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Islands and Integrals

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Islands and Integrals
Processes of Diversification in an Island Archipelago and
Bayesian Methods of Comparative Phylogeogra...

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Southeast Asia
Islands and Integrals J. Oaks, University of Kansas 2/53

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Southeast Asia
Islands and Integrals J. Oaks, University of Kansas 3/53

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Islands and Integrals

  1. 1. Islands and Integrals Processes of Diversification in an Island Archipelago and Bayesian Methods of Comparative Phylogeographical Model Choice Jamie R. Oaks1 1Department of Ecology and Evolutionary Biology, University of Kansas October 16, 2013 Islands and Integrals J. Oaks, University of Kansas 1/53
  2. 2. Southeast Asia Islands and Integrals J. Oaks, University of Kansas 2/53
  3. 3. Southeast Asia Islands and Integrals J. Oaks, University of Kansas 3/53
  4. 4. Philippine Archipelago Islands and Integrals J. Oaks, University of Kansas 4/53
  5. 5. Philippine Archipelago Islands and Integrals J. Oaks, University of Kansas 4/53
  6. 6. Climate-driven diversification model Repeated coalescence and fragmentation of island complexes Islands and Integrals J. Oaks, University of Kansas 5/53
  7. 7. Climate-driven diversification model Repeated coalescence and fragmentation of island complexes Prominent paradigm for explaining Philippine biodiversity Islands and Integrals J. Oaks, University of Kansas 5/53
  8. 8. Climate-driven diversification model Repeated coalescence and fragmentation of island complexes Prominent paradigm for explaining Philippine biodiversity Proposed as model of diversification Islands and Integrals J. Oaks, University of Kansas 5/53
  9. 9. Testing climate-driven diversification Did repeated fragmentation of islands during inter-glacial rises in sea level promote diversification? Islands and Integrals J. Oaks, University of Kansas 6/53
  10. 10. Testing climate-driven diversification Did repeated fragmentation of islands during inter-glacial rises in sea level promote diversification? Model has testable prediction: Temporally clustered divergences among taxa co-distributed across fragmented islands Islands and Integrals J. Oaks, University of Kansas 6/53
  11. 11. Climate-driven model: Prediction 0100200300400500 Time (kya) 0 -50 -100 Sealevel(m) Islands and Integrals J. Oaks, University of Kansas 7/53
  12. 12. Climate-driven model: Prediction 0100200300400500 Time (kya) 0 -50 -100 Sealevel(m) Islands and Integrals J. Oaks, University of Kansas 7/53
  13. 13. Climate-driven model: Prediction T2 T3 T5 τ2 τ1 T1 T4 0100200300400500 Time (kya) 0 -50 -100 Sealevel(m) Islands and Integrals J. Oaks, University of Kansas 7/53
  14. 14. Divergence model choice T2 T3 T5 τ2 τ1 T1 T4 0100200300400500 Time (kya) 0 -50 -100 Sealevel(m) Islands and Integrals J. Oaks, University of Kansas 8/53
  15. 15. Divergence model choice T = (T1, T2, T3, T4, T5) τ = {τ1, τ2} |τ| = 2 T2 T3 T5 τ2 τ1 T1 T4 0100200300400500 Time (kya) 0 -50 -100 Sealevel(m) Islands and Integrals J. Oaks, University of Kansas 8/53
  16. 16. Divergence model choice T = (330, 330, 125, 125, 125) τ = {125, 330} |τ| = 2 T2 T3 T5 τ2 τ1 T1 T4 0100200300400500 Time (kya) 0 -50 -100 Sealevel(m) Islands and Integrals J. Oaks, University of Kansas 8/53
  17. 17. Divergence model choice T = (330, 330, 125, 330, 125) τ = {125, 330} |τ| = 2 T2 T3 T5 τ2 τ1 T1 T4 0100200300400500 Time (kya) 0 -50 -100 Sealevel(m) Islands and Integrals J. Oaks, University of Kansas 8/53
  18. 18. Divergence model choice T = (375, 330, 125, 330, 125) τ = {125, 330, 375} |τ| = 3 T2 T3 T5 τ2 τ1 T1 τ3 T4 0100200300400500 Time (kya) 0 -50 -100 Sealevel(m) Islands and Integrals J. Oaks, University of Kansas 8/53
  19. 19. Divergence model choice T = (T1, T2, T3, T4, T5) τ = {τ1, τ2, τ3} |τ| = 3 T2 T3 T5 τ2 τ1 T1 τ3 T4 0100200300400500 Time (kya) 0 -50 -100 Sealevel(m) Islands and Integrals J. Oaks, University of Kansas 8/53
  20. 20. Divergence model choice T = (T1, T2, . . . , TY) τ = {τ1, . . . , τ|τ|} |τ| T2 T3 T5 τ2 τ1 T1 τ3 T4 0100200300400500 Time (kya) 0 -50 -100 Sealevel(m) Islands and Integrals J. Oaks, University of Kansas 8/53
  21. 21. Divergence model choice T = (T1, T2, . . . , TY) τ = {τ1, . . . , τ|τ|} |τ| We want to infer T given DNA sequence alignments X Islands and Integrals J. Oaks, University of Kansas 8/53
