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Bayesian Phylogenetics. Bayes Theorem Pr(Tree|Data) = Pr(Data|Tree) x Pr(Tree) Pr(Data)

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Presentation on theme: "Bayesian Phylogenetics. Bayes Theorem Pr(Tree|Data) = Pr(Data|Tree) x Pr(Tree) Pr(Data)"— Presentation transcript:

1 Bayesian Phylogenetics

2 Bayes Theorem Pr(Tree|Data) = Pr(Data|Tree) x Pr(Tree) Pr(Data)

3 Bayes Theorem Pr(Tree) = Prior probability of the tree Pr(Data) = Prior probability of the data –Pr(Data|Tree) over all trees, weighted by their prior probabilities Pr(Data|Tree) = Likelihood of the data given the tree Pr(Tree|Data) = Posterior probability of the tree Pr(Tree|Data) = Pr(Data|Tree) x Pr(Tree) Pr(Data)

4 Assuming the prior probability of two trees (i.e., tree topologies) equal, the ratio of their posterior probabilities equals the ratio of their likelihood scores. True or false? Pr(Tree1|Data) = Pr(Data|Tree1) x Pr(Tree1) Pr(Data) Pr(Tree2|Data) = Pr(Data|Tree2) x Pr(Tree2) Pr(Data) Pr(Tree1|Data) Pr(Tree2|Data) = Pr(Data) Pr(Data|Tree2) x Pr(Tree2) Pr(Data|Tree1) x Pr(Tree1) Pr(Data) ( )( )

5 Assuming the prior probability of two trees (i.e., tree topologies) equal, the ratio of their posterior probabilities equals the ratio of their likelihood scores. True or false? Pr(Tree1|Data) = Pr(Data|Tree1) x Pr(Tree1) Pr(Data) Pr(Tree2|Data) = Pr(Data|Tree2) x Pr(Tree2) Pr(Data) Pr(Tree1|Data) Pr(Tree2|Data) = Pr(Data) Pr(Data|Tree2) x Pr(Tree2) Pr(Data|Tree1) x Pr(Tree1) Pr(Data) ( )( )

6 Assuming the prior probability of two trees (i.e., tree topologies) equal, the ratio of their posterior probabilities equals the ratio of their likelihood scores. True or false? Pr(Tree1|Data) = Pr(Data|Tree1) x Pr(Tree1) Pr(Data) Pr(Tree2|Data) = Pr(Data|Tree2) x Pr(Tree2) Pr(Data) Pr(Tree1|Data) Pr(Tree2|Data) = Pr(Data) Pr(Data|Tree2) x Pr(Tree2) Pr(Data|Tree1) x Pr(Tree1) Pr(Data) ( )( )

7 Assuming the prior probability of two trees (i.e., tree topologies) equal, the ratio of their posterior probabilities equals the ratio of their likelihood scores. True or false? Pr(Tree1|Data) = Pr(Data|Tree1) x Pr(Tree1) Pr(Data) Pr(Tree2|Data) = Pr(Data|Tree2) x Pr(Tree2) Pr(Data) Pr(Tree1|Data) Pr(Tree2|Data) = Pr(Data|Tree1) Pr(Data|Tree2)

8 If tree topology is the parameter of interest, what are some “nuisance parameters” that need to be accommodated in a Bayesian or Maximum Likelihood analysis? Branch lengths Substitution rates Base frequencies Rate heterogeneity parameters

9 How do Bayesian and Likelihood approaches differ in their treatment of nuisance parameters? Likelihood –Find the value of each parameter that maximizes the likelihood of the data Bayesian –Integrate over all possible values of each parameter (weighted by the prior probability distribution)

10 How does MCMC get around the problem of not being able to calculate Pr(Data)? The probability of staying/leaving a tree is determined by its posterior probability relative to other nearby trees The overall time spent on a tree will converge to its absolute posterior probability

11 Like walking-over tree space at a rate governed by altitude..

12 What happens at each “step” of a Bayesian, phylogenetic MCMC analysis? A new parameter (topology, branch lengths, substitution parameters etc.) is proposed Whether the new parameter is accepted is governed by the metropolis-hasting equation Pr(Data|x= j) Pr(Data|x = i Pr(x = j) Pr(x = i Pr(proposing x=i| x=j) Pr(proposing x=j| x=i) xx 1, min Posterior ratioPrior ratioProposal ratio

13 Why is the output of MCMC called a “posterior distribution?” What does it contain? It contains a list of parameters in effect at a sampled set of generations (after burnin) The frequency of a parameter in this sample should be proportional to its posterior probability

14 How can the posterior distribution be queried to find the posterior probability of a clade? Just see the proportion of trees in the distribution that have the clade

15 How can the posterior distribution be queried to evaluate other parameters, for example the transition:transversions ratio? Generate a histogram (or fit to a probability density function) Establish a credibility interval – the range than encompasses, say, 95% of the distribution

16 A dice example A manufacturer makes regular dice and trick dice (with 2 sixes) – in equal numbers You are given a die from this manufacturer and are not allowed to look at all the sides – you can only look at the side that is up after a roll You roll this die 10 times and get these numbers What is the probability that the die is one with two sixes?

17 Bayesian approach Pr(H|D) = Pr(D|H) x Pr(H) Pr(D) 1-six2-sixSum Pr(H) Pr(D|H)

18 Bayesian approach Pr(H|D) = Pr(D|H) x Pr(H) Pr(D) 1-six2-sixSum Pr(H)1.0 Pr(D|H)

19 Bayesian approach Pr(H|D) = Pr(D|H) x Pr(H) Pr(D) 1-six2-sixSum Pr(H)0.5 1.0 Pr(D|H)

20 Bayesian approach Pr(H|D) = Pr(D|H) x Pr(H) Pr(D) 1-six2-sixSum Pr(H)0.5 1.0 Pr(D|H) (1/6) 5 *(5/6) 3 (2/6) 5 *(4/6) 3

21 Bayesian approach Pr(H|D) = Pr(D|H) x Pr(H) Pr(D) 1-six2-sixSum Pr(H)0.5 1.0 Pr(D|H) 7.44E-050.001230.00129

22 Bayesian approach Pr(H|D) = 0.00123 x 0.5= 0.953 0.00129 x 0.5 1-six2-sixSum Pr(H)0.5 1.0 Pr(D|H) 7.44E-050.001230.00129 Pr(H|D) = 0.00006 x 0.5= 0.047 0.00129 x 0.5

23 MCMC approach 1 six2 sixes Pr(Data|x= j) Pr(Data|x = i Pr(x = j) Pr(x = i Pr(proposing x=i| x=j) Pr(proposing x=j| x=i) xx 1, min 1 1

24 Pr(Data|one six) Pr(Data|two sixes MCMC approach 1 six2 sixes 1, min = 1

25 Pr(Data|two sixes) Pr(Data|one six MCMC approach 1 six2 sixes 1, min = Pr(Data|two sixes) Pr(Data|one six = 0.047 If you run this long enough, you will spend only 4.7% of the time on “1 six.” This is the PP of that hypothesis.

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