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Statistical Rethinking 2022 Lecture 16

Statistical Rethinking 2022 Lecture 16

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Richard McElreath

February 23, 2022
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  1. Statistical Rethinking 16: Space, Time and Phylogeny 2022

  2. though tough trough through thought thorough bough hough hiccough lough

  3. None
  4. None
  5. Oceanic Technology data(Kline2) Number of tool types associated with population

    size Spatial covariation: Islands close together share unobserved confounds and innovations
  6. Oceanic Technology data(Kline2) Number of tool types associated with population

    size Spatial covariation: Islands close together share unobserved confounds and innovations T P C U Population Tools Contact Unobserved
 confounds
  7. ̂ T = αPβ γ T i ∼ Poisson(λ i

    ) λ i = ̂ T ΔT = αPβ − γT change in tools innovation rate diminishing returns
 (elasticity) rate of loss T P C U Population Tools Contact Unobserved
 confounds
  8. ̂ T = αPβ γ T i ∼ Poisson(λ i

    ) λ i = ̂ T T P C U Population Tools Contact Unobserved
 confounds Spatial covariation: Islands close together share unobserved confounds and innovations Effect of U is to make closer islands have more similar ̂ T
  9. T i ∼ Poisson(λ i ) log λ i =

    ¯ α + α S[i] deviation log-tools
 in society i
  10. T i ∼ Poisson(λ i ) α 1 α 2

    ⋮ α 10 ∼ MVNormal 0 0 ⋮ 0 , K vector of all varying effects log λ i = ¯ α + α S[i]
  11. T i ∼ Poisson(λ i ) α 1 α 2

    ⋮ α 10 ∼ MVNormal 0 0 ⋮ 0 , K vector of all varying effects vector of zeros covariance matrix,
 the “Kernel” log λ i = ¯ α + α S[i]
  12. log λ i = ¯ α + α S[i] T

    i ∼ Poisson(λ i ) α 1 α 2 ⋮ α 10 ∼ MVNormal 0 0 ⋮ 0 , K K = σ2 k 1,2 k 1,3 k 1,4 k 1,5 k 1,6 k 1,7 k 1,8 k 1,9 k 1,10 σ2 k 2,3 k 2,4 k 2,5 k 2,6 k 2,7 k 2,8 k 2,9 k 2,10 σ2 k 3,4 k 3,5 k 3,6 k 3,7 k 3,8 k 3,9 k 3,10 σ2 k 4,5 k 4,6 k 4,7 k 4,8 k 4,9 k 4,10 σ2 k 5,6 k 5,7 k 5,8 k 5,9 k 5,10 σ2 k 6,7 k 6,8 k 6,9 k 6,10 σ2 k 7,8 k 7,9 k 7,10 σ2 k 8,9 k 8,10 σ2 k 9,10 σ2
  13. K = σ2 k 1,2 k 1,3 k 1,4 k

    1,5 k 1,6 k 1,7 k 1,8 k 1,9 k 1,10 σ2 k 2,3 k 2,4 k 2,5 k 2,6 k 2,7 k 2,8 k 2,9 k 2,10 σ2 k 3,4 k 3,5 k 3,6 k 3,7 k 3,8 k 3,9 k 3,10 σ2 k 4,5 k 4,6 k 4,7 k 4,8 k 4,9 k 4,10 σ2 k 5,6 k 5,7 k 5,8 k 5,9 k 5,10 σ2 k 6,7 k 6,8 k 6,9 k 6,10 σ2 k 7,8 k 7,9 k 7,10 σ2 k 8,9 k 8,10 σ2 k 9,10 σ2 Hawaii Santa Cruz Malekula Tikopia Yap Fiji Trobriand Chuuk Manus Tonga Malekula Tikopia Santa Cruz Yap Fiji Trobriand Chuuk Manus Tonga Hawaii 45 covariances
  14. Gaussian Processes A Gaussian Process is “an infinite- dimensional generalization

    of multivariate normal distributions” What does this mean? Instead of conventional covariance matrix, use a kernel function that generalizes to infinite dimensions/ observations/predictions
  15. Gaussian Processes Instead of conventional covariance matrix, use a kernel

    function The kernel gives the covariance between any pair of points as a function of their distance Distance can be difference, space, time, etc Continuous, ordered categories
  16. Kernel function Possible functions mapping x to y

  17. Kernel function Possible functions mapping x to y

  18. Kernel function Possible functions mapping x to y

  19. Kernel function Possible functions mapping x to y

  20. None
  21. None
  22. None
  23. k(x 1 , x 2 ) = α2 exp (

    − (x 1 − x 2 )2 σ2 ) Quadratic (L2) 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.4 0.8 |x1 - x2| covariance
  24. k(x 1 , x 2 ) = α2 exp (

    − (x 1 − x 2 )2 σ2 ) k(x 1 , x 2 ) = α2 exp ( − |x 1 − x 2 | σ ) Quadratic (L2) Ornstein-Uhlenbeck 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.4 0.8 |x1 - x2| covariance 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.4 0.8 |x1 - x2| covariance
  25. k(x 1 , x 2 ) = α2 exp (

    − (x 1 − x 2 )2 σ2 ) k(x 1 , x 2 ) = α2 exp ( − |x 1 − x 2 | σ ) k(x 1 , x 2 ) = α2 exp ( − 2 sin2 ((x 1 − x 2 )/2) σ2 ) Quadratic (L2) Ornstein-Uhlenbeck Periodic 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.4 0.8 |x1 - x2| covariance 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.4 0.8 |x1 - x2| covariance 0 1 2 3 4 5 6 0.0 0.4 0.8 |x1 - x2| covariance
  26. log λ i = ¯ α + α S[i] T

    i ∼ Poisson(λ i ) α 1 α 2 ⋮ α 10 ∼ MVNormal 0 0 ⋮ 0 , K k i,j = η2 exp ( −ρ2d2 i,j) covariance maximum covariance rate of decline distance i,j
  27. log λ i = ¯ α + α S[i] T

    i ∼ Poisson(λ i ) α 1 α 2 ⋮ α 10 ∼ MVNormal 0 0 ⋮ 0 , K ¯ α ∼ Normal(3,0.5) η2 ∼ Exponential(2) ρ2 ∼ Exponential(0.5) What do these priors imply? Ml Ti SC Ya Fi Tr Ch Mn To Ha Malekula 0.0 0.5 0.6 4.4 1.2 2.0 3.2 2.8 1.9 5.7 Tikopia 0.5 0.0 0.3 4.2 1.2 2.0 2.9 2.7 2.0 5.3 Santa Cruz 0.6 0.3 0.0 3.9 1.6 1.7 2.6 2.4 2.3 5.4 Yap 4.4 4.2 3.9 0.0 5.4 2.5 1.6 1.6 6.1 7.2 Lau Fiji 1.2 1.2 1.6 5.4 0.0 3.2 4.0 3.9 0.8 4.9 Trobriand 2.0 2.0 1.7 2.5 3.2 0.0 1.8 0.8 3.9 6.7 Chuuk 3.2 2.9 2.6 1.6 4.0 1.8 0.0 1.2 4.8 5.8 Manus 2.8 2.7 2.4 1.6 3.9 0.8 1.2 0.0 4.6 6.7 Tonga 1.9 2.0 2.3 6.1 0.8 3.9 4.8 4.6 0.0 5.0 Hawaii 5.7 5.3 5.4 7.2 4.9 6.7 5.8 6.7 5.0 0.0 Distance matrix (thousand km) k i,j = η2 exp ( −ρ2d2 i,j)
  28. 0 1 2 3 4 5 6 7 0.0 0.5