  22. 22. Divergence model choice T = (T1, T2, . . . , TY) τ = {τ1, . . . , τ|τ|} |τ| We want to infer T given DNA sequence alignments X p(T | X) = p(X | T)p(T) p(X) Islands and Integrals J. Oaks, University of Kansas 8/53
  23. 23. Divergence model choice T = (T1, T2, . . . , TY) τ = {τ1, . . . , τ|τ|} |τ| We want to infer T given DNA sequence alignments X p(T | X) = p(X | T)p(T) p(X) This approach implemented in msBayes Islands and Integrals J. Oaks, University of Kansas 8/53
  24. 24. Divergence model choice T = (T1, T2, . . . , TY) τ = {τ1, . . . , τ|τ|} |τ| We want to infer T given DNA sequence alignments X p(T | X) = p(X | T)p(T) p(X) This approach implemented in msBayes Not that simple Islands and Integrals J. Oaks, University of Kansas 8/53
  25. 25. The msBayes model T2 T3 T5 τ2 τ1 T1 T4 0100200300400500 Time (kya) 0 -50 -100 Sealevel(m) Islands and Integrals J. Oaks, University of Kansas 9/53
  26. 26. The msBayes model T1 T2 τ2 0100200300400500 Time (kya) 0 -50 -100 Sealevel(m) Islands and Integrals J. Oaks, University of Kansas 9/53
  27. 27. The msBayes model T1 T2 τ2 0100200300400500 Time (kya) 0 -50 -100 Sealevel(m) Islands and Integrals J. Oaks, University of Kansas 9/53
  28. 28. The msBayes model T1 T2 τ2 0100200300400500 Time (kya) 0 -50 -100 Sealevel(m) Islands and Integrals J. Oaks, University of Kansas 10/53
  29. 29. The msBayes model X Sequence alignments G Gene trees T Divergence times Θ Demographic parameters T1 T2 τ2 0100200300400500 Time (kya) 0 -50 -100 Sealevel(m) Islands and Integrals J. Oaks, University of Kansas 10/53
  30. 30. The msBayes model X Sequence alignments G Gene trees T Divergence times Θ Demographic parameters Islands and Integrals J. Oaks, University of Kansas 11/53
  31. 31. The msBayes model Full Model: p(G, T, Θ | X) = p(X | G, T, Θ)p(G, T, Θ) p(X) X Sequence alignments G Gene trees T Divergence times Θ Demographic parameters Islands and Integrals J. Oaks, University of Kansas 11/53
  32. 32. The msBayes model Full Model: p(G, T, Θ | X) = p(X | G, T, Θ)p(G, T, Θ) p(X) p(G, T, θA, θD1, θD2, τB, ζD1, ζD2, m, α, υ | X, φ, ρ, ν) = 1 p(X) p(T)f (α) Y i=1 p(θA,i )p(θD1,i , θD2,i )p(τB,i )p(ζD1,i )f (ζD2,i )p(mi ) ki j=1 p(Xi,j | Gi,j , φi,j )p(Gi,j | Ti , θA,i , θD1,i , θD2,i , ρi,j , νi,j , υj , τB,i , ζD1,i , ζD2,i , mi ) K j=1 f (υj | α) Islands and Integrals J. Oaks, University of Kansas 11/53
  33. 33. The msBayes model Full Model: p(G, T, Θ | X) = p(X | G, T, Θ)p(G, T, Θ) p(X) Approximate Bayesian computation (ABC) X → S∗ → B (S∗ ) Islands and Integrals J. Oaks, University of Kansas 11/53
  34. 34. The msBayes model Full Model: p(G, T, Θ | X) = p(X | G, T, Θ)p(G, T, Θ) p(X) Approximate Bayesian computation (ABC) X → S∗ → B (S∗ ) Approximate Model: p(G, T, Θ | B (S∗ )) = p(X | G, T, Θ)p(G, T, Θ) p(B (S∗ )) Islands and Integrals J. Oaks, University of Kansas 11/53
  35. 35. The msBayes model Full Model: p(G, T, Θ | X) = p(X | G, T, Θ)p(G, T, Θ) p(X) T Vector of divergence times across pairs of populations |τ| Number of divergence parameters DT The variance of T Islands and Integrals J. Oaks, University of Kansas 11/53
  36. 36. Species n1 n2 Mammals Crocidura beatus 12 11 Crocidura negrina-panayensis 12 6 Hipposideros obscurus 19 9 Hipposideros pygmaeus 3 12 Cynopterus brachyotis 20 8 Cynopterus brachyotis 8 14 Haplonycteris fischeri 29 8 Haplonycteris fischeri 9 21 Macroglossus minimus 19 4 Macroglossus minimus 8 10 Ptenochirus jagori 4 7 Ptenochirus jagori 8 8 Ptenochirus minor 30 9 Squamates Cyrtodactylus gubaot-sumuroi 29 6 Cyrtodactylus annulatus 14 3 Cyrtodactylus philippinicus 6 14 Gekko mindorensis 8 11 Insulasaurus arborens 22 10 Pinoyscincus jagori 8 8 Dendrelaphis marenae 6 6 Anurans Limnonectes leytensis 4 2 Limnonectes magnus 2 3 Islands and Integrals J. Oaks, University of Kansas 12/53
  37. 37. Species n1 n2 Mammals Crocidura beatus 12 11 Crocidura negrina-panayensis 12 6 Hipposideros obscurus 19 9 Hipposideros pygmaeus 3 12 Cynopterus brachyotis 20 8 Cynopterus brachyotis 8 14 Haplonycteris fischeri 29 8 Haplonycteris fischeri 9 21 Macroglossus minimus 19 4 Macroglossus minimus 8 10 Ptenochirus jagori 4 7 Ptenochirus jagori 8 8 Ptenochirus minor 30 9 Squamates Cyrtodactylus gubaot-sumuroi 29 6 Cyrtodactylus annulatus 14 3 Cyrtodactylus philippinicus 6 14 Gekko mindorensis 8 11 Insulasaurus arborens 22 10 Pinoyscincus jagori 8 8 Dendrelaphis marenae 6 6 Anurans Limnonectes leytensis 4 2 Limnonectes magnus 2 3 Islands and Integrals J. Oaks, University of Kansas 13/53
