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phylo3 - Ancestral Reconstruction Selection Pressure...

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Ancestral Reconstruction & Selection Pressure Christopher Lee December 3, 2009
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Evolutionary Trees as Markov Chains Assume we’re given binary tree as a directed graph G with nodes Θ u , and branch lengths t uv for each edge Θ u Θ v . Assume leaf-node Θ u emits observation X u with likelihood p ( X u = x | Θ u = x ) = 1 . No obs for internal nodes! How to perform HMM calculations on this tree? e.g. p ( X ) 1
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Maximum Posterior Tree Find Θ * that maximizes p ( Θ , X ) . Algorithm? Computational complexity? 2
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Viterbi on Binary Trees Recursion rule for maximizing p ( X ) ? How do we take into account priors? Match to intuitive ancestor inference rules? 3
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Posterior for Ancestral States p ( Θ u | X ) ? p ( X ) ? Really a question of how to split the tree. 4
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“Above-Below” Algorithm Split tree into descendants X u of Θ u and non-descendants X ¬ u . “below”: b ui = p ( X u | Θ u = s i ) “above”: a ui = p ( X ¬ u , Θ u = s i ) p ( Θ u | X ) = p ( X u | Θ u = s i ) p ( X ¬ u , Θ u = s i ) p ( X ) 5
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“Below” Probability Summation Probability of descendants X u of Θ u : b ui = p ( X u | Θ u = s i ) = ˆ j p ( Θ v = s j | Θ u = s i ) b v j k p ( Θ w = s k | Θ u = s i ) b wk ! where Θ v , Θ w are daughter nodes of Θ u . Computational complexity? 6
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Obs Likelihood p ( X ) ? 7
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Observation Likelihood For root node Θ R and prior p ( Θ = s i ) = π i : p ( X ) = i p ( X | Θ R = s i ) p ( Θ R = s i ) p ( X ) = i b Ri π i 8
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Non-Descendants Define Θ p as the parent node of Θ u , Θ q as its “sister” node: a ui = p ( Θ u = s i , X ¬ u ) 9
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