Reconstruction on trees and Phylogeny 4

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Reconstruction on trees and Phylogeny 4
Elchanan Mossel, U.C. Berkeley mossel_at_stat.berke
ley.edu, http//www.cs.berkeley.edu/mossel/ Sup
ported by Microsoft Research and the Miller
Institute
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Future research in Phylogeny

• Applications
• Check validity of currently published
  phylogenies.
• See if new reconstruction methods out-perform
  existing ones.
• See if standard methods (e.g. Maximum likelihood)
  behave differently in different phases.
• CFN and R.C models
• Prove logarithmic reconstruction under average
  low mutation.
• Try and remove clock/balance condition for CFN
  model.
• Other Markov modes on trees.
• Prove logarithmic reconstruction for low mutation
  rate.
• Critical value?

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Future research in Phylogeny

• Network tomography.
• Check how methods do for recovering
  tree-networks.
• Other tree models (e.g. coming from delays).
• Other graphical models.
• Is phase transition playing a similar role in
  reconstructing tree-like and non-tree-like
  graphical models?
• Importance of phase transition for other
  problems in graphical models.

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Reconstruction for other Markov models

• Leaving the Ising model
• Reconstruction for other models is more
  interesting.
• The natural bound for reconstruction is b ?2
  gt 1, where ? is the second eigen-value of M (in
  absolute value).
• In count reconstruction we reconstruct from Yn
  (Yn(i))i 2 A, where Y_n(i) of times color i
  appears at the nth level.
• Theorem M-Peres 2002 The count reconstruction
  problem is solvable if b ?2 gt 1 and unsolvable
  if b ?2 lt 1.

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Reconstruction for other Markov models

• Theorem M-Peres 2002 The count reconstruction
  problem is solvable if b ?2 gt 1 and
• unsolvable if b ?2 lt 1.
• Proof uses Kesten-Stigum-66 theorem.

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Robust Reconstruction for Markov models

• So the threshold b ?2 1 is important.
• But M-2000 it is not the threshold for the
  reconstruction problem.
• Not even for 2 2 markov chains,
• Or symmetric markov chains on q symbols.
• Moreover, there exists a markov chain M s.t. ?
  0, but the reconstruction problem is solvable for
  some b.

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Reconstruction for Markov models

• In robust reconstruction, instead of the nth
  level, ?n, we are given ?n, where for each v at
  level n
• ?n(v) ?n(v) with probability ?,
• ?n(v) an independent color from a
  distribution ? with probability 1 - ?.
• Similar to robust phase transition
  Pemantle-Steif 99.
• Easy if b ?2 gt 1 then robust reconstruction is
  solvable for all ? gt 0.
• Theorem Janson-M 2003 If b ?2 lt 1, then for ?
  gt 0 small, robust reconstruction is unsolvable.
• Same is true with br(T) instead of b.