    1.0 1.5 2.0 distance (thousand km) covariance # sim priors for distance model n <- 30 etasq <- rexp(n,2) rhosq <- rexp(n,0.5) plot( NULL , xlim=c(0,7) , ylim=c(0,2) , xlab="distance (thousand km)" , ylab="covariance" ) for ( i in 1:n ) curve( etasq[i]*exp(-rhosq[i]*x^2) , add=TRUE , lwd=4 , col=col.alpha(2,0.5) ) η2 ∼ Exponential(2) ρ2 ∼ Exponential(0.5) k i,j = η2 exp ( −ρ2d2 i,j)
  29. log λ i = ¯ α + α S[i] T

    i ∼ Poisson(λ i ) α 1 α 2 ⋮ α 10 ∼ MVNormal 0 0 ⋮ 0 , K ¯ α ∼ Normal(3,0.5) η2 ∼ Exponential(2) ρ2 ∼ Exponential(0.5) 0 1 2 3 4 5 6 7 0.0 0.5 1.0 1.5 2.0 distance (thousand km) covariance Ml Ti SC Ya Fi Tr Ch Mn To Ha Malekula 0.0 0.5 0.6 4.4 1.2 2.0 3.2 2.8 1.9 5.7 Tikopia 0.5 0.0 0.3 4.2 1.2 2.0 2.9 2.7 2.0 5.3 Santa Cruz 0.6 0.3 0.0 3.9 1.6 1.7 2.6 2.4 2.3 5.4 Yap 4.4 4.2 3.9 0.0 5.4 2.5 1.6 1.6 6.1 7.2 Lau Fiji 1.2 1.2 1.6 5.4 0.0 3.2 4.0 3.9 0.8 4.9 Trobriand 2.0 2.0 1.7 2.5 3.2 0.0 1.8 0.8 3.9 6.7 Chuuk 3.2 2.9 2.6 1.6 4.0 1.8 0.0 1.2 4.8 5.8 Manus 2.8 2.7 2.4 1.6 3.9 0.8 1.2 0.0 4.6 6.7 Tonga 1.9 2.0 2.3 6.1 0.8 3.9 4.8 4.6 0.0 5.0 Hawaii 5.7 5.3 5.4 7.2 4.9 6.7 5.8 6.7 5.0 0.0 Distance matrix (thousand km) k i,j = η2 exp ( −ρ2d2 i,j)
  30. data(Kline2) d <- Kline2 data(islandsDistMatrix) dat_list <- list( T =

    d$total_tools, S = 1:10, D = islandsDistMatrix ) mTdist <- ulam( alist( T ~ dpois(lambda), log(lambda) <- abar + a[S], vector[10]:a ~ multi_normal( 0 , K ), matrix[10,10]:K <- cov_GPL2(D,etasq,rhosq,0.01), abar ~ normal(3,0.5), etasq ~ dexp( 2 ), rhosq ~ dexp( 0.5 ) ), data=dat_list , chains=4 , cores=4 , iter=4000 ) log λ i = ¯ α + α S[i] T i ∼ Poisson(λ i ) α 1 α 2 ⋮ α 10 ∼ MVNormal 0 0 ⋮ 0 , K ¯ α ∼ Normal(3,0.5) η2 ∼ Exponential(2) ρ2 ∼ Exponential(0.5) k i,j = η2 exp ( −ρ2d2 i,j)
  31. data(Kline2) d <- Kline2 data(islandsDistMatrix) dat_list <- list( T =

    d$total_tools, S = 1:10, D = islandsDistMatrix ) mTdist <- ulam( alist( T ~ dpois(lambda), log(lambda) <- abar + a[S], vector[10]:a ~ multi_normal( 0 , K ), matrix[10,10]:K <- cov_GPL2(D,etasq,rhosq,0.01), abar ~ normal(3,0.5), etasq ~ dexp( 2 ), rhosq ~ dexp( 0.5 ) ), data=dat_list , chains=4 , cores=4 , iter=4000 ) log λ i = ¯ α + α S[i] T i ∼ Poisson(λ i ) α 1 α 2 ⋮ α 10 ∼ MVNormal 0 0 ⋮ 0 , K ¯ α ∼ Normal(3,0.5) η2 ∼ Exponential(2) ρ2 ∼ Exponential(0.5) k i,j = η2 exp ( −ρ2d2 i,j)
  32. data(Kline2) d <- Kline2 data(islandsDistMatrix) dat_list <- list( T =

    d$total_tools, S = 1:10, D = islandsDistMatrix ) mTdist <- ulam( alist( T ~ dpois(lambda), log(lambda) <- abar + a[S], vector[10]:a ~ multi_normal( 0 , K ), matrix[10,10]:K <- cov_GPL2(D,etasq,rhosq,0.01), abar ~ normal(3,0.5), etasq ~ dexp( 2 ), rhosq ~ dexp( 0.5 ) ), data=dat_list , chains=4 , cores=4 , iter=2000 ) log λ i = ¯ α + α S[i] T i ∼ Poisson(λ i ) α 1 α 2 ⋮ α 10 ∼ MVNormal 0 0 ⋮ 0 , K ¯ α ∼ Normal(3,0.5) η2 ∼ Exponential(2) ρ2 ∼ Exponential(0.5) k i,j = η2 exp ( −ρ2d2 i,j) > precis(mTdist,2) mean sd 5.5% 94.5% n_eff Rhat4 a[1] -0.67 0.31 -1.16 -0.18 1385 1 a[2] -0.43 0.30 -0.89 0.06 1075 1 a[3] -0.38 0.30 -0.87 0.10 1137 1 a[4] 0.24 0.28 -0.20 0.71 1133 1 a[5] 0.02 0.29 -0.44 0.49 1124 1 a[6] -0.48 0.30 -0.96 -0.01 1303 1 a[7] 0.16 0.29 -0.28 0.63 1154 1 a[8] -0.17 0.30 -0.65 0.29 1164 1 a[9] 0.45 0.28 0.01 0.90 1053 1 a[10] 0.73 0.27 0.30 1.18 1027 1 abar 3.49 0.25 3.08 3.88 896 1 etasq 0.38 0.27 0.12 0.87 1989 1 rhosq 1.09 1.39 0.08 3.68 3302 1
  33. 0 1 2 3 4 5 6 7 0.0 0.5

    1.0 1.5 2.0 distance (thousand km) covariance prior posterior -40 -20 0 20 -20 -10 0 10 20 longitude latitude Malekula Tikopia Santa Cruz Yap Lau Fiji Trobriand Chuuk Manus Tonga Hawaii
  34. -40 -20 0 20 -20 -10 0 10 20 longitude

    latitude Malekula Tikopia Santa Cruz Yap Lau Fiji Trobriand Chuuk Manus Tonga Hawaii
  35. dat_list <- list( T = d$total_tools, P = d$population, S

    = 1:10, D = islandsDistMatrix ) mTDP <- ulam( alist( T ~ dpois(lambda), lambda <- (abar*P^b/g)*exp(a[S]), vector[10]:a ~ multi_normal( 0 , K ), transpars> matrix[10,10]:K <- cov_GPL2(D,etasq,rhosq,0.01), c(abar,b,g) ~ dexp( 1 ), etasq ~ dexp( 2 ), rhosq ~ dexp( 0.5 ) ), data=dat_list , chains=4 , cores=4 , iter=4000 ) λ i = ¯ αPβ γ exp(α S[i] ) T i ∼ Poisson(λ i ) α 1 α 2 ⋮ α 10 ∼ MVNormal 0 0 ⋮ 0 , K ¯ α, β, γ ∼ Exponential(1) η2 ∼ Exponential(2) ρ2 ∼ Exponential(0.5) k i,j = η2 exp ( −ρ2d2 i,j)
  36. 0 1 2 3 4 5 6 7 0.0 0.5

    1.0 1.5 2.0 distance (thousand km) covariance empty model population model 7 8 9 10 11 12 20 30 40 50 60 70 log population tools
  37. PAUSE

