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Fair-Balance Paradox, Star-tree Paradox, and Bayesian Phylogenetics

Fair-Balance Paradox, Star-tree Paradox, and Bayesian Phylogenetics. Ziheng Yang , Mol. Biol. Evol . 2007. Presented by Caroline Uhler and Anna- Sapfo Malaspinas. Outline. What is the star-tree paradox? Simulations Explanation: the fair-coin paradox. Solutions to the star-tree paradox:

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Fair-Balance Paradox, Star-tree Paradox, and Bayesian Phylogenetics

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  1. Fair-Balance Paradox, Star-tree Paradox, and Bayesian Phylogenetics Ziheng Yang , Mol. Biol. Evol. 2007 Presented by Caroline Uhler and Anna-SapfoMalaspinas

  2. Outline • What is the star-tree paradox? • Simulations • Explanation: the fair-coin paradox. • Solutions to the star-tree paradox: • Data size dependent prior • Degenerate-model prior • Discussion

  3. Paradoxes • Star tree paradox 3 species rooted tree. If data is generated using a star tree the probability of each resolved tree does not approch 1/3 in large data sets. • Fair coin paradox Assuming you flip a fair coin n times and observe y number of heads. The posterior P+=P(q > 1/2) does not approach ½ (but rather the uniform distribution).

  4. Simulations P1 P3 P1 P3 P2 P2 Pi: posterior probability of seeing tree topology ti

  5. Solution to the paradox(es) • Specification of the prior: • Data Size-Dependant prior • Degenerate-Model Prior (non zero probability of to the degenerate model)

  6. Fair-coin paradox: Behavior of posterior with data size dependent prior

  7. Star-tree paradox: Standard deviation g = 0 g = 0.5 g = 0.51 g = 0.707 g = 0.8

  8. Fair-coin paradox: Effect of prior a: q0 = 0 g = 2 b: q0 = 0.1 g = 2 c: q0 = 0 p0 = 1/3 d: q0 = 0.1 p0 = 1/3

  9. Discussion • Does the star-tree occur in nature? • Are there other ways of resolving the paradox in practice? • Should priors in existing programs (e.g. MrBayes) be modified accordingly? • Use (features of) the data to define the prior? • Phylogenetics: isthatpriorappropriate in general? • A different approach: Steel and Matsen (2007)

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