  38. 38. Species n1 n2 Mammals Crocidura beatus 12 11 Crocidura negrina-panayensis 12 6 Hipposideros obscurus 19 9 Hipposideros pygmaeus 3 12 Cynopterus brachyotis 20 8 Cynopterus brachyotis 8 14 Haplonycteris fischeri 29 8 Haplonycteris fischeri 9 21 Macroglossus minimus 19 4 Macroglossus minimus 8 10 Ptenochirus jagori 4 7 Ptenochirus jagori 8 8 Ptenochirus minor 30 9 Squamates Cyrtodactylus gubaot-sumuroi 29 6 Cyrtodactylus annulatus 14 3 Cyrtodactylus philippinicus 6 14 Gekko mindorensis 8 11 Insulasaurus arborens 22 10 Pinoyscincus jagori 8 8 Dendrelaphis marenae 6 6 Anurans Limnonectes leytensis 4 2 Limnonectes magnus 2 3 Islands and Integrals J. Oaks, University of Kansas 13/53
  39. 39. Species n1 n2 Mammals Crocidura beatus 12 11 Crocidura negrina-panayensis 12 6 Hipposideros obscurus 19 9 Hipposideros pygmaeus 3 12 Cynopterus brachyotis 20 8 Cynopterus brachyotis 8 14 Haplonycteris fischeri 29 8 Haplonycteris fischeri 9 21 Macroglossus minimus 19 4 Macroglossus minimus 8 10 Ptenochirus jagori 4 7 Ptenochirus jagori 8 8 Ptenochirus minor 30 9 Squamates Cyrtodactylus gubaot-sumuroi 29 6 Cyrtodactylus annulatus 14 3 Cyrtodactylus philippinicus 6 14 Gekko mindorensis 8 11 Insulasaurus arborens 22 10 Pinoyscincus jagori 8 8 Dendrelaphis marenae 6 6 Anurans Limnonectes leytensis 4 2 Limnonectes magnus 2 3 Islands and Integrals J. Oaks, University of Kansas 13/53
  40. 40. Species n1 n2 Mammals Crocidura beatus 12 11 Crocidura negrina-panayensis 12 6 Hipposideros obscurus 19 9 Hipposideros pygmaeus 3 12 Cynopterus brachyotis 20 8 Cynopterus brachyotis 8 14 Haplonycteris fischeri 29 8 Haplonycteris fischeri 9 21 Macroglossus minimus 19 4 Macroglossus minimus 8 10 Ptenochirus jagori 4 7 Ptenochirus jagori 8 8 Ptenochirus minor 30 9 Squamates Cyrtodactylus gubaot-sumuroi 29 6 Cyrtodactylus annulatus 14 3 Cyrtodactylus philippinicus 6 14 Gekko mindorensis 8 11 Insulasaurus arborens 22 10 Pinoyscincus jagori 8 8 Dendrelaphis marenae 6 6 Anurans Limnonectes leytensis 4 2 Limnonectes magnus 2 3 Islands and Integrals J. Oaks, University of Kansas 13/53
  41. 41. Species n1 n2 Mammals Crocidura beatus 12 11 Crocidura negrina-panayensis 12 6 Hipposideros obscurus 19 9 Hipposideros pygmaeus 3 12 Cynopterus brachyotis 20 8 Cynopterus brachyotis 8 14 Haplonycteris fischeri 29 8 Haplonycteris fischeri 9 21 Macroglossus minimus 19 4 Macroglossus minimus 8 10 Ptenochirus jagori 4 7 Ptenochirus jagori 8 8 Ptenochirus minor 30 9 Squamates Cyrtodactylus gubaot-sumuroi 29 6 Cyrtodactylus annulatus 14 3 Cyrtodactylus philippinicus 6 14 Gekko mindorensis 8 11 Insulasaurus arborens 22 10 Pinoyscincus jagori 8 8 Dendrelaphis marenae 6 6 Anurans Limnonectes leytensis 4 2 Limnonectes magnus 2 3 Islands and Integrals J. Oaks, University of Kansas 13/53
  42. 42. Species n1 n2 Mammals Crocidura beatus 12 11 Crocidura negrina-panayensis 12 6 Hipposideros obscurus 19 9 Hipposideros pygmaeus 3 12 Cynopterus brachyotis 20 8 Cynopterus brachyotis 8 14 Haplonycteris fischeri 29 8 Haplonycteris fischeri 9 21 Macroglossus minimus 19 4 Macroglossus minimus 8 10 Ptenochirus jagori 4 7 Ptenochirus jagori 8 8 Ptenochirus minor 30 9 Squamates Cyrtodactylus gubaot-sumuroi 29 6 Cyrtodactylus annulatus 14 3 Cyrtodactylus philippinicus 6 14 Gekko mindorensis 8 11 Insulasaurus arborens 22 10 Pinoyscincus jagori 8 8 Dendrelaphis marenae 6 6 Anurans Limnonectes leytensis 4 2 Limnonectes magnus 2 3 Islands and Integrals J. Oaks, University of Kansas 13/53
  43. 43. Empirical results Strong support for simultaneous divergence of all 22 taxon pairs pp > 0.96 ∼100,000–250,000 years ago Islands and Integrals J. Oaks, University of Kansas 14/53
  44. 44. Simulation-based power analyses What is “simultaneous”? Islands and Integrals J. Oaks, University of Kansas 15/53
  45. 45. Simulation-based power analyses What is “simultaneous”? Simulate datasets in which all 22 divergence times are random Islands and Integrals J. Oaks, University of Kansas 15/53
  46. 46. Simulation-based power analyses What is “simultaneous”? Simulate datasets in which all 22 divergence times are random τ ∼ U(0, 0.5 MGA) τ ∼ U(0, 1.5 MGA) τ ∼ U(0, 2.5 MGA) τ ∼ U(0, 5.0 MGA) MGA = Millions of Generations Ago Islands and Integrals J. Oaks, University of Kansas 15/53
  47. 47. Simulation-based power analyses What is “simultaneous”? Simulate datasets in which all 22 divergence times are random τ ∼ U(0, 0.5 MGA) τ ∼ U(0, 1.5 MGA) τ ∼ U(0, 2.5 MGA) τ ∼ U(0, 5.0 MGA) MGA = Millions of Generations Ago Simulate 1000 datasets for each τ distribution Analyze all 4000 datasets as we did the empirical data Islands and Integrals J. Oaks, University of Kansas 15/53