  38. Allenopithecus nigroviridis Cercopithecus albogularis Cercopithecus ascanius Cercopithecus campbelli Cercopithecus campbelli

    lowei Cercopithecus cephus Cercopithecus cephus cephus Cercopithecus cephus ngottoensis Cercopithecus diana Cercopithecus erythrogaster Cercopithecus erythrogaster erythrogaster Cercopithecus erythrotis Cercopithecus hamlyni Cercopithecus lhoesti Cercopithecus mitis Cercopithecus mona Cercopithecus neglectus Cercopithecus nictitans Cercopithecus petaurista Cercopithecus pogonias Cercopithecus preussi Cercopithecus solatus Cercopithecus wolfi Chlorocebus aethiops Chlorocebus pygerythrus Chlorocebus pygerythrus cynosurus Chlorocebus sabaeus Chlorocebus tantalus Erythrocebus patas Miopithecus talapoin Allocebus trichotis Archaeolemur majori Avahi cleesei Avahi laniger Avahi occidentalis Avahi unicolor Cheirogaleus crossleyi Cheirogaleus m ajor Cheirogaleus m edius Daubentonia madagascariensis Eulemur coronatus Eulemur fulvus albifrons Eulemur fulvus albocollaris Eulemur fulvus collaris Eulemur fulvus fulvus Eulemur fulvus mayottensis Eulemur fulvus rufus Eulemur fulvus sanfordi Eulemur macaco flavifrons Eulemur macaco macaco Eulemur mongoz Eulemur rubriventer Hapalemur aureus Hapalemur griseus Hapalemur griseus alaotrensis Hapalemur griseus griseus Hapalemur griseus meridionalis Hapalemur griseus occidentalis Hapalemur simus Indri indri Lemur catta Lepilem ur aeeclis Lepilem ur ankaranensis Lepilem ur dorsalis Lepilem ur edw ardsi Lepilem ur hubbardorum Lepilem ur leucopus Lepilemur manasamody Lepilemur microdon Lepilem ur m itsinjoensis Lepilemur mustelinus Lepilemur otto Lepilem ur randrianasoli Lepilem ur ruficaudatus Lepilem ur saham alazensis Lepilemur seali Lepilemur septentrionalis Microcebus berthae Microcebus bongolavensis Microcebus danfossi Microcebus griseorufus Microcebus jollyae Microcebus lehilahytsara Microcebus lokobensis Microcebus macarthurii Microcebus mamiratra Microcebus mittermeieri Microcebus murinus Microcebus myoxinus Microcebus ravelobensis Microcebus rufus Microcebus sambiranensis Microcebus simmonsi Microcebus tavaratra Mirza coquereli Mirza zaza Phaner furcifer Phaner furcifer pallescens Propithecus coquereli Propithecus deckenii Propithecus diadema Propithecus edwardsi Propithecus tattersalli Propithecus verreauxi Varecia rubra Varecia variegata variegata Alouatta belzebul Alouatta caraya Alouatta guariba Alouatta palliata Alouatta pigra Alouatta sara Alouatta seniculus Ateles belzebuth Ateles fusciceps Ateles geoffroyi Ateles paniscus Brachyteles arachnoides Lagothrix lagotricha Aotus azarai Aotus azarai boliviensis Aotus brumbacki Aotus infulatus Aotus lemurinus Aotus lemurinus griseimembra Aotus nancymaae Aotus nigriceps Aotus trivirgatus Aotus vociferans Callimico goeldii Callithrix argentata Callithrix aurita Callithrix emiliae Callithrix geoffroyi Callithrix humeralifera Callithrix jacchus Callithrix kuhli Callithrix mauesi Callithrix penicillata Callithrix pygmaea Cebus albifrons Cebus apella Cebus capucinus Cebus olivaceus Cebus xanthosternos Leontopithecus chrysomelas Leontopithecus chrysopygus Leontopithecus rosalia Saguinus bicolor Saguinus fuscicollis Saguinus fuscicollis m elanoleucus Saguinus geoffroyi Saguinus im perator Saguinus leucopus Saguinus midas Saguinus m ystax Saguinus niger Saguinus oedipus Saguinus tripartitus Saimiri boliviensis Saimiri oerstedii Saimiri sciureus Saimiri ustus Arctocebus aureus Arctocebus calabarensis Loris lydekkerianus Loris tardigradus Nycticebus bengalensis Nycticebus coucang Nycticebus javanicus Nycticebus menagensis Nycticebus pygmaeus Perodicticus potto Bunopithecus hoolock Gorilla beringei Gorilla gorilla gorilla Gorilla gorilla graueri Homo sapiens Homo sapiens neanderthalensis Hylobates agilis Hylobates klossii Hylobates lar Hylobates moloch Hylobates muelleri Hylobates pileatus Nomascus concolor Nomascus gabriellae Nomascus leucogenys Nomascus nasutus Nomascus siki Pan paniscus Pan troglodytes schweinfurthii Pan troglodytes troglodytes Pan troglodytes vellerosus Pan troglodytes verus Pongo abelii Pongo pygmaeus Symphalangus syndactylus Cacajao calvus Cacajao melanocephalus Callicebus donacophilus Callicebus hoffmannsi Callicebus moloch Callicebus personatus Callicebus torquatus Chiropotes satanas Pithecia irrorata Pithecia pithecia Cercocebus agilis Cercocebus galeritus Cercocebus torquatus Cercocebus torquatus atys Lophocebus albigena Lophocebus aterrimus Macaca arctoides Macaca assamensis M acaca brunnescens Macaca cyclopis Macaca fascicularis Macaca fuscata M acaca hecki M acaca leonina M acaca m aura Macaca mulatta Macaca munzala Macaca nemestrina M acaca nem estrina leonina M acaca nem estrina siberu M acaca nigra M acaca nigrescens M acaca ochreata M acaca pagensis Macaca radiata Macaca silenus Macaca sinica Macaca sylvanus Macaca thibetana M acaca tonkeana Mandrillus leucophaeus Mandrillus sphinx Papio anubis Papio cynocephalus Papio hamadryas Papio papio Papio ursinus Rungwecebus kipunji Theropithecus gelada Colobus angolensis Colobus angolensis palliatus Colobus guereza Colobus polykomos Colobus satanas Colobus vellerosus Nasalis larvatus Piliocolobus badius Piliocolobus foai Piliocolobus gordonorum Piliocolobus kirkii Piliocolobus pennantii Piliocolobus preussi Piliocolobus rufomitratus Piliocolobus tephrosceles Piliocolobus tholloni Presbytis comata Presbytis melalophos Procolobus verus Pygathrix cinerea Pygathrix nemaeus Rhinopithecus avunculus Rhinopithecus bieti Rhinopithecus brelichi Rhinopithecus roxellana Sem nopithecus entellus Trachypithecus auratus Trachypithecus cristatus Trachypithecus delacouri Trachypithecus francoisi Trachypithecus geei Trachypithecus germ aini Trachypithecus johnii Trachypithecus laotum Trachypithecus obscurus Trachypithecus phayrei Trachypithecus pileatus Trachypithecus poliocephalus Trachypithecus vetulus Euoticus elegantulus Galago alleni Galago gallarum Galago granti Galago matschiei Galago moholi Galago senegalensis Galagoides demidoff Galagoides zanzibaricus Otolemur crassicaudatus Otolemur garnettii Tarsius bancanus Tarsius dentatus Tarsius lariang Tarsius syrichta Primates
  39. Callithrix argentata Callithrix emiliae Callithrix humeralifera Callithrix Callithrix kuhli Callithrix

    mauesi Callithrix penicillata Callithrix pygmaea Bunopithecus hoolock Gorilla beringei Gorilla gorilla gorilla Gorilla gorilla graueri Homo sapiens Homo sapiens neanderthalensis Hylobates agilis Hylobates klossii H ylobates lar Hylobates m oloch Hylobates m uelleri H ylobates pileatus Nom ascus concolor Nom ascus gabriellae Nom ascus leucogenys Nom ascus nasutus Nom ascus siki Pan paniscus Pan troglodytes schweinfurthii Pan troglodytes troglodytes Pan troglodytes vellerosus Pan troglodytes verus Pongo abelii Pongo pygmaeus Sym phalangus syndactylus Colobus angolensis obus angolensis palliatus Colobus guereza Colobus polykomos Colobus satanas Colobus vellerosus arvatus ocolobus badius iliocolobus foai ordonorum olobus kirkii bus pennantii olobus preussi omitratus hrosceles us tholloni comata alophos Procolobus verus nerea maeus ulus bieti elichi lana Apes
  40. Allenopithecus nigroviridis Cercopithecus albogularis Cercopithecus ascanius Cercopithecus campbelli Cercopithecus campbelli

