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

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(  > 1/2) does not approach ½ (but rather the uniform distribution).

4. Simulations P1P1 P2P2 P3P3 P2P2 P3P3 P1P1 P i : posterior probability of seeing tree topology  i

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  = 0  = 0.5  = 0.51  = 0.707  = 0.8

8. Fair-coin paradox: Effect of prior a:  0 = 0  = 2 b:  0 = 0.1  = 2 c:  0 = 0  0 = 1/3 d:  0 = 0.1  0 = 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. Mr Bayes) be modified accordingly? Use (features of) the data to define the prior? Phylogenetics: is that prior appropriate in general? A different approach: Steel and Matsen (2007)