  48. 48. Simulation-based power analyses: Results 1 3 5 7 9 11 13 15 17 19 210.0 0.2 0.4 0.6 0.8 1.0 p( ˆ|τ| =1)=1.0 τ∼U(0, 0.5 MGA) 1 3 5 7 9 11 13 15 17 19 210.0 0.2 0.4 0.6 0.8 1.0 p( ˆ|τ| =1)=1.0 τ∼U(0, 1.5 MGA) 1 3 5 7 9 11 13 15 17 19 210.0 0.2 0.4 0.6 0.8 1.0 p( ˆ|τ| =1)=1.0 τ∼U(0, 2.5 MGA) 1 3 5 7 9 11 13 15 17 19 210.0 0.2 0.4 0.6 0.8 1.0 p( ˆ|τ| =1)=1.0 τ∼U(0, 5.0 MGA) Estimated number of divergence events (mode) Density Islands and Integrals J. Oaks, University of Kansas 16/53
  49. 49. Simulation-based power analyses: Results 1 3 5 7 9 11 13 15 17 19 210.0 0.2 0.4 0.6 0.8 1.0 p( ˆ|τ| =1)=1.0 τ∼U(0, 0.5 MGA) 1 3 5 7 9 11 13 15 17 19 210.0 0.2 0.4 0.6 0.8 1.0 p( ˆ|τ| =1)=1.0 τ∼U(0, 1.5 MGA) 1 3 5 7 9 11 13 15 17 19 210.0 0.2 0.4 0.6 0.8 1.0 p( ˆ|τ| =1)=1.0 τ∼U(0, 2.5 MGA) 1 3 5 7 9 11 13 15 17 19 210.0 0.2 0.4 0.6 0.8 1.0 p( ˆ|τ| =1)=1.0 τ∼U(0, 5.0 MGA) Estimated number of divergence events (mode) Density 0.05 0.25 0.45 0.65 0.850 5 10 15 20 0.05 0.25 0.45 0.65 0.850 5 10 15 20 0.05 0.25 0.45 0.65 0.850 2 4 6 8 10 12 0.05 0.25 0.45 0.65 0.850 2 4 6 8 10 Posterior probability of one divergence Density Islands and Integrals J. Oaks, University of Kansas 16/53
  50. 50. Simulation-based power analyses: Results Strong support for highly clustered divergences when divergence times are random over 5 million generations Our empirical results are likely spurious Islands and Integrals J. Oaks, University of Kansas 17/53
  51. 51. Why the bias? Potential causes of the bias: 1. The prior on divergence models 2. Broad uniform priors on many of the model’s parameters, including divergence times Islands and Integrals J. Oaks, University of Kansas 18/53
  52. 52. Causes of bias: Prior on divergence models T = (375, 330, 125, 330, 125) τ = {125, 330, 375} |τ| = 3 T2 T3 T5 τ2 τ1 T1 τ3 T4 0100200300400500 Time (kya) 0 -50 -100 Sealevel(m) Islands and Integrals J. Oaks, University of Kansas 19/53
  53. 53. Causes of bias: Prior on divergence models msBayes uses a discrete uniform prior on the number of divergence events, |τ| #ofdivergencemodels 020406080100120 1 3 5 7 9 11 13 15 17 19 21 A p(M|τ|,i) 0.000.010.020.030.04 1 3 5 7 9 11 13 15 17 19 21 B # of divergence events, |τ| Islands and Integrals J. Oaks, University of Kansas 20/53
  54. 54. Causes of bias: Broad priors msBayes uses uniform priors on most model parameters, including divergence times Islands and Integrals J. Oaks, University of Kansas 21/53
  55. 55. Causes of bias: Broad priors msBayes uses uniform priors on most model parameters, including divergence times This requires the use of broad priors Islands and Integrals J. Oaks, University of Kansas 21/53
  56. 56. Causes of bias: Broad priors msBayes uses uniform priors on most model parameters, including divergence times This requires the use of broad priors Models with more divergence-time parameters have much greater parameter space, much of it with low likelihood Islands and Integrals J. Oaks, University of Kansas 21/53
  57. 57. Causes of bias: Broad priors msBayes uses uniform priors on most model parameters, including divergence times This requires the use of broad priors Models with more divergence-time parameters have much greater parameter space, much of it with low likelihood This vast space can cause problems with Bayesian model choice Reduced marginal likelihoods Islands and Integrals J. Oaks, University of Kansas 21/53
  58. 58. Causes of bias: Marginal likelihoods p(X) = θ p(X | θ)p(θ)dθ Islands and Integrals J. Oaks, University of Kansas 22/53
  59. 59. Causes of bias: Marginal likelihoods p(X) = θ p(X | θ)p(θ)dθ 0.0 0.2 0.4 0.6 0.8 1.0 θ 0 5 10 15 20 25 30 Density p(X| θ) Islands and Integrals J. Oaks, University of Kansas 22/53
  60. 60. Causes of bias: Marginal likelihoods p(X) = θ p(X | θ)p(θ)dθ 0.0 0.2 0.4 0.6 0.8 1.0 θ 0 5 10 15 20 25 30 Density p(X| θ) p(θ) Islands and Integrals J. Oaks, University of Kansas 22/53
  61. 61. Causes of bias: Marginal likelihoods Islands and Integrals J. Oaks, University of Kansas 23/53
  62. 62. Causes of bias: Marginal likelihoods Islands and Integrals J. Oaks, University of Kansas 23/53
  63. 63. Causes of bias: Marginal likelihoods Islands and Integrals J. Oaks, University of Kansas 23/53
  64. 64. Causes of bias: Marginal likelihoods p(θ | X) = p(X | θ)p(θ) p(X) p(X) = θ p(X | θ)p(θ)dθ Islands and Integrals J. Oaks, University of Kansas 24/53