    lowei Cercopithecus cephus Cercopithecus cephus cephus Cercopithecus cephus ngottoensis Cercopithecus diana Cercopithecus erythrogaster Cercopithecus erythrogaster erythrogaster Cercopithecus erythrotis Cercopithecus hamlyni Cercopithecus lhoesti Cercopithecus mitis Cercopithecus mona Cercopithecus neglectus Cercopithecus nictitans Cercopithecus petaurista Cercopithecus pogonias Cercopithecus preussi Cercopithecus solatus Cercopithecus wolfi Chlorocebus aethiops Chlorocebus pygerythrus Chlorocebus pygerythrus cynosurus Chlorocebus sabaeus Chlorocebus tantalus Erythrocebus patas Miopithecus talapoin Arctocebus aureus Arctocebus calabarensis Loris lydekkerianus Loris tardigradus Nycticebus bengalensis Nycticebus coucang Nycticebus javanicus Nycticebus menagensis Nycticebus pygmaeus Perodicticus potto Bunopithecus hoolock Gorilla beringei orilla gor orilla gorilla graueri Homo sapiens s neanderthalensis Hylobates agilis Hylobates klossii Hylobates lar Hylobates moloch Hylobates muelleri Hylobates pileatus Nomascus concolor Nomascus gabriellae Nomascus leucogenys Nomascus nasutus Nomascus siki Pan paniscus glodytes schweinfurthii oglodytes troglodytes roglodytes vellerosus an troglodytes verus Symphalangus syndactylus Cercocebus agilis Cercocebus galeritus Cercocebus torquatus Cercocebus torquatus atys Lophocebus albigena Lophocebus aterrimus M acaca arctoides M acaca assam ensis Macaca brunnescens M acaca cyclopis M acaca fascicularis M acaca fuscata Macaca hecki Macaca leonina Macaca maura M acaca m ulatta M acaca m unzala Macaca nemestrina Macaca nemestrina leonina Macaca nemestrina siberu Macaca nigra Macaca nigrescens Macaca ochreata Macaca pagensis Macaca radiata Macaca silenus Macaca sinica M acaca sylvanus M acaca thibetana Macaca tonkeana Mandrillus leucophaeus Mandrillus sphinx Papio anubis Papio cynocephalus Papio ham adryas Papio papio Papio ursinus Rungwecebus kipunji Theropithecus gelada Colobus angolensis Colobus angolensis palliatus Colobus guereza Colobus polykom os Colobus satanas Colobus vellerosus Nasalis larvatus Piliocolobus badius Piliocolobus foai Piliocolobus gordonorum Piliocolobus kirkii Piliocolobus pennantii Piliocolobus preussi Piliocolobus rufom itratus Piliocolobus tephrosceles Piliocolobus tholloni Presbytis comata Presbytis m elalophos Procolobus verus Pygathrix cinerea Pygathrix nemaeus Rhinopithecus avunculus Rhinopithecus bieti Rhinopithecus brelichi Rhinopithecus roxellana Semnopithecus entellus Trachypithecus auratus Trachypithecus cristatus Trachypithecus delacouri Trachypithecus francoisi Trachypithecus geei Trachypithecus germaini Trachypithecus johnii Trachypithecus laotum Trachypithecus obscurus Trachypithecus phayrei Trachypithecus pileatus Trachypithecus poliocephalus Trachypithecus vetulus oides demidoff African &
 Asian
 monkeys
  41. ae crocebus bongolavensis icrocebus danfossi ufus ocebus jollyae obensis cebus

    m acarthurii iratra ttermeieri nus s rocebus ravelobensis cebus rufus nensis simmonsi cebus tavaratra Alouatta belzebul Alouatta caraya Alouatta guariba Alouatta palliata Alouatta pigra Alouatta sara Alouatta seniculus Ateles belzebuth Ateles fusciceps Ateles geoffroyi Ateles paniscus Brachyteles arachnoides Lagothrix lagotricha Aotus azarai Aotus azarai boliviensis Aotus brumbacki Aotus infulatus Aotus lemurinus Aotus lemurinus griseimembra Aotus nancymaae Aotus nigriceps Aotus trivirgatus Aotus vociferans Callimico goeldii Callithrix argentata Callithrix aurita Callithrix em iliae Callithrix geoffroyi Callithrix hum eralifera Callithrix jacchus Callithrix kuhli Callithrix mauesi Callithrix penicillata Callithrix pygmaea Cebus albifrons Cebus apella Cebus capucinus Cebus olivaceus Cebus xanthosternos Leontopithecus chrysomelas Leontopithecus chrysopygus Leontopithecus rosalia Saguinus bicolor Saguinus fuscicollis Saguinus fuscicollis melanoleucus Saguinus geoffroyi Saguinus imperator Saguinus leucopus Saguinus midas Saguinus mystax Saguinus niger Saguinus oedipus Saguinus tripartitus Saimiri boliviensis Saimiri oerstedii Saimiri sciureus Saimiri ustus ecus hoolock G orilla beringei G orilla gorilla gorilla G orilla gorilla graueri Hom o sapiens sapiens neanderthalensis ii s lar ch eri atus s concolor abriellae ogenys cus nasutus cus siki Pan paniscus schweinfurthii ytes troglodytes odytes vellerosus n troglodytes verus Pongo abelii Pongo pygm aeus ctylus Cacajao calvus Cacajao melanocephalus Callicebus donacophilus Callicebus hoffmannsi Callicebus moloch Callicebus personatus Callicebus torquatus Chiropotes satanas Pithecia irrorata Pithecia pithecia Tarsius bancanus Tarsius dentatus Tarsius lariang Tarsius syrichta American monkeys
  42. Allocebus trichotis Archaeolemur majori Avahi cleesei Avahi laniger Avahi occidentalis

    Avahi unicolor Cheirogaleus crossleyi Cheirogaleus major Cheirogaleus medius Eulem ur cor Eulem ur fulvus albifrons Eulem ur fulvus albocollaris Eulem ur fulvus collaris Eulem ur fulvus fulvus Eulem ur fulvus m ayottensis Eulem ur fulvus rufus Eulem ur fulvus sanfordi Eulem ur m acac Eulem ur m aca Eulem ur m ongoz Eulem ur rubriventer Hap Hapalem u Hapal Hapa Hapalem Hapalem Ha Indri indri L Lepilemur aeeclis Lepilemur ankaranensis Lepilemur dorsalis Lepilemur edwardsi Lepilemur hubbardorum Lepilemur leucopus Lepilemur manasamody Lepilemur microdon Lepilemur mitsinjoensis Lepilemur mustelinus Lepilemur otto Lepilemur randrianasoli Lepilemur ruficaudatus Lepilemur sahamalazensis Lepilemur seali Lepilemur septentrionalis Microcebus berthae Microcebus bongolavensis Microcebus danfossi Microcebus griseorufus Microcebus jollyae Microcebus lehilahytsara Microcebus lokobensis Microcebus macarthurii Microcebus mamiratra Microcebus mittermeieri Microcebus murinus Microcebus myoxinus Microcebus ravelobensis Microcebus rufus Microcebus sambiranensis Microcebus simmonsi Microcebus tavaratra Mirza coquereli Mirza zaza Phaner furcifer Phaner furcifer pallescens Propithecus coquereli Propithecus deckenii Propithecus diadema Propithecus edwardsi Propithecus tattersalli Propithecus verreauxi V us bus donacophilus hoffmannsi icebus moloch allicebus personatus Callicebus torquatus nas rrorata pithecia Tarsius bancanus Tarsius dentatus Tarsius lariang Tarsius syrichta Lemurs etc Tarsiers
  43. Archaeolemur majori Avahi cleesei Avahi laniger Avahi occidentalis Avahi unicolor

