1. Reconstruction on trees and Phylogeny 4
Elchanan Mossel,
U.C. Berkeley
Supported by Microsoft Research and the Miller Institute
11/22/2018

2. 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?
11/22/2018

3. 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.
11/22/2018

4. Reconstruction for other Markov models
Leaving the Ising model …
Reconstruction for other models is more interesting.
The “natural” bound for reconstruction is b ||2 > 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 n’th level.
Theorem [M-Peres 2002]: The count reconstruction problem is solvable if b ||2 > 1 and unsolvable if b ||2 < 1.
11/22/2018

5. Reconstruction for other Markov models
Theorem [M-Peres 2002]: The count reconstruction problem is solvable if b ||2 > 1 and
unsolvable if b ||2 < 1.
Proof uses [Kesten-Stigum-66] theorem.
11/22/2018

6. 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.
11/22/2018

7. Reconstruction for Markov models
In “robust reconstruction”, instead of the n’th 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 > 1 then robust reconstruction is solvable for all  > 0.
Theorem [Janson-M 2003]: If b 2 < 1, then for  > 0 small, robust reconstruction is unsolvable.
Same is true with br(T) instead of b.
11/22/2018