Experimentally derived distances are estimates of
  true distances
• We want to fit them to a mathematical model
  (additive tree) and find the optimal value for
  the adjustable parameters
• Branching pattern
• Branch lenghts
• Some methods based on this optimality criterion
  are Fitch Margoliash, minumum evolution
• Other methods fit trees to distances
  algorithmically (NJ, UPGMA)

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UPGMA an algorithmic method

• Cluster analysis Unweighted pair group method
  using arithmetic averages (Sneath and Sokal 1973)
• Assumes ultrametricity

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UPGMA example

• Given a matrix of pairwise distances, find the
  clusters (taxa) i and j such that dij is the Min
  value in the table
• Define the depth of the branching between i and j
  (lij) to be dij/2
• If i and j were the last two clusters, the tree
  is complete. Otherwise, create a new cluster
  called u
• Define a distance from u to each other cluster k
  (with k ? i or j) to be the average of the
  distances dki and dkj
• Go back to step 1 with one less cluster clusters
  i and j have been eliminated, and cluster u has
  been added

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Distance Matrices and phenogram
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Classification of phylogenetic methods
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Parsimony methods

• The most widely-used method, familiar notion in
  science (simplicity)
• Shared attributes among taxa are inherited from
  common ancestors
• When character conflicts occur, ad hoc hypotheses
  cannot be avoided if you want to explain all the
  data, and assumptions of homoplasy must be invoked

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Parsimony

• From all sets of possible trees, find all trees ?
  such that L(?) is minimal
• B is the number of branches
• N is the number of characters
• k and k are the two nodes incident to each
  branch k
• xkj and xkj represent either elements of the
  input matrix or optimal-character assignments
  made to internal nodes
• Diff(y,z) is a function specifying the cost of
  transformation from y to z along any branch (for
  unrooted trees diff(y,z)diff(z,y)
• The coefficient w is the weight assigned to each
  character

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Parsimony
From the set of all possible trees, find all
trees ? such that L(?) is minimal

• Two problems
• How do you actually implement the objective
  function to evaluate a particular tree?
• How do you find all minimal trees when you have
  many taxa?
• Objective function and tree-searches can be
  formalized into algorithms

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Fitch and Wagner Parsimony

• Simplest parsimony methods
• Fitch unordered multistate characters
• Wagner binary,ordered multistate and continuous
• In describing the algorithm we will consider a
  single character (j) in isolation (independence)
• The tree (strictly bifurcating) is a given, just
  any tree we wish to evaluate (tree-search
  algorithms, later), we only need to assign an
  arbitrary root (any taxon), denoted r

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Fitch parsimony The algorithm

• To each terminal node i (including the one at
  the root), assign a state set Si containing the
  character state assigned to the corresponding
  taxon in the input data matrix (Xij)

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Post-order traversal

• Visit an internal node k for which a state set Sk
  has not been defined but for which the state sets
  of ks two immediate descendants (Si and Sj) has
  been defined. Assign to k a state set Sk
  according to the following rules

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then

• if

then

• Otherwise

and increase the length by 1
The intersection of Sb and Sc is empty, so we use
the union as Sx and increase L by 1 (L 1)
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Post-order traversal

• If node k is located at the basal fork of the
  tree (I.e., the immediate descendant of the
  terminal node placed at the root). The traversal
  has been completed proceed to step 4. Otherwise
  return to step 2.

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Return to step 2
The intersection of Sx and Sd is not empty, so we
use the intersection as Sy and do not increase
L (L remains 1)
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Return to step 2
The intersection of Sy and Se is empty, so we use
the union as Sz and increase L by 1 (now, L 2)
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Post-order traversal

• If the state assigned to the terminal node at the
  root of the tree (Xr) is not contained in the
  state set just assigned to the node at the basal
  fork of the tree (Sk), increase the tree length
  by 1

In this case state C is not included in state set
A,G, so we increase L by 1 to (now, L3) This
procedure concludes the evaluation but does not
assign optimal character states to the internal
nodes
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Pre-order traversal

• To obtain a most parsimonious reconstruction
  (MPR) and assign optimal character states to each
  node we need to make a second pass over the tree
• This time from the root to the tips