    Daubentonia madagascariensis Eulemur coronatus Eulemur fulvus albifrons Eulemur fulvus albocollaris Eulemur fulvus collaris Eulemur fulvus fulvus Eulemur fulvus mayottensis Eulemur fulvus rufus Eulemur fulvus sanfordi Eulemur macaco flavifrons Eulemur macaco macaco Eulemur mongoz Eulemur rubriventer Hapalemur aureus Hapalemur griseus Hapalemur griseus alaotrensi Hapalemur griseus griseus Hapalemur griseus meridiona Hapalemur griseus occidenta Hapalemur simus Indri indri Lemur catta Lepilemur Lepilemur do Le Lepilem Lepilemur manasamod Lepilemur microdon Lepilemur mustelinus Lepilemur otto Lepil Lepilemur saha Lepilemur seali Lepilemur septentri Phaner furcifer Phaner furcifer pallescens Propithecus coquereli Propithecus deckenii Propithecus diadema Propithecus edwardsi Propithecus tattersalli Propithecus verreauxi Varecia rubra Varecia variegata variegata Arctocebus aureus Arctocebus calabarensis Loris lydekkerianus Loris tardigradus Nycticebus bengalensis Nycticebus coucang Nycticebus javanicus Nycticebus menagensis Nycticebus pygmaeus Perodicticus potto M acaca brunnescens M acaca hecki M acaca le M acaca m aura Ma Macac Macaca M acaca nigra M acaca nig M acaca ochreata Maca M Mac M M acaca tonkean Euoticus elegantulus Galago alleni Galago gallarum Galago granti Galago matschiei Galago moholi Galago senegalensis Galagoides demidoff Galagoides zanzibaricus Otolemur crassicaudatus Otolemur garnettii Galagos
 Lorises
  44. Allenopithecus nigroviridis Cercopithecus albogularis Cercopithecus ascanius Cercopithecus campbelli Cercopithecus campbelli

    lowei Cercopithecus cephus Cercopithecus cephus cephus Cercopithecus cephus ngottoensis Cercopithecus diana Cercopithecus erythrogaster Cercopithecus erythrogaster erythrogaster Cercopithecus erythrotis Cercopithecus hamlyni Cercopithecus lhoesti Cercopithecus mitis Cercopithecus mona Cercopithecus neglectus Cercopithecus nictitans Cercopithecus petaurista Cercopithecus pogonias Cercopithecus preussi Cercopithecus solatus Cercopithecus wolfi Chlorocebus aethiops Chlorocebus pygerythrus Chlorocebus pygerythrus cynosurus Chlorocebus sabaeus Chlorocebus tantalus Erythrocebus patas Miopithecus talapoin Allocebus trichotis Archaeolemur majori Avahi cleesei Avahi laniger Avahi occidentalis Avahi unicolor Cheirogaleus crossleyi Cheirogaleus major Cheirogaleus medius Daubentonia madagascariensis Eulem ur coronatus Eulem ur fulvus albifrons Eulem ur fulvus albocollaris Eulem ur fulvus collaris Eulem ur fulvus fulvus Eulem ur fulvus m ayottensis Eulem ur fulvus rufus Eulem ur fulvus sanfordi Eulem ur m acaco flavifrons Eulem ur m acaco m acaco Eulem ur m ongoz Eulem ur rubriventer Hapalemur aureus Hapalem ur griseus Hapalemur griseus alaotrensis Hapalemur griseus griseus Hapalem ur griseus m eridionalis Hapalemur griseus occidentalis Hapalemur simus Indri indri Lemur catta Lepilemur aeeclis Lepilemur ankaranensis Lepilemur dorsalis Lepilemur edwardsi Lepilemur hubbardorum Lepilemur leucopus Lepilemur manasamody Lepilemur microdon Lepilemur mitsinjoensis Lepilemur mustelinus Lepilemur otto Lepilemur randrianasoli Lepilemur ruficaudatus Lepilemur sahamalazensis Lepilemur seali Lepilemur septentrionalis Microcebus berthae Microcebus bongolavensis Microcebus danfossi Microcebus griseorufus Microcebus jollyae Microcebus lehilahytsara Microcebus lokobensis Microcebus macarthurii Microcebus mamiratra Microcebus mittermeieri Microcebus murinus Microcebus myoxinus Microcebus ravelobensis Microcebus rufus Microcebus sambiranensis Microcebus simmonsi Microcebus tavaratra Mirza coquereli Mirza zaza Phaner furcifer Phaner furcifer pallescens Propithecus coquereli Propithecus deckenii Propithecus diadema Propithecus edwardsi Propithecus tattersalli Propithecus verreauxi Varecia rubra Varecia variegata variegata Alouatta belzebul Alouatta caraya Alouatta guariba Alouatta palliata Alouatta pigra Alouatta sara Alouatta seniculus Ateles belzebuth Ateles fusciceps Ateles geoffroyi Ateles paniscus Brachyteles arachnoides Lagothrix lagotricha Aotus azarai Aotus azarai boliviensis Aotus brumbacki Aotus infulatus Aotus lemurinus Aotus lemurinus griseimembra Aotus nancymaae Aotus nigriceps Aotus trivirgatus Aotus vociferans Callimico goeldii Callithrix argentata Callithrix aurita Callithrix emiliae Callithrix geoffroyi Callithrix humeralifera Callithrix jacchus Callithrix kuhli Callithrix mauesi Callithrix penicillata Callithrix pygmaea Cebus albifrons Cebus apella Cebus capucinus Cebus olivaceus Cebus xanthosternos Leontopithecus chrysomelas Leontopithecus chrysopygus Leontopithecus rosalia Saguinus bicolor Saguinus fuscicollis Saguinus fuscicollis melanoleucus Saguinus geoffroyi Saguinus imperator Saguinus leucopus Saguinus midas Saguinus mystax Saguinus niger Saguinus oedipus Saguinus tripartitus Saimiri boliviensis Saimiri oerstedii Saimiri sciureus Saimiri ustus Arctocebus aureus Arctocebus calabarensis Loris lydekkerianus Loris tardigradus Nycticebus bengalensis Nycticebus coucang Nycticebus javanicus Nycticebus menagensis Nycticebus pygmaeus Perodicticus potto Bunopithecus hoolock Gorilla beringei Gorilla gorilla gorilla Gorilla gorilla graueri Homo sapiens Homo sapiens neanderthalensis Hylobates agilis Hylobates klossii H ylobates lar Hylobates m oloch Hylobates m uelleri H ylobates pileatus Nom ascus concolor Nom ascus gabriellae Nom ascus leucogenys Nom ascus nasutus Nom ascus siki Pan paniscus Pan troglodytes schweinfurthii Pan troglodytes troglodytes Pan troglodytes vellerosus Pan troglodytes verus Pongo abelii Pongo pygmaeus Sym phalangus syndactylus Cacajao calvus Cacajao m elanocephalus Callicebus donacophilus Callicebus hoffmannsi Callicebus moloch Callicebus personatus Callicebus torquatus Chiropotes satanas Pithecia irrorata Pithecia pithecia Cercocebus agilis Cercocebus galeritus Cercocebus torquatus Cercocebus torquatus atys Lophocebus albigena Lophocebus aterrim us Macaca arctoides Macaca assamensis Macaca brunnescens Macaca cyclopis Macaca fascicularis Macaca fuscata Macaca hecki Macaca leonina Macaca maura Macaca mulatta Macaca munzala Macaca nemestrina Macaca nemestrina leonina Macaca nemestrina siberu Macaca nigra Macaca nigrescens Macaca ochreata Macaca pagensis Macaca radiata Macaca silenus Macaca sinica Macaca sylvanus Macaca thibetana Macaca tonkeana M andrillus leucophaeus M andrillus sphinx Papio anubis Papio cynocephalus Papio hamadryas Papio papio Papio ursinus Rungwecebus kipunji Theropithecus gelada Colobus angolensis Colobus angolensis palliatus Colobus guereza Colobus polykomos Colobus satanas Colobus vellerosus Nasalis larvatus Piliocolobus badius Piliocolobus foai Piliocolobus gordonorum Piliocolobus kirkii Piliocolobus pennantii Piliocolobus preussi Piliocolobus rufomitratus Piliocolobus tephrosceles Piliocolobus tholloni Presbytis comata Presbytis melalophos Procolobus verus Pygathrix cinerea Pygathrix nemaeus Rhinopithecus avunculus Rhinopithecus bieti Rhinopithecus brelichi Rhinopithecus roxellana Semnopithecus entellus Trachypithecus auratus Trachypithecus cristatus Trachypithecus delacouri Trachypithecus francoisi Trachypithecus geei Trachypithecus germaini Trachypithecus johnii Trachypithecus laotum Trachypithecus obscurus Trachypithecus phayrei Trachypithecus pileatus Trachypithecus poliocephalus Trachypithecus vetulus Euoticus elegantulus Galago alleni Galago gallarum Galago granti Galago matschiei Galago moholi Galago senegalensis Galagoides demidoff Galagoides zanzibaricus Otolemur crassicaudatus Otolemur garnettii Tarsius bancanus Tarsius dentatus Tarsius lariang Tarsius syrichta
  45. Phylogenetic regression data(Primates301) Life history traits Mass g, brain cc,