  65. 65. Causes of bias: Marginal likelihoods p(θ1 | X, M1) = p(X | θ1, M1)p(θ1 | M1) p(X | M1) p(X | M1) = θ1 p(X | θ1, M1)p(θ | M1)dθ1 Islands and Integrals J. Oaks, University of Kansas 24/53
  66. 66. Causes of bias: Marginal likelihoods p(θ1 | X, M1) = p(X | θ1, M1)p(θ1 | M1) p(X | M1) p(X | M1) = θ1 p(X | θ1, M1)p(θ | M1)dθ1 p(M1 | X) = p(X | M1)p(M1) p(X | M1)p(M1) + p(X | M2)p(M2) Islands and Integrals J. Oaks, University of Kansas 24/53
  67. 67. Causes of bias: Marginal likelihoods Predictions: Posterior estimates should be sensitive to priors As prior converges to distribution underlying the data, the bias should disappear Islands and Integrals J. Oaks, University of Kansas 25/53
  68. 68. Causes of bias: Marginal likelihoods Predictions: Posterior estimates should be sensitive to priors As prior converges to distribution underlying the data, the bias should disappear Testing prior sensitivity: Islands and Integrals J. Oaks, University of Kansas 25/53
  69. 69. Causes of bias: Marginal likelihoods Predictions: Posterior estimates should be sensitive to priors As prior converges to distribution underlying the data, the bias should disappear Testing prior sensitivity: 1. Analyze empirical data under several different prior settings Results are very sensitive Islands and Integrals J. Oaks, University of Kansas 25/53
  70. 70. Causes of bias: Marginal likelihoods Predictions: Posterior estimates should be sensitive to priors As prior converges to distribution underlying the data, the bias should disappear Testing prior sensitivity: 1. Analyze empirical data under several different prior settings Results are very sensitive 2. Use simulations to assess behavior when priors are correct Islands and Integrals J. Oaks, University of Kansas 25/53
  71. 71. Simulation results: Performance when priors are correct 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Posterior probability of one divergence Trueprobabilityofonedivergence msBayes performs well when all assumptions are met Islands and Integrals J. Oaks, University of Kansas 26/53
  72. 72. Causes of bias: Marginal likelihoods Predictions: Posterior estimates should be sensitive to priors As prior converges to distribution underlying the data, the bias should disappear Testing prior sensitivity: 1. Analyze empirical data under several different prior settings Results are very sensitive 2. Use simulations to assess behavior when priors are correct Islands and Integrals J. Oaks, University of Kansas 27/53
  73. 73. Causes of bias: Marginal likelihoods Predictions: Posterior estimates should be sensitive to priors As prior converges to distribution underlying the data, the bias should disappear Testing prior sensitivity: 1. Analyze empirical data under several different prior settings Results are very sensitive 2. Use simulations to assess behavior when priors are correct 3. Use simulations to assess behavior under “ideal” real-world priors Islands and Integrals J. Oaks, University of Kansas 27/53
  74. 74. Simulation results: Power with informed priors 1 3 5 7 9 11 13 15 17 19 210.0 0.2 0.4 0.6 0.8 1.0 p( ˆ|τ| =1)=1.0 τ∼U(0, 0.5 MGA) 1 3 5 7 9 11 13 15 17 19 210.0 0.2 0.4 0.6 0.8 1.0 p( ˆ|τ| =1)=1.0 τ∼U(0, 1.5 MGA) 1 3 5 7 9 11 13 15 17 19 210.0 0.2 0.4 0.6 0.8 1.0 p( ˆ|τ| =1)=1.0 τ∼U(0, 2.5 MGA) 1 3 5 7 9 11 13 15 17 19 210.0 0.2 0.4 0.6 0.8 1.0 p( ˆ|τ| =1)=1.0 τ∼U(0, 5.0 MGA) Estimated number of divergence events (mode) Density 1 3 5 7 9 11 13 15 17 19 210.0 0.2 0.4 0.6 0.8 1.0 p( ˆ|τ| =1)=1.0 1 3 5 7 9 11 13 15 17 19 210.0 0.2 0.4 0.6 0.8 1.0 p( ˆ|τ| =1)=1.0 1 3 5 7 9 11 13 15 17 19 210.0 0.2 0.4 0.6 0.8 1.0 p( ˆ|τ| =1)=0.997 1 3 5 7 9 11 13 15 17 19 210.0 0.1 0.2 0.3 0.4 0.5 0.6 p( ˆ|τ| =1)=0.473 Estimated number of divergence events (mode) Density Islands and Integrals J. Oaks, University of Kansas 28/53
  75. 75. Simulation results: Power with informed priors 0.05 0.25 0.45 0.65 0.850 5 10 15 20 τ∼U(0, 0.5 MGA) 0.05 0.25 0.45 0.65 0.850 5 10 15 20 τ∼U(0, 1.5 MGA) 0.05 0.25 0.45 0.65 0.850 2 4 6 8 10 12 τ∼U(0, 2.5 MGA) 0.05 0.25 0.45 0.65 0.850 2 4 6 8 10 τ∼U(0, 5.0 MGA) Posterior probability of one divergence Density 0.05 0.25 0.45 0.65 0.850 2 4 6 8 10 12 14 0.05 0.25 0.45 0.65 0.850 1 2 3 4 5 6 7 8 9 0.05 0.25 0.45 0.65 0.850 1 2 3 4 5 6 0.05 0.25 0.45 0.65 0.850.0 0.5 1.0 1.5 2.0 Posterior probability of one divergence Density Islands and Integrals J. Oaks, University of Kansas 29/53