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Pre-order traversal

• Visit an internal node k for which an optimal
  state assignment Xk has not yet been made but for
  which an assignment has been made to ks
  immediate ancestor, denoted m
• If Xm is contained in the state set assigned to k
  in the first pass (Sk), assign this state to k as
  well. Otherwise arbitrarily assign any state from
  Sk to k

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Pre-order traversal

• Two MPRs exist for this case according to which
  state we arbitrarily choose for the set A,G at
  the basal fork
• A third option is to assign C to all three
  internal nodes

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Other parsimony variants

• Dollo parsimony every derived character must be
  uniquely derived (originate only once in the
  tree)
• Homoplasy only reversals are allowed (no
  parallelism or convergence)
• In practice, Dollo parsimony does not require
  inclusion of hypothetical ancestors just
  character polarity (unrooted Dollo)
• Convenient for restriction-site characters
  (easier to loose that to gain a site)

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Dollo parsimony and RFLP data
Relaxed Dollo criterion, may be applied using
generalized parsimony
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Generalized Parsimony

• All parsimony variants can be subsumed into a
  generalized method that assigns a cost for each
  possible transformation
• Costs are represented in a m-by-m cost matrix S,
  where each element Sij represents the increase in
  tree length due to a transformation from state i
  to j
• The cost of each transformation (weight) can be
  determined a priori (e.g. for RFLPs or for
  transition/transversion changes) or a posteriori
  (using the same data, e.g. successive
  approximations method)

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Generalized Parsimony Cost matrices
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Protein parsimony

• A 20x20 matrix specifies the cost for each
  possible transformation
• The matrix may be based on the genetic code
  (PROTPARS matrix) and/or the biochemical
  properties of the amino acids themselves (Dayhoff
  matrices)

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Parsimony

• From all sets of possible trees, find all trees ?
  such that L(?) is minimal
• B is the number of branches
• N is the number of characters
• Diff(y,z) is a function specifying the cost of
  transformation from y to z along any branch (for
  unrooted trees diff(y,z)diff(z,y) -- Cost matrix
• The coefficient w is the weight assigned to each
  character (weighted parsimony)
• A priori or a posteriori

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Difference in perspective MP and ML

• Parsimony seeks solutions that minimize the
  amount of change required to explain the data
  (underestimates superimposed changes)
• ML attempts to estimate the actual amount of
  change (by specifying the evolutionary model that
  will account for the data with the highest
  likelihood)
• Methods that incorporate models of evolutionary
  change can make more efficient use of the data

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Searching for optimal trees

• Methods with explicit optimality criteria
• Parsimony
• Maximum likelihood
• Additive-tree distance
• Separate the problem of
• evaluating the tree
• finding the optimal tree(s)
• Can we evaluate all possible trees for a
  particular problem?

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Searching for optimal trees

• For small to moderate data sets, with as many as
  8-20 taxa, we can use exact methods
• Exact methods guarantee the discovery of all
  optimal trees
• Exact methods include
• Exhaustive search
• Branch-and-bound search

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How many trees?
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Exhaustive search enumerate al possible trees
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• Branch-and-bound search
• Does not require exhaustive search and yet
  provides an exact solution.
• Traverse a search tree in a depth-first sequence
  (to get an initial tree, could be a random tree
  but better use heuristics to make BB more
  efficient)
• Use this tree score as the upper bound (L) on
  optimal value of chosen criterion.

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Branch-and-bound 3. Move along path to tips and
evaluate trees. If tree is gtL then dispense the
rest of that path. 4. If a better tree is found
with L lt L, we now have improved the upper bound
on the score of the optimal tree. This may enable
us to dispense of other paths and finish the
search more quickly
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Approximate methods

• For larger data sets computing time becomes
  prohibitive and we only explore some subset of
  all possible trees (hoping that the optimal trees
  will be found in the subset explored)
• Heuristic approaches sacrifice the guarantee of
  optimality in favor of reduced computer time
• Use hill climbing methods. Initial tree starts
  the process, then we seek to improve its score
• When we can find no way to further improve the
  score, we stop.We dont know if we reached a
  local or a global optimum

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Initial trees

• May be obtained by stepwise addition, the most
  commonly used method
• Similar to exhaustive search but evaluate trees
  at every step, each time you add a new taxon and
  only follow the path derived from the optimal
  tree
• Which taxa do you choose first? Which do you
  connect next?
• These are greedy algorithms