    group size Much missing data, measurement error, unobserved confounding Street et al 2017 Coevolution of cultural intelligence, extended life history, sociality, and brain size in primates Allenopithecus nigroviridis Cercopithecus albogularis Cercopithecus ascanius Cercopithecus campbelli Cercopithecus campbelli lowei Cercopithecus cephus Cercopithecus cephus cephus Cercopithecus cephus ngottoensis Cercopithecus diana Cercopithecus erythrogaster Cercopithecus erythrogaster erythrogaster Cercopithecus erythrotis Cercopithecus hamlyni Cercopithecus lhoesti Cercopithecus mitis Cercopithecus mona Cercopithecus neglectus Cercopithecus nictitans Cercopithecus petaurista Cercopithecus pogonias Cercopithecus preussi Cercopithecus solatus Cercopithecus wolfi Chlorocebus aethiops Chlorocebus pygerythrus Chlorocebus pygerythrus cynosurus Chlorocebus sabaeus Chlorocebus tantalus Erythrocebus patas Miopithecus talapoin Allocebus trichotis Archaeolemur majori Avahi cleesei Avahi laniger Avahi occidentalis Avahi unicolor Cheirogaleus crossleyi Cheirogaleus major Cheirogaleus medius Daubentonia madagascariensis Eulem ur coronatus Eulem ur fulvus albifrons Eulem ur fulvus albocollaris Eulem ur fulvus collaris Eulem ur fulvus fulvus Eulem ur fulvus m ayottensis Eulem ur fulvus rufus Eulem ur fulvus sanfordi Eulem ur m acaco flavifrons Eulem ur m acaco m acaco Eulem ur m ongoz Eulem ur rubriventer Hapalem ur aureus Hapalem ur griseus Hapalemur griseus alaotrensis Hapalemur griseus griseus Hapalemur griseus meridionalis Hapalemur griseus occidentalis Hapalemur simus Indri indri Lemur catta Lepilemur aeeclis Lepilemur ankaranensis Lepilemur dorsalis Lepilemur edwardsi Lepilemur hubbardorum Lepilemur leucopus Lepilemur manasamody Lepilemur microdon Lepilemur mitsinjoensis Lepilemur mustelinus Lepilemur otto Lepilemur randrianasoli Lepilemur ruficaudatus Lepilemur sahamalazensis Lepilemur seali Lepilemur septentrionalis Microcebus berthae Microcebus bongolavensis Microcebus danfossi Microcebus griseorufus Microcebus jollyae Microcebus lehilahytsara Microcebus lokobensis Microcebus macarthurii Microcebus mamiratra Microcebus mittermeieri Microcebus murinus Microcebus myoxinus Microcebus ravelobensis Microcebus rufus Microcebus sambiranensis Microcebus simmonsi Microcebus tavaratra Mirza coquereli Mirza zaza Phaner furcifer Phaner furcifer pallescens Propithecus coquereli Propithecus deckenii Propithecus diadema Propithecus edwardsi Propithecus tattersalli Propithecus verreauxi Varecia rubra Varecia variegata variegata Alouatta belzebul Alouatta caraya Alouatta guariba Alouatta palliata Alouatta pigra Alouatta sara Alouatta seniculus Ateles belzebuth Ateles fusciceps Ateles geoffroyi Ateles paniscus Brachyteles arachnoides Lagothrix lagotricha Aotus azarai Aotus azarai boliviensis Aotus brumbacki Aotus infulatus Aotus lemurinus Aotus lemurinus griseimembra Aotus nancymaae Aotus nigriceps Aotus trivirgatus Aotus vociferans Callimico goeldii Callithrix argentata Callithrix aurita Callithrix emiliae Callithrix geoffroyi Callithrix humeralifera Callithrix jacchus Callithrix kuhli Callithrix mauesi Callithrix penicillata Callithrix pygmaea Cebus albifrons Cebus apella Cebus capucinus Cebus olivaceus Cebus xanthosternos Leontopithecus chrysomelas Leontopithecus chrysopygus Leontopithecus rosalia Saguinus bicolor Saguinus fuscicollis Saguinus fuscicollis melanoleucus Saguinus geoffroyi Saguinus imperator Saguinus leucopus Saguinus midas Saguinus mystax Saguinus niger Saguinus oedipus Saguinus tripartitus Saim iri boliviensis Saim iri oerstedii Saim iri sciureus Saim iri ustus Arctocebus aureus Arctocebus calabarensis Loris lydekkerianus Loris tardigradus Nycticebus bengalensis Nycticebus coucang Nycticebus javanicus Nycticebus menagensis Nycticebus pygmaeus Perodicticus potto Bunopithecus hoolock Gorilla beringei Gorilla gorilla gorilla Gorilla gorilla graueri Homo sapiens Homo sapiens neanderthalensis Hylobates agilis Hylobates klossii H ylobates lar Hylobates m oloch H ylobates m uelleri Hylobates pileatus Nom ascus concolor Nom ascus gabriellae Nom ascus leucogenys Nomascus nasutus Nom ascus siki Pan paniscus Pan troglodytes schweinfurthii Pan troglodytes troglodytes Pan troglodytes vellerosus Pan troglodytes verus Pongo abelii Pongo pygmaeus Sym phalangus syndactylus Cacajao calvus Cacajao m elanocephalus Callicebus donacophilus Callicebus hoffmannsi Callicebus moloch Callicebus personatus Callicebus torquatus Chiropotes satanas Pithecia irrorata Pithecia pithecia Cercocebus agilis Cercocebus galeritus Cercocebus torquatus Cercocebus torquatus atys Lophocebus albigena Lophocebus aterrim us Macaca arctoides Macaca assamensis Macaca brunnescens Macaca cyclopis Macaca fascicularis Macaca fuscata Macaca hecki Macaca leonina Macaca maura Macaca mulatta Macaca munzala Macaca nemestrina Macaca nemestrina leonina Macaca nemestrina siberu Macaca nigra Macaca nigrescens Macaca ochreata Macaca pagensis Macaca radiata Macaca silenus Macaca sinica Macaca sylvanus Macaca thibetana Macaca tonkeana M andrillus leucophaeus M andrillus sphinx Papio anubis Papio cynocephalus Papio ham adryas Papio papio Papio ursinus Rungwecebus kipunji Theropithecus gelada Colobus angolensis Colobus angolensis palliatus Colobus guereza Colobus polykomos Colobus satanas Colobus vellerosus Nasalis larvatus Piliocolobus badius Piliocolobus foai Piliocolobus gordonorum Piliocolobus kirkii Piliocolobus pennantii Piliocolobus preussi Piliocolobus rufomitratus Piliocolobus tephrosceles Piliocolobus tholloni Presbytis comata Presbytis m elalophos Procolobus verus Pygathrix cinerea Pygathrix nemaeus Rhinopithecus avunculus Rhinopithecus bieti Rhinopithecus brelichi Rhinopithecus roxellana Semnopithecus entellus Trachypithecus auratus Trachypithecus cristatus Trachypithecus delacouri Trachypithecus francoisi Trachypithecus geei Trachypithecus germaini Trachypithecus johnii Trachypithecus laotum Trachypithecus obscurus Trachypithecus phayrei Trachypithecus pileatus Trachypithecus poliocephalus Trachypithecus vetulus Euoticus elegantulus Galago alleni Galago gallarum Galago granti Galago matschiei Galago moholi Galago senegalensis Galagoides demidoff Galagoides zanzibaricus Otolemur crassicaudatus Otolemur garnettii Tarsius bancanus Tarsius dentatus Tarsius lariang Tarsius syrichta
  46. 301 species Allenopithecus nigroviridis Cercopithecus albogularis Cercopithecus ascanius Cercopithecus campbelli

