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by: Rupert Davis

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# PHYLOGENETIC INFRNC GENOME 570

Rupert Davis
UW
GPA 3.79

Joseph Felsenstein

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COURSE
PROF.
Joseph Felsenstein
TYPE
Class Notes
PAGES
27
WORDS
KARMA
25 ?

## Popular in Genome Sciences

This 27 page Class Notes was uploaded by Rupert Davis on Wednesday September 9, 2015. The Class Notes belongs to GENOME 570 at University of Washington taught by Joseph Felsenstein in Fall. Since its upload, it has received 18 views. For similar materials see /class/192399/genome-570-university-of-washington in Genome Sciences at University of Washington.

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Date Created: 09/09/15
Week 7 Bayesian inference Testing trees Bootstraps Genome 570 May 2008 Week 7 Bayesian inference Testing trees Bootstraps 7 p154 Bayes Theorem Conditional probability of hypothesis given data is Prob H 84 D Prob H l D Prob D Since Prob H 84 D Prob H Prob D l H Substituting this in Prob H Prob D l H Prob H l D Prob D The denominator Prob D is the sum of the numerators over all possible hypotheses H Prob H PrOb D l H Prob H l D EH Prob H Prob D l H Week 7 Bayesian inference7 Testing trees Bootstraps 7 p254 A dramatic if not real example Example Space probe photos show no Little Green Men on Mars priors 0 likelihoods no o is O of 4 posteriors Week 7 Bayesian inference7 Testing trees Bootstraps 7 p354 Calculations for that example Using the Odds Ratio form of Bayes Theorem Prob H1D Prob DH1 Prob H1 Prob H2D Prob DH2 Prob H2 AH V V posterior likelihood prior odds ratio ratio odds ratio For the odds favoring their existence the calculation is for the optimist about Little Green Men 4 13 43 43 1lt 1 1 While for the pessimist it is lx 112 2112 4 1 1 Week 7 Bayesian inference7 Testing trees Bootstraps 7 p454 With repeated observation the prior matters less If we send 5 space probes and all fail to see LGMs since the probability of this observation is 1 35 if there are LGMs and 1 if there aren t we get for the optimist about Little Green Men 5 X m3 4243 1 1 I lgt while for the pessimist about Little Green Men 15 12 gtlt 71z972 lgtI Week 7 Bayesian inference7 Testing trees Bootstraps 7 p554 A coin tossing example 00 02 04 06 08 10 00 02 04 06 08 10 P P n 00 02 04 06 08 10 00 02 04 06 08 10 00 02 04 06 08 10 00 02 04 06 08 10 P P 11 tosses 5 heads 44 tosses 20 heads Week 7 Bayesian inference7 Testing trees Bootstraps 7 p654 Markov Chain Monte Carlo sampling To draw trees from a distribution whose probabilities are proportional to f t we can use the Metropolis algorithm 1 2 Start at some tree Call this Ti Pick a tree that is a neighbor of this tree in the graph of trees Call this the proposal Tj Compute the ratio of the probabilities or probability density functions of the proposed new tree and the old tree trees If R 2 1 accept the new tree as the current tree If R lt 1 draw a uniform random number a random fraction between 0 and 1 If it is less than R accept the new tree as the current tree Otherwise reject the new tree and continue with tree Ti as the current tree Return to step 2 Week 7 Bayesian inference7 Testing trees Bootstraps 7 p754 Reversibility again Week 7 Bayesian inference Testing trees Bootstraps 7 p854 Does it achieve the desired equilibrium distribution If gt T0 then PI Ob Ti TJ39 I 1 and PI Ob TJ39 Ti I so that PI Ob TJ39 PI Ob the same formula can be shown to hold when fTi lt fTJ Then in both cases 1CTi Prob T1 W Prob Ti T1 fTj Summing over all Ti the righthand side sums up to fTJ so ZN Prob T1 Ti HTJ Ti This shows that applying this algorithm if we start in the desired distribution we stay