Reconstruction on trees and Phylogeny 2

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Reconstruction on trees and Phylogeny 2
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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Reconstruction on Ising-CFN model

• We study the reconstruction problem for the
  Ising-CFN model on regular trees.

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Markov models on trees

• Finite set A of information values.
• Tree T(V,E) rooted at r.
• Vertex v 2 V, has information sv 2 A.
• Edge e(v, u), where v is the parent of u, has a
  mutation matrix Me of size A A
• Mi,j (v,u) P?u j ?v i
• For each character ?, we are given ??T (?v)v 2
  ?T,
• where ?T is the boundary of the tree.
• We will focus on the Ising-CFN model

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Statistical physics

• Statistical physics is a sub-field of
  mathematical physics where we study complex
  systems with simple microscopic interactions.
• The Ising model on a graph is a probability
  measure (Gibbs distribution) on the space of
  configurations s from vertices to -1,1 such
  that
• Ps exp(S(v, w) e E s(v)s(w)/T).
• Traditionally studied on cubes in Zd.

The Ising model on 200 x 200 grid
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Statistical physics on trees

• The Ising model on the binary tree can be
  defined
• Set sr, the root spin, to be /- with probability
  ½.
• For all pairs of (parent, child) (v, w), set sw
  sv, with probability ?, otherwise sw /-
  with probability ½.
• This is exactly the CFN model.

• Studied in statistical physics Spitzer 75,
  Higuchi 77, Bleher-Ruiz-Zagrebnov 95,
  Evans-Kenyon-Peres-Schulman 2000, Ioffe 99, M 98,
  Haggstrom-M 2000, Kenyon-M-Peres 2001,
  Martinelli-Sinclair Weitz 2003, Martine 2003

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Reconstruction solvability

• Let T be an infinite rooted tree and Tn denote
  the first n levels of T.
• We say that the reconstruction problem is
  solvable if one of the following equivalent
  conditions hold
• 9 ? s.t. (8 non-degenerate ?) limn ! 1 I(X0,Xn)
  gt 0, where I(X0,Xn) H(X0) H(Xn) H(X0,Xn) H
  is the entropy operator, H(X) -?x PX x log2
  PX x.
• 9 i,j s.t. limn ! 1 Pni - Pnj gt 0, where Pnj
  denotes the distribution of Xn conditional on X0
  j.
• If X0 has the uniform distribution then,
  liminfn ! 1 ?n gt 1/m, where ?n is the
  probability of correct reconstruction of X0 given
  Xn.
• 9 ? (8 non-degenerate ?) liminfn ! 1
  VarEX0Xn gt 0.

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The Ising model on the 3-regular tree
mutual information H(s?) H(sr)) - H(sr,s?)
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Reconstruction for the CFN model

• Thm The reconstruction problem for the Ising
  model on the (b1)-regular tree is solvable if
  and only if b ?2 gt 1.
• Easy direction Higuchi 77 prove that a
  certain reconstruction algorithm works when b ?2
  gt 1.
• Higuchi argument extends to general chains and
  general trees.
• Will also show an argument from M98 useful for
  phylogeny.
• Hard direction 95 Non-reconstruction?
• 6 different proofs!
• All involve a magic.
• None extends to other markov models.
• Will follow a coupling proof Martinelli-Siclair-W
  eitz

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Non-reconstruction - Coupling down

• Copying rule. For i ,-
• Pi ! i ?.
• Pi ! Uniform 1 ?.
• Continuing down the tree, non-coupled elements
  form a branching process with parameter ?.

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• If b ? 1, branching process dies ) coupling.
• More generally, at level n, the expected number
  of uncoupled sites is bn?n.
• (Doesnt work all the way to b ?2 1).

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Non-reconstruction - Coupling up

• We try to couple two configurations which differ
  at level n so that they agree at the root.
• First consider the case where they differ at
  exactly one site.

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• Lemma Mossel-Kenyon-Peres Among all boundary
  conditions ?, E? ?u 1 ?v 1 E??u -1
  ?v 1 is maximized for the free boundary.
• ) Pnot coupling at u ?.
• ) Pnot coupling at the root ?n.