    Cercopithecus campbelli lowei Cercopithecus cephus Cercopithecus cephus cephu Cercopithecus cephus ngotto Cercopithecus diana Cercopithecus erythrogaster us erythrogaster erythrogaster Cercopithecus erythrotis Cercopithecus hamlyni Cercopithecus lhoesti Cercopithecus mitis Cercopithecus mona Cercopithecus neglectus Cercopithecus nictitans Cercopithecus petaurista Cercopithecus pogonias Cercopithecus preussi Cercopithecus solatus Cercopithecus wolfi Chlorocebus aethiops Chlorocebus pygerythrus Chlorocebus pygerythrus cynosurus Chlorocebus sabaeus Chlorocebus tantalus Erythrocebus patas Miopithecus talapoin Allocebus trichotis Archaeolemur majori Avahi cleesei Avahi laniger Avahi occidentalis Avahi unicolor Cheirogaleus crossleyi Cheirogaleus major Cheirogaleus m edius Daubentonia madagascariensis Eulemur coronatus Eulemur fulvus albifrons Eulemur fulvus albocollaris Eulemur fulvus collaris Eulemur fulvus fulvus Eulemur fulvus mayottensis Eulemur fulvus rufus Eulemur fulvus sanfordi Eulemur macaco flavifrons Eulemur macaco macaco Eulemur mongoz Eulemur rubriventer Hapalemur aureus Hapalemur griseus Hapalemur griseus alaotrensis Hapalemur griseus griseus Hapalemur griseus meridionalis Hapalemur griseus occidentalis Hapalemur simus Indri indri Lemur catta Lepilem ur aeeclis Lepilem ur ankaranensis Lepilem ur dorsalis Lepilem ur edwardsi Lepilem ur hubbardorum Lepilem ur leucopus Lepilem ur m anasam ody Lepilemur microdon Lepilem ur m itsinjoensis Lepilemur mustelinus Lepilem ur otto Lepilem ur randrianasoli Lepilem ur ruficaudatus Lepilem ur saham alazensis Lepilemur seali Lepilem ur septentrionalis Microcebus berthae Microcebus bongolavensis Microcebus danfossi Microcebus griseorufus Microcebus jollyae Microcebus lehilahytsara Microcebus lokobensis Microcebus macarthurii Microcebus mamiratra Microcebus mittermeieri Microcebus murinus Microcebus myoxinus Microcebus ravelobensis Microcebus rufus Microcebus sambiranensis Microcebus simmonsi Microcebus tavaratra Mirza coquereli Mirza zaza Phaner furcifer Phaner furcifer pallescens Propithecus coquereli Propithecus deckenii Propithecus diadema Propithecus edwardsi Propithecus tattersalli Propithecus verreauxi Varecia rubra Varecia variegata variegata Alouatta belzebul Alouatta caraya Alouatta guariba Alouatta palliata Alouatta pigra Alouatta sara Alouatta seniculus Ateles belzebuth Ateles fusciceps Ateles geoffroyi Ateles paniscus Brachyteles arachnoides Lagothrix lagotricha Aotus azarai Aotus azarai boliviensis Aotus brum backi Aotus infulatus Aotus lemurinus Aotus lemurinus griseimembra Aotus nancymaae Aotus nigriceps Aotus trivirgatus Aotus vociferans Callimico goeldii Callithrix argentata Callithrix aurita Callithrix emiliae Callithrix geoffroyi Callithrix humeralifera Callithrix jacchus Callithrix kuhli Callithrix mauesi Callithrix penicillata Callithrix pygmaea Cebus albifrons Cebus apella Cebus capucinus Cebus olivaceus Cebus xanthosternos Leontopithecus chrysomelas Leontopithecus chrysopygus Leontopithecus rosalia Saguinus bicolor Saguinus fuscicollis Saguinus fuscicollis m elanoleucus Saguinus geoffroyi Saguinus im perator Saguinus leucopus Saguinus midas Saguinus m ystax Saguinus niger Saguinus oedipus Saguinus tripartitus Saimiri boliviensis Saimiri oerstedii Saimiri sciureus Saimiri ustus Arctocebus aureus Arctocebus calabarensis Loris lydekkerianus Loris tardigradus Nycticebus bengalensis Nycticebus coucang Nycticebus javanicus Nycticebus menagensis Nycticebus pygmaeus Perodicticus potto Bunopithecus hoolock Gorilla beringei Gorilla gorilla gorilla Gorilla gorilla graueri Homo sapiens Homo sapiens neanderthalensis Hylobates agilis Hylobates klossii Hylobates lar Hylobates moloch Hylobates muelleri Hylobates pileatus Nomascus concolor Nomascus gabriellae Nomascus leucogenys Nomascus nasutus Nomascus siki Pan paniscus Pan troglodytes schweinfurthii Pan troglodytes troglodytes Pan troglodytes vellerosus Pan troglodytes verus Pongo abelii Pongo pygmaeus Symphalangus syndactylus Cacajao calvus Cacajao melanocephalus Callicebus donacophilus Callicebus hoffmannsi Callicebus moloch Callicebus personatus Callicebus torquatus Chiropotes satanas Pithecia irrorata Pithecia pithecia Cercocebus agilis Cercocebus galeritus Cercocebus torquatus Cercocebus torquatus atys Lophocebus albigena Lophocebus aterrimus Macaca arctoides Macaca assamensis M acaca brunnescens Macaca cyclopis Macaca fascicularis Macaca fuscata M acaca hecki M acaca leonina M acaca m aura Macaca mulatta Macaca munzala M acaca nem estrina M acaca nem estrina leonina M acaca nem estrina siberu M acaca nigra M acaca nigrescens M acaca ochreata M acaca pagensis M acaca radiata M acaca silenus Macaca sinica Macaca sylvanus Macaca thibetana M acaca tonkeana Mandrillus leucophaeus Mandrillus sphinx Papio anubis Papio cynocephalus Papio hamadryas Papio papio Papio ursinus Rungwecebus kipunji Theropithecus gelada Colobus angolensis Colobus angolensis palliatus Colobus guereza Colobus polykomos Colobus satanas Colobus vellerosus Nasalis larvatus Piliocolobus badius Piliocolobus foai Piliocolobus gordonorum Piliocolobus kirkii Piliocolobus pennantii Piliocolobus preussi Piliocolobus rufomitratus Piliocolobus tephrosceles Piliocolobus tholloni Presbytis comata Presbytis melalophos Procolobus verus Pygathrix cinerea Pygathrix nem aeus Rhinopithecus avunculus Rhinopithecus bieti Rhinopithecus brelichi Rhinopithecus roxellana Sem nopithecus entellus Trachypithecus auratus Trachypithecus cristatus Trachypithecus delacouri Trachypithecus francoisi Trachypithecus geei Trachypithecus germaini Trachypithecus johnii Trachypithecus laotum Trachypithecus obscurus Trachypithecus phayrei Trachypithecus pileatus Trachypithecus poliocephalus Trachypithecus vetulus Euoticus elegantulus Galago alleni Galago gallarum Galago granti Galago matschiei Galago moholi Galago senegalensis Galagoides demidoff Galagoides zanzibaricus Otolemur crassicaudatus Otolemur garnettii Tarsius bancanus Tarsius dentatus Tarsius lariang Tarsius syrichta
  47. Allenopithecus nigroviridis Cercopithecus albogularis Cercopithecus ascanius Cercopithecus campbelli Cercopithecus campbelli