  76. 76. Causes of bias: Simulation results Broad uniform priors are reducing marginal likelihoods of models with more divergence events Even when uniform priors are informed by the data the bias remains Islands and Integrals J. Oaks, University of Kansas 30/53
  77. 77. Causes of bias: Simulation results Broad uniform priors are reducing marginal likelihoods of models with more divergence events Even when uniform priors are informed by the data the bias remains Potential solution: More flexible priors Islands and Integrals J. Oaks, University of Kansas 30/53
  78. 78. Mitigating the bias Potential solution: More flexible priors 0.0 0.2 0.4 0.6 0.8 1.0 θ 0 5 10 15 20 25 30 Density p(X| θ) p(θ) Islands and Integrals J. Oaks, University of Kansas 31/53
  79. 79. Mitigating the bias Potential solution: More flexible priors 0.0 0.2 0.4 0.6 0.8 1.0 θ 0 5 10 15 20 25 30 Density p(X| θ) p(θ) Islands and Integrals J. Oaks, University of Kansas 31/53
  80. 80. Mitigating the bias Potential solution: More flexible priors #ofdivergencemodels 020406080100120 1 3 5 7 9 11 13 15 17 19 21 A p(M|τ|,i) 0.000.010.020.030.04 1 3 5 7 9 11 13 15 17 19 21 B # of divergence events, |τ| Islands and Integrals J. Oaks, University of Kansas 31/53
  81. 81. Mitigating the bias Potential solution: More flexible priors Potential solution: Alternative prior over divergence models (e.g., uniform or Dirichlet process) Islands and Integrals J. Oaks, University of Kansas 31/53
  82. 82. New method: dpp-msbayes Reparameterized the model implemented in msBayes Islands and Integrals J. Oaks, University of Kansas 32/53
  83. 83. New method: dpp-msbayes Reparameterized the model implemented in msBayes Replaced uniform priors on continuous parameters with gamma and beta distributions Islands and Integrals J. Oaks, University of Kansas 32/53
  84. 84. New method: dpp-msbayes Reparameterized the model implemented in msBayes Replaced uniform priors on continuous parameters with gamma and beta distributions Dirichlet process prior (DPP) over all possible discrete divergence models Islands and Integrals J. Oaks, University of Kansas 32/53
  85. 85. New method: dpp-msbayes Reparameterized the model implemented in msBayes Replaced uniform priors on continuous parameters with gamma and beta distributions Dirichlet process prior (DPP) over all possible discrete divergence models Uniform prior over divergence models Islands and Integrals J. Oaks, University of Kansas 32/53
  86. 86. dpp-msbayes: Simulation-based assessment Simulate 50,000 datasets under four models MmsBayes U-shaped prior on divergence models Uniform priors on continuous parameters MUshaped U-shaped prior on divergence models Gamma priors on continuous parameters MUniform Uniform prior on divergence models Gamma priors on continuous parameters MDPP DPP prior on divergence models Gamma priors on continuous parameters Analyze all datasets under each of the models Islands and Integrals J. Oaks, University of Kansas 33/53
  87. 87. dpp-msbayes: Simulation-based assessment Assess power Simulate datasets in which all 22 divergence times are random τ ∼ U(0, 0.5 MGA) τ ∼ U(0, 1.5 MGA) τ ∼ U(0, 2.5 MGA) τ ∼ U(0, 5.0 MGA) MGA = Millions of Generations Ago Simulate 1000 datasets for each τ distribution Analyze all 4000 datasets as we did the empirical data Islands and Integrals J. Oaks, University of Kansas 34/53
  88. 88. dpp-msbayes: Simulation results 0.0 0.2 0.4 0.6 0.8 1.0 MmsBayes MDPP MmsBayes 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 MDPP Posterior probability of one divergence Trueprobabilityofonedivergence Analysismodel Data model Islands and Integrals J. Oaks, University of Kansas 35/53
  89. 89. dpp-msbayes: Simulation results 0.0 0.2 0.4 0.6 0.8 1.0 MmsBayes MDPP MUniform MUshaped MmsBayes 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 MDPP Posterior probability of one divergence Trueprobabilityofonedivergence Analysismodel Data model Islands and Integrals J. Oaks, University of Kansas 36/53
  90. 90. dpp-msbayes: Simulation results 1 3 5 7 9 11 13 15 17 19 210.0 0.2 0.4 0.6 0.8 1.0 p( ˆ|τ| =1)=1.0 τ∼U(0, 0.5 MGA) 1 3 5 7 9 11 13 15 17 19 210.0 0.2 0.4 0.6 0.8 1.0 p( ˆ|τ| =1)=1.0 τ∼U(0, 1.5 MGA) 1 3 5 7 9 11 13 15 17 19 210.0 0.2 0.4 0.6 0.8 1.0 p( ˆ|τ| =1)=0.999 τ∼U(0, 2.5 MGA) 1 3 5 7 9 11 13 15 17 19 210.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 p( ˆ|τ| =1)=0.83 τ∼U(0, 5.0 MGA) MmsBayes Estimated number of divergence events (mode) Density 1 3 5 7 9 11 13 15 17 19 210.0 0.2 0.4 0.6 0.8 1.0 p( ˆ|τ| =1)=0.926 1 3 5 7 9 11 13 15 17 19 210.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 p( ˆ|τ| =1)=0.605 1 3 5 7 9 11 13 15 17 19 210.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 p( ˆ|τ| =1)=0.187 1 3 5 7 9 11 13 15 17 19 210.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 p( ˆ|τ| =1)=0.003 MDPP Estimated number of divergence events (mode) Density Islands and Integrals J. Oaks, University of Kansas 37/53