in it It is also true under most conditions that if you start in any other distribution you converge to this one Week 7 Bayesian inference7 Testing trees Bootstraps 7 p954 Bayesian MCMC We try to achieve the posterior Prob T Prob D l T denominator and this turns out to need the acceptance ratio Prob T new Prob D l TneW Prob Told Prob D l Told R the denominators are the same and cancel out This is a great convenience as we often cannot evaluate the deonominator but we can usually evaluate the numerators Note that we could also have a prior on model parameters too Week 7 Bayesian inference Testing trees Bootstraps 7 p1054 Mau and Newton s proposal mechanism ACDFBE BDFCAE BDFCAE m gt I gt 39L39 BDFCAE BDFCAE Tim D gt Pig Week 7 Bayesian inference Testing trees Bootstraps 7 p1154 0 m o o gt O m Using MrBayes on the primates data 0416 0856 0938 0899 0949 0999 100 100 0905 0997 0986 Bovine Lemur Tarsier Crab EMac Rhesus Mac Jpn Macaq BarbMacaq Orang Chimp Human Gorilla Gibbon Squir Monk Mouse Frequencies of partitions posterior clade probabilities Week 7 Bayesian inference Testing trees Bootstraps 7 p1254 An example Suppose we have two species with a JukesCantor model so that the estimation of the unrooted tree is simply the estimation of the branch length between the two species We can express the result either as branch length t or as the net probability of base change p 2 1 exp t Week 7 Bayesian inference Testing trees Bootstraps 7 p1354 A at prior on p 00 025 05 P 075 Week 7 Bayesian inference Testing trees7 Bootstraps 7 p1454 The corresponding prior on t Week 7 Bayesian inference Testing trees Bootstraps 7 p1554 Flat prior for t between 0 and 5 Week 7 Bayesian inference Testing trees Bootstraps 7 p1654 The corresponding prior on p 00 025 05 P 075 Week 7 Bayesian inference Testing trees Bootstraps 7 p1754 Likelihood curve for t 6 8 A s1 10 12 140 39 392 393 4 5 L t t0383112 1903 When 3 sites differ out of 10 Week 7 Bayesian inference Testing trees Bootstraps 7 p1854 Likelihood curve for p 6 8 A q 10 v 12 4400 02 04 06 08 1903 A t 0383112 When 3 sites differ out of 10 Week 7 Bayesian inference Testing trees Bootstraps 7 p1954 When t has a wide at prior T The 95 twotailed credible interval for t with various truncation points on a at prior for t Week 7 Bayesian inference Testing trees Bootstraps 7 132054 What is going on in that case is p T is so large this is lt 25 of the area Week 7 Bayesian inference Testing trees Bootstraps 7 p2154 The likelihood curve is nearly a normal distribution for large amounts of data tX the quotsufficient statisticquot Week 7 Bayesian inference Testing trees Bootstraps 7 p2254 Curvatures and covariances of ML estimates ML estimates have covariances computable from curvatures of the expected loglikelihood Var 3 2 1 W The same is true when there are multiple parameters 39Var cz39V72 43l 8210gL CU E M where Week 7 Bayesian inference Testing trees Bootstraps 7 p2354 With large amounts of data asymptotically When the true value of 6 is 60 A 6 60 Since 1v is the negative of the curvature of the loglikelihood 90 2 A 1 lnL60 ln L6 E V so that twice the difference of loglikelihoods is the square of a normal 21nL 1nL60 Xi Week 7 Bayesian inference Testing trees Bootstraps 7 p2454 Corresponding results for multiple parameters In L60 g lnL60 g 60 6T C 60 6 6 60TClt6 60 XE so that the loglikelihood difference is 21nL 1nL60 XE When q of the p parameters are constrained 21nL 1nL60 X3 Week 7 Bayesian inference Testing trees Bootstraps 7 p2554 Likelihood ratio interval for a parameter lnL 2620 2625 2630 2635 2640 5 10 20 50 100 200 Transition l transversion ratio Week 7 Bayesian inference Testing trees Bootstraps 7 p2654

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