Remarks on the Limiting GIbbs States on a ($d+1$)-Tree

Higuchi, Yasunari

doi:10.2977/prims/1195189812

Cited by 48 publications

(52 citation statements)

References 4 publications

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“…Note that in terms of θ we have reconstruction solvability whenever θ(e) ≥ θ > θ * for all e where 2θ 2 * = 1. For the CFN model both the majority algorithm [10] and recursive majority algorithms [14] are effective in reconstructing the root value (for other models in general, most simple reconstruction algorithms are not effective all the way to the reconstruction threshold [15,18,12]). …”

Section: Properties Of the Majority Functionmentioning

confidence: 99%

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Optimal phylogenetic reconstruction

Daskalakis

Mossel

Roch

2006

Proceedings of the Thirty-Eighth Annual ACM Symposium on Theory of Computing

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One of the major tasks of evolutionary biology is the reconstruction of phylogenetic trees from molecular data. This problem is of critical importance in almost all areas of biology and has a very clear mathematical formulation. The evolutionary model is given by a Markov chain on the true evolutionary tree. Given samples from this Markov chain at the leaves of the tree, the goal is to reconstruct the evolutionary tree. It is crucial to minimize the number of samples, i.e., the length of genetic sequences, as it is constrained by the underlying biology, the price of sequencing etc.It is well known that in order to reconstruct a tree on n leaves, sequences of length Ω(log n) are needed. It was conjectured by M. Steel that for the CFN evolutionary model, if the mutation probability on all edges of the tree is less than p * = ( √ 2 −1)/2 3/2 than the tree can be recovered from sequences of length O(log n). This was proven by the second author in the special case where the tree is "balanced". The second author also proved that if all edges have mutation probability larger than p * then the length needed is n Ω(1) . This "phasetransition " in the number of samples needed is closely related to the phase transition for the reconstruction problem (or extremality of free measure) studied extensively in statistical physics and probability.Here we complete the proof of Steel's conjecture and give a reconstruction algorithm using optimal (up to a multiplicative constant) sequence length. Our results further extend to obtain optimal reconstruction algorithm for the Jukes-Cantor model with short edges. All reconstruction algorithms run in time polynomial in the sequence length.The algorithm and the proofs are based on a novel combination of combinatorial, metric and probabilistic arguments.Keywords optimal phylogenetic reconstruction, mutation probability, second author, markov chain, phylogenetic tree, underlying biology, special case, statistical physic, phase transition, reconstruction problem, evolutionary tree, genetic sequence, molecular data, cfn evolutionary model, evolutionary model, clear mathematical formulation, true evolutionary tree, major task, evolutionary biology, critical importance, free measure Disciplines Statistics and Probability | Theory and Algorithms AbstractOne of the major tasks of evolutionary biology is the reconstruction of phylogenetic trees from molecular data. This problem is of critical importance in almost all areas of biology and has a very clear mathematical formulation. The evolutionary model is given by a Markov chain on the true evolutionary tree. Given samples from this Markov chain at the leaves of the tree, the goal is to reconstruct the evolutionary tree. It is crucial to minimize the number of samples, i.e., the length of genetic sequences, as it is constrained by the underlying biology, the price of sequencing etc.It is well known that in order to reconstruct a tree on n leaves, sequences of length Ω(log n) are needed. It was conjectured by M. Steel that for the CFN evol...

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Section: Properties Of the Majority Functionmentioning

confidence: 99%

“…This problem was studied earlier in statistical physics, probability and computer science under the title of the reconstruction problem, or the "extremality of the free Gibbs measure", see [21,10,9]. The reconstruction problem for the CFN model was analyzed in [2,6,11,1,13].…”

Section: Introductionmentioning

confidence: 99%

Optimal phylogenetic reconstruction

Daskalakis

Mossel

Roch

2006

Proceedings of the Thirty-Eighth Annual ACM Symposium on Theory of Computing

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“…The extremality of the free measure for the Ising model on the regular tree plays a crucial role in this paper. The study of the extremality of the free measure begins with [30], [16]. Later papers include [4], [10], [17], [23] (see [10] for more detailed background).…”

Section: Where D(t ) = θ(Depth Of T ) and C = C(δ)mentioning

confidence: 99%

Phase transitions in phylogeny

Mossel

2003

Trans. Amer. Math. Soc.

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Abstract. We apply the theory of Markov random fields on trees to derive a phase transition in the number of samples needed in order to reconstruct phylogenies.We consider the Cavender-Farris-Neyman model of evolution on trees, where all the inner nodes have degree at least 3, and the net transition on each edge is bounded by . Motivated by a conjecture by M. Steel, we show that if 2(1 − 2 ) 2 > 1, then for balanced trees, the topology of the underlying tree, having n leaves, can be reconstructed from O(log n) samples (characters) at the leaves. On the other hand, we show that if 2(1 − 2 ) 2 < 1, then there exist topologies which require at least n Ω(1) samples for reconstruction.Our results are the first rigorous results to establish the role of phase transitions for Markov random fields on trees, as studied in probability, statistical physics and information theory, for the study of phylogenies in mathematical biology.

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“…It is known ( [6], [9]) that the set (J) of all Gibbs measures for a fixed J is a nonempty, compact convex set. A limiting Gibbs measure is a Gibbs measure for the same J. Conversely, every extremal point of (J) is a limiting Gibbs measure with a suitable boundary condition for the same J.…”

Section: Definitionsmentioning

confidence: 99%

Boundary Conditions for Translation-Invariant Gibbs Measures of the Potts Model on Cayley Trees

Gandolfo

Рахматуллаев²,

Rozikov³

2017

J Stat Phys

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Abstract. We consider translation-invariant splitting Gibbs measures (TISGMs) for the qstate Potts model on a Cayley tree of order two. Recently a full description of the TISGMs was obtained, and it was shown in particular that at sufficiently low temperatures their number is 2 q − 1. In this paper for each TISGM µ we explicitly give the set of boundary conditions such that limiting Gibbs measures with respect to these boundary conditions coincide with µ.Mathematics Subject Classifications (2010). 82B26; 60K35.

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Remarks on the Limiting GIbbs States on a ($d+1$)-Tree

Cited by 48 publications

References 4 publications

Optimal phylogenetic reconstruction

Optimal phylogenetic reconstruction

Phase transitions in phylogeny

Boundary Conditions for Translation-Invariant Gibbs Measures of the Potts Model on Cayley Trees

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