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Stepwise addition
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• Initial trees also may be obtained by star
  decomposition, another greedy algorithm

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Branch swapping

• To improve the initial estimate we can perform
  sets of predefined rearrangements on the tree
• Any of these rearrangements amounts to a stab in
  the dark
• Globally optimal trees may be several
  rearrangements away from the starting tree
• If a better tree is found, a new round of
  rearrangements is then performed in the new tree
• Several branch-swapping algorithms are available

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Branch swapping by tree bisection and
reconnection (TBR) 1. Tree is bisected along a
branch, yielding two disjunct subtrees 2. The
subtrees are reconnected by joining a pair of
branches, one from each subtree 3. All possible
bisections and pairwise reconnections are
evaluated
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Branch swapping by subtree prunning and
regrafting (SPR) 1. A subtree is pruned from the
tree (e.g. A,B) 2. The subtree is then regrafted
to a different location on the tree 3. All
possible subtree removals and reattachment points
are evaluated
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Branch swapping by nearest-neighbor interchanges
(NNI) 1. Each interior branch of the tree
defines a local region of four subtrees
2. Interchanging a subtree on one side of the
branch with one from the other constitutes an
NNI 3. Two such rearrangements are possible for
each interior branch (all interior branches are
swapped)
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Landscapes and the problem of islands of trees
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Bayesian Inference of Phylogenies

• Closely related to ML methods, differing only in
  the use of a PRIOR DISTRIBUTION (which would
  typically be a tree)
• Use of a prior enables us to interpret the result
  as the distribution of the tree given the data
• Bayes described this in 1790, and controversy
  among statisticians over its appropriateness is
  almost that old
• Recently, the introduction of Markov Chain Monte
  Carlos (MCMC) methods has given a new impetus to
  Bayesian inference
• The latest silver bullet for phylogenetic
  analysis?

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Simple example of Bayesian inference
Box with 90 fair and 10 biased dice
Take a die at random from the box and roll it
twice get a 4 and a 6 What is the probability
that the die is biased?
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A Bayesian analysis combines ones prior beliefs
about the probability of a hypothesis with its
likelihood
Likelihood assuming a fair die
Likelihood assuming a biased die
Probability of observing the data is 1.96 times
greater under the hypothesis that the die is
biased
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Bayesian inference is based upon the POSTERIOR
probability of a hypothesis
The posterior probability that the die is biased
can be obtained using Bayess formula
Our opinion of the die being biased changed from
0.1 to 0.179 after observing a 4 and a 6 Priors
are a strength or a weakness of the method?
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Bayesian inference of phylogeny
Based upon the posterior probability of a
phylogenetic tree ( )
Posterior probability of the ith tree, can be
interpreted as the probability that this tree is
the correct tree given the data
prior probability of the ith tree, typically
The summation in the denominator is over all B
trees possible for s species (taxa)
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Markov Chain Monte Carlo
Typically, the posterior probability
cannot be calculated analytically. However,
the posterior probability of phylogeniers can be
approximated by sampling trees from the posterior
probability distribution. MCMC can be used to
sample phylogenie according to their posterior
probabilities Let
be a specific tree, combination of branch
lengths, substitution parameters, and gamma shape
parameter The MH algorithm is an MCMC allgoritm
that has been successfully used to approximate
the posterior probability of trees
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The MH algorithm constructs a Markov chain that
has as its stationary frequency the posterior
probability of interest (in this case the joint
posterior prob of ) The
current state of the chain is denoted A
new state is then proposed The new state is
accepted with probability
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A uniform random number is drawn and if this
number is lt R, then the proposed change is
accepted. Otherwise the chain remains in the
original state. This process of proposing a new
state, calculating the acceptance probability,
and accepting or rejecting the move is repeated
thousands of times. The sequence of states
visited forms the Markov Chain. The chain is
sampled after it reached stationarity and the
sampled trees represent the posterior prob
distribution
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The proportion of times a single tree is found
among these samples is the posteriror prob of
that tree A majority rule consensus can be
derived from the sample and the proportions
obtained for each clade are an approximation of
the posterior prob of the clades