  91. 91. dpp-msbayes: Simulation results 0.05 0.25 0.45 0.65 0.850 2 4 6 8 10 12 14 16 τ∼U(0, 0.5 MGA) 0.05 0.25 0.45 0.65 0.850 1 2 3 4 5 6 7 8 9 τ∼U(0, 1.5 MGA) 0.05 0.25 0.45 0.65 0.850 1 2 3 4 5 6 7 τ∼U(0, 2.5 MGA) 0.05 0.25 0.45 0.65 0.850.0 0.5 1.0 1.5 2.0 2.5 3.0 τ∼U(0, 5.0 MGA) MmsBayes Posterior probability of one divergence Density 0.05 0.25 0.45 0.65 0.850 1 2 3 4 5 6 7 0.05 0.25 0.45 0.65 0.850.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 0.05 0.25 0.45 0.65 0.850 1 2 3 4 5 0.05 0.25 0.45 0.65 0.850 5 10 15 20 MDPP Posterior probability of one divergence Density Islands and Integrals J. Oaks, University of Kansas 38/53
  92. 92. dpp-msbayes: Simulation results 0.0 0.02 0.04 0.06 0.08 0.1 0.120 50 100 150 200 p( ˆDT <0.01)=1.0 τ∼U(0, 0.5 MGA) 0.0 0.02 0.04 0.06 0.08 0.1 0.120 50 100 150 200 p( ˆDT <0.01)=0.999 τ∼U(0, 1.5 MGA) 0.0 0.02 0.04 0.06 0.08 0.1 0.120 50 100 150 200 p( ˆDT <0.01)=0.996 τ∼U(0, 2.5 MGA) 0.0 0.02 0.04 0.06 0.08 0.1 0.120 20 40 60 80 100 120 140 160 180 p( ˆDT <0.01)=0.637 τ∼U(0, 5.0 MGA) MmsBayes Estimated variance in divergence times (median) Density 0.0 0.1 0.2 0.3 0.4 0.50 2 4 6 8 10 p( ˆDT <0.01)=0.002 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.40.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 p( ˆDT <0.01)=0.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.40.0 0.5 1.0 1.5 2.0 2.5 p( ˆDT <0.01)=0.0 0.0 0.4 0.8 1.2 1.60.0 0.5 1.0 1.5 2.0 2.5 3.0 p( ˆDT <0.01)=0.0 MDPP Estimated variance in divergence times (median) Density Islands and Integrals J. Oaks, University of Kansas 39/53
  93. 93. dpp-msbayes: Simulation results 0.0 0.02 0.04 0.06 0.08 0.1 0.120 50 100 150 200 p( ˆDT <0.01)=1.0 τ∼U(0, 0.5 MGA) 0.0 0.02 0.04 0.06 0.08 0.1 0.120 50 100 150 200 p( ˆDT <0.01)=0.999 τ∼U(0, 1.5 MGA) 0.0 0.02 0.04 0.06 0.08 0.1 0.120 50 100 150 200 p( ˆDT <0.01)=0.996 τ∼U(0, 2.5 MGA) 0.0 0.02 0.04 0.06 0.08 0.1 0.120 20 40 60 80 100 120 140 160 180 p( ˆDT <0.01)=0.637 τ∼U(0, 5.0 MGA) MmsBayes Estimated variance in divergence times (median) Density 0.0 0.05 0.1 0.15 0.2 0.25 0.3 0.350 10 20 30 40 50 60 70 p( ˆDT <0.01)=0.914 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80 5 10 15 20 25 p( ˆDT <0.01)=0.626 0.0 0.2 0.4 0.6 0.80 1 2 3 4 5 6 7 8 9 p( ˆDT <0.01)=0.235 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.40.0 0.5 1.0 1.5 2.0 2.5 p( ˆDT <0.01)=0.004 MUshaped Estimated variance in divergence times (median) Density 0.0 0.1 0.2 0.3 0.4 0.50 2 4 6 8 10 p( ˆDT <0.01)=0.002 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.40.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 p( ˆDT <0.01)=0.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.40.0 0.5 1.0 1.5 2.0 2.5 p( ˆDT <0.01)=0.0 0.0 0.4 0.8 1.2 1.60.0 0.5 1.0 1.5 2.0 2.5 3.0 p( ˆDT <0.01)=0.0 MDPP Estimated variance in divergence times (median) Density Islands and Integrals J. Oaks, University of Kansas 40/53
  94. 94. dpp-msbayes: Simulation results Results confirm the bias of msBayes was caused by 1. Broad uniform priors 2. U-shaped prior on divergence models The new model shows improved model-choice accuracy, power, and robustness Islands and Integrals J. Oaks, University of Kansas 41/53
  95. 95. Testing climate-driven diversification Did repeated fragmentation of islands during inter-glacial rises in sea level promote diversification? Islands and Integrals J. Oaks, University of Kansas 42/53
  96. 96. Species n1 n2 Mammals Crocidura beatus 12 11 Crocidura negrina-panayensis 12 6 Hipposideros obscurus 19 9 Hipposideros pygmaeus 3 12 Cynopterus brachyotis 20 8 Cynopterus brachyotis 8 14 Haplonycteris fischeri 29 8 Haplonycteris fischeri 9 21 Macroglossus minimus 19 4 Macroglossus minimus 8 10 Ptenochirus jagori 4 7 Ptenochirus jagori 8 8 Ptenochirus minor 30 9 Squamates Cyrtodactylus gubaot-sumuroi 29 6 Cyrtodactylus annulatus 14 3 Cyrtodactylus philippinicus 6 14 Gekko mindorensis 8 11 Insulasaurus arborens 22 10 Pinoyscincus jagori 8 8 Dendrelaphis marenae 6 6 Anurans Limnonectes leytensis 4 2 Limnonectes magnus 2 3 Islands and Integrals J. Oaks, University of Kansas 43/53