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Coupling up path coupling

• We got that if ? and ? are two boundary
  conditions which differ in one position at level
  n, then
• E??(?) E??(?) 2 ?n, where ? is the root.
• ) if ? and ? are two boundary conditions which
  differ at k sites, then
• E??(?) E??(?) 2 k ?n.
• Pf If ? and ? differ at k sites, then we can
  find a sequence ? ?(0),?(1),,?(k) ?, such
  that ?i and ?i1 differ in exactly one site.
• E??(?) E??(?)
• ?i1k E?(i)?(?) E?(i-1)?(?) 2 k ?n.

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Non reconstruction for b ?2 lt 1

• Fix ? such that b ?2 lt 1.
• We will show that EE?(?) ? E-E?(?)
  ?- ! 0,
• where ? boundary conditions conditioned on
  ?(?) .
• Let (?,?-) be given by the down coupling.
• Let K(?,?-) number of disagreements between
  ?,?-.
• EE?(?) ? E- E?(?) ?-
• E_,-E?(?) ? - E?(?) ?-
• E,-2 K(?,?-) ?n 2 ?n E,-K(?,?-) (up
  coupling).
• 2 ?n bn ?n (down coupling)
• 2 (b ?2)n ! 0 exp. fast in n.

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Where we stopped

• Thm The reconstruction problem for the Ising
  model on the (b1)-regular tree is solvable if
  and only if b ?2 gt 1.
• We showed that if b ?2 lt 1, it is impossible to
  reconstruct (hard direction).
• We now show that if b ?2 gt 1, we can reconstruct.

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Reconstruction via majority

• Fix ? such that b ?2 gt 1.
• Let X Xn () - (-) at level n.
• We claim that Xn is a good estimator of ?(?).
• EXn bn ?n E-Xn -bn ?n.
• We show that E/-Xn2 c(E/-Xn)2 c b2n
  ?2n.

• Let f fn (g gn) be the density of the (-)
  measure with respect to some reference measure ?.
• 2 bn ?n EX E-X s X (f g) d ?
• s X (f1/2 g1/2) (f1/2 g1/2) d ?
• (s X2 (f1/2 g1/2)2 d?)1/2 (s (f1/2
  g1/2)2 d? )1/2
• (4 s X2 f d? 4 s X2 g d?)1/2 (s f g
  d?)1/2
• (8 c b2n ?2n)1/2 (DTV(,-))1/2.

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Bounds on the second moment

• Write Xn ?v ?(v), where the sum is over all v
  in level n.
• EXn2 ?v,w E?(v) ?(w).
• For each edge with prob. ? the two end points are
  the same and with prob. 1-? the two points are
  independent.
• If there is a red edge on the path between v and
  w, then E?(v) ?(w) 0.

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• Otherwise, ?(v) ?(w).
• E?(v) ?(w) ?d(v,w).
• EXn2 bn(1 ?i1n (bi bi-1)?2i)
• bn(1 (b-1) ?2 ?i0n-1 bi ?2i).
• O(b2n ?2n) iff b ?2 gt 1.

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Remarks on the second moment

• Kamea/ Higuchi argument is very robust.
• Works for general trees when br(T) ?2 gt 1.
• Works for general markov chains, where ? 2nd
  eigenvalue of M (M-Peres 2002).
• Kesten-Stigum (1966!) proved that for all markov
  chains
• if b ?2 gt 1, then the limiting law of the count
  depends on the root.
• If b ?2 lt 1, then the limiting law is normal for
  all root values.
• M-Peres (2002) count reconstruction is impossible
  if b ?2 lt 1.

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Recursive reconstruction for Ising models
?

• An alternative proof for reconstruction for b ?2
  gt 1 M98
• Advantage Works also when we have lower bound on
  ?. Majority doesnt.
• Blue edges have ?1 , black ?2, ?1 lt ?2 1.
• Maj(s?) Maj of black tree.
• Maj of black tree sv .
• sv and s? have exp. small correlation.
• Phylogeny reconstruction given bounds.

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• Instead we will use recursive-majority.