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:– Data size dependent prior– Degenerate-model prior

• Discussion

Paradoxes

• Star tree paradox3 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 paradoxAssuming 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).

Simulations

P1

P2

P3

P2

P3P1

Pi: posterior probability of seeing tree topology ti

Solution to the paradox(es)

• Specification of the prior:

• Data Size-Dependant prior

• Degenerate-Model Prior (non zero probability of to the degenerate model)

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

Star-tree paradox: Standard deviation

g = 0

g = 0.5

g = 0.51g = 0.707 g = 0.8

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

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)