  97. 97. dpp-msbayes: Philippine diversification 1 3 5 7 9 11 13 15 17 19 21 Number of divergence events 0.0 0.1 0.2 0.3 0.4 0.5 Posteriorprobability msBayes 1 3 5 7 9 11 13 15 17 19 21 Number of divergence events dpp-msbayes Islands and Integrals J. Oaks, University of Kansas 44/53
  98. 98. dpp-msbayes: Philippine diversification 1 3 5 7 9 11 13 15 17 19 21 Number of divergence events 0.00 0.02 0.04 0.06 0.08 0.10 0.12 Probability Prior 1 3 5 7 9 11 13 15 17 19 21 Number of divergence events Posterior 0100200300400500 Time (kya) 0 -50 -100 Sealevel(m) Islands and Integrals J. Oaks, University of Kansas 45/53
  99. 99. Conclusions Our new approximate-Bayesian method of phylogeographical model choice shows improved behavior Improved accuracy, robustness, and power More “honest” estimates regarding uncertainty Islands and Integrals J. Oaks, University of Kansas 46/53
  100. 100. Conclusions Our new approximate-Bayesian method of phylogeographical model choice shows improved behavior Improved accuracy, robustness, and power More “honest” estimates regarding uncertainty Philippine climate-driven diversification model? Results consistent with prediction of clustered divergences Results suggest multiple co-divergences However, there is a lot of uncertainty Islands and Integrals J. Oaks, University of Kansas 46/53
  101. 101. Future directions: Full-Bayesian phylogenetic framework T2 T3 T5 τ2 τ1 T1 τ3 T4 0100200300400500 Time (kya) 0 -50 -100 Sealevel(m) Islands and Integrals J. Oaks, University of Kansas 47/53
  102. 102. Future directions: Full-Bayesian phylogenetic framework 0100200300400500 Time (kya) 0 -50 -100 Sealevel(m) Islands and Integrals J. Oaks, University of Kansas 47/53
  103. 103. Software Everything is on GitHub. . . dpp-msbayes: https://github.com/joaks1/dpp-msbayes PyMsBayes: https://github.com/joaks1/PyMsBayes ABACUS: Approximate BAyesian C UtilitieS. https://github.com/joaks1/abacus Islands and Integrals J. Oaks, University of Kansas 48/53
  104. 104. Open Notebook Science Everything is on GitHub. . . msbayes-experiments: https://github.com/joaks1/msbayes-experiments joaks1@gmail.com Islands and Integrals J. Oaks, University of Kansas 49/53
  105. 105. Acknowledgments Ideas and feedback: KU Herpetology Holder Lab Melissa Callahan Mike Hickerson Laura Kubatko My committee Computation: KU ITTC KU Computing Center iPlant Funding: NSF KU Grad Studies, EEB & BI SSB Sigma Xi Photo credits: Rafe Brown, Cam Siler, & Jake Esselstyn FMNH Philippine Mammal Website: D.S. Balete, M.R.M. Duya, & J. Holden Islands and Integrals J. Oaks, University of Kansas 50/53
  106. 106. Acknowledgments Friends & Family Islands and Integrals J. Oaks, University of Kansas 51/53
  107. 107. Acknowledgments Friends & Family Islands and Integrals J. Oaks, University of Kansas 52/53
  108. 108. Questions? Islands and Integrals J. Oaks, University of Kansas 53/53
  109. 109. Gene tree divergences Age (mybp) Split(Taxon:Island1−Island2) Crocidura beatus: Leyte−Samar Crocidura negrina−panayensis: Negros−Panay Cynopterus brachyotis: Biliran−Mindanao Cynopterus brachyotis: Negros−Panay Cyrtodactylus annulatus: Bohol−Mindanao Cyrtodactylus gubaot−sumuroi: Leyte−Samar Cyrtodactylus philippinicus: Negros−Panay Dendrelaphis marenae: Negros−Panay Gekko mindorensis: Negros−Panay Haplonycteris fischeri: Biliran−Mindanao Haplonycteris fischeri: Negros−Panay Hipposideros obscurus: Leyte−Mindanao Hipposideros pygmaeus: Bohol−Mindanao Limnonectes leytensis: Bohol−Mindanao Limnonectes magnus: Bohol−Mindanao Macroglossus minimus: Biliran−Mindanao Macroglossus minimus: Negros−Panay Ptenochirus jagori: Leyte−Mindanao Ptenochirus jagori: Negros−Panay Ptenochirus minor: Biliran−Mindanao Insulasaurus arborens: Negros−Panay Pinoyscincus jagori: Mindanao−Samar ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.5 1.0 1.5 2.0 2.5 3.0 Islands and Integrals J. Oaks, University of Kansas 53/53
  110. 110. Causes of bias: Insufficient sampling Models with more parameter space are less densely sampled Could explain bias toward small models in extreme cases Predicts large variance in posterior estimates We explored empirical and simulation-based analyses with 2, 5, and 10 million prior samples, and estimates were very similar 0.0 0.2 0.4 0.6 0.8 1.0 1e8 0.0 0.2 0.4 0.6 0.8 1.0 1.2 95%HPDDT UnadjustedA 0.0 0.2 0.4 0.6 0.8 1.0 1e8 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 GLM-adjustedB Number of prior samples Islands and Integrals J. Oaks, University of Kansas 53/53
  111. 111. Geological history Islands and Integrals J. Oaks, University of Kansas 53/53

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