    lowei Cercopithecus cephus Cercopithecus cephus cephu Cercopithecus cephus ngotto Cercopithecus diana Cercopithecus erythrogaster us erythrogaster erythrogaster Cercopithecus erythrotis Cercopithecus hamlyni Cercopithecus lhoesti Cercopithecus mitis Cercopithecus mona Cercopithecus neglectus Cercopithecus nictitans Cercopithecus petaurista Cercopithecus pogonias Cercopithecus preussi Cercopithecus solatus Cercopithecus wolfi Chlorocebus aethiops Chlorocebus pygerythrus Chlorocebus pygerythrus cynosurus Chlorocebus sabaeus Chlorocebus tantalus Erythrocebus patas Miopithecus talapoin Allocebus trichotis Archaeolemur majori Avahi cleesei Avahi laniger Avahi occidentalis Avahi unicolor Cheirogaleus crossleyi Cheirogaleus major Cheirogaleus m edius Daubentonia madagascariensis Eulemur coronatus Eulemur fulvus albifrons Eulemur fulvus albocollaris Eulemur fulvus collaris Eulemur fulvus fulvus Eulemur fulvus mayottensis Eulemur fulvus rufus Eulemur fulvus sanfordi Eulemur macaco flavifrons Eulemur macaco macaco Eulemur mongoz Eulemur rubriventer Hapalemur aureus Hapalemur griseus Hapalemur griseus alaotrensis Hapalemur griseus griseus Hapalemur griseus meridionalis Hapalemur griseus occidentalis Hapalemur simus Indri indri Lemur catta Lepilem ur aeeclis Lepilem ur ankaranensis Lepilem ur dorsalis Lepilem ur edwardsi Lepilem ur hubbardorum Lepilem ur leucopus Lepilem ur m anasam ody Lepilemur microdon Lepilem ur m itsinjoensis Lepilemur mustelinus Lepilem ur otto Lepilem ur randrianasoli Lepilem ur ruficaudatus Lepilem ur saham alazensis Lepilemur seali Lepilem ur septentrionalis Microcebus berthae Microcebus bongolavensis Microcebus danfossi Microcebus griseorufus Microcebus jollyae Microcebus lehilahytsara Microcebus lokobensis Microcebus macarthurii Microcebus mamiratra Microcebus mittermeieri Microcebus murinus Microcebus myoxinus Microcebus ravelobensis Microcebus rufus Microcebus sambiranensis Microcebus simmonsi Microcebus tavaratra Mirza coquereli Mirza zaza Phaner furcifer Phaner furcifer pallescens Propithecus coquereli Propithecus deckenii Propithecus diadema Propithecus edwardsi Propithecus tattersalli Propithecus verreauxi Varecia rubra Varecia variegata variegata Alouatta belzebul Alouatta caraya Alouatta guariba Alouatta palliata Alouatta pigra Alouatta sara Alouatta seniculus Ateles belzebuth Ateles fusciceps Ateles geoffroyi Ateles paniscus Brachyteles arachnoides Lagothrix lagotricha Aotus azarai Aotus azarai boliviensis Aotus brum backi Aotus infulatus Aotus lemurinus Aotus lemurinus griseimembra Aotus nancymaae Aotus nigriceps Aotus trivirgatus Aotus vociferans Callimico goeldii Callithrix argentata Callithrix aurita Callithrix emiliae Callithrix geoffroyi Callithrix humeralifera Callithrix jacchus Callithrix kuhli Callithrix mauesi Callithrix penicillata Callithrix pygmaea Cebus albifrons Cebus apella Cebus capucinus Cebus olivaceus Cebus xanthosternos Leontopithecus chrysomelas Leontopithecus chrysopygus Leontopithecus rosalia Saguinus bicolor Saguinus fuscicollis Saguinus fuscicollis m elanoleucus Saguinus geoffroyi Saguinus im perator Saguinus leucopus Saguinus midas Saguinus m ystax Saguinus niger Saguinus oedipus Saguinus tripartitus Saimiri boliviensis Saimiri oerstedii Saimiri sciureus Saimiri ustus Arctocebus aureus Arctocebus calabarensis Loris lydekkerianus Loris tardigradus Nycticebus bengalensis Nycticebus coucang Nycticebus javanicus Nycticebus menagensis Nycticebus pygmaeus Perodicticus potto Bunopithecus hoolock Gorilla beringei Gorilla gorilla gorilla Gorilla gorilla graueri Homo sapiens Homo sapiens neanderthalensis Hylobates agilis Hylobates klossii Hylobates lar Hylobates moloch Hylobates muelleri Hylobates pileatus Nomascus concolor Nomascus gabriellae Nomascus leucogenys Nomascus nasutus Nomascus siki Pan paniscus Pan troglodytes schweinfurthii Pan troglodytes troglodytes Pan troglodytes vellerosus Pan troglodytes verus Pongo abelii Pongo pygmaeus Symphalangus syndactylus Cacajao calvus Cacajao melanocephalus Callicebus donacophilus Callicebus hoffmannsi Callicebus moloch Callicebus personatus Callicebus torquatus Chiropotes satanas Pithecia irrorata Pithecia pithecia Cercocebus agilis Cercocebus galeritus Cercocebus torquatus Cercocebus torquatus atys Lophocebus albigena Lophocebus aterrimus Macaca arctoides Macaca assamensis M acaca brunnescens Macaca cyclopis Macaca fascicularis Macaca fuscata M acaca hecki M acaca leonina M acaca m aura Macaca mulatta Macaca munzala M acaca nem estrina M acaca nem estrina leonina M acaca nem estrina siberu M acaca nigra M acaca nigrescens M acaca ochreata M acaca pagensis M acaca radiata M acaca silenus Macaca sinica Macaca sylvanus Macaca thibetana M acaca tonkeana Mandrillus leucophaeus Mandrillus sphinx Papio anubis Papio cynocephalus Papio hamadryas Papio papio Papio ursinus Rungwecebus kipunji Theropithecus gelada Colobus angolensis Colobus angolensis palliatus Colobus guereza Colobus polykomos Colobus satanas Colobus vellerosus Nasalis larvatus Piliocolobus badius Piliocolobus foai Piliocolobus gordonorum Piliocolobus kirkii Piliocolobus pennantii Piliocolobus preussi Piliocolobus rufomitratus Piliocolobus tephrosceles Piliocolobus tholloni Presbytis comata Presbytis melalophos Procolobus verus Pygathrix cinerea Pygathrix nem aeus Rhinopithecus avunculus Rhinopithecus bieti Rhinopithecus brelichi Rhinopithecus roxellana Sem nopithecus entellus Trachypithecus auratus Trachypithecus cristatus Trachypithecus delacouri Trachypithecus francoisi Trachypithecus geei Trachypithecus germaini Trachypithecus johnii Trachypithecus laotum Trachypithecus obscurus Trachypithecus phayrei Trachypithecus pileatus Trachypithecus poliocephalus Trachypithecus vetulus Euoticus elegantulus Galago alleni Galago gallarum Galago granti Galago matschiei Galago moholi Galago senegalensis Galagoides demidoff Galagoides zanzibaricus Otolemur crassicaudatus Otolemur garnettii Tarsius bancanus Tarsius dentatus Tarsius lariang Tarsius syrichta 301 species 151 complete cases
  48. 0.0 0.5 1.0 Index Brain vol (cc) Mass (log kg)

    Group size (log)
  49. 4 6 8 10 12 1 2 3 4 5

    6 log(dstan$body) log(dstan$brain) log body mass log brain volume 0 1 2 3 4 1 2 3 4 5 6 log(dstan$group_size) log(dstan$brain) log group size 4 6 8 10 12 0 1 2 3 4 log(dstan$body) log(dstan$group_size) log body mass log group size log brain volume
  50. Causal Salad in Evolutionary Ecology Phylogenetic comparative methods dominated by

    causal salad Causal salad: Tossing factors into regression and interpreting every coefficient as causal “Controlling for phylogeny”: Required but mindless Regression + phylogeny still requires causal model Illustration by Julia Suits
  51. Brain Group size B M G Mass Social brain hypothesis

  52. B M G B M G B M G No

    interpretation without causal representation
  53. Brain Group size B M G Mass

  54. Brain Group size B M G Mass u

  55. Brain Group size B M G Mass u h history

  56. G1 B1

  57. G1 G2 B1 B2 G1 B1

  58. G1 G2 B1 B2 G1 G2 B1 B2 G1 B1

  59. G1 G3 G2 B1 B3 B2 G1 G2 B1 B2

    G1 G2 B1 B2 G1 B1
  60. G1 G3 G2 B1 B3 B2 G1 G2 B1 B2

    G1 G2 B1 B2 G1 B1 Observed
  61. G1 G3 G2 B1 B3 B2 G1 G2 B1 B2

    G1 G2 B1 B2 G1 B1 Observed
  62. Phylogenetic regression Two conjoint problems (1) What is the history

    (phylogeny)? (2) How to use it to model causes?
  63. Phylogenetic regression (1) What is the history (phylogeny)? Gotten much

    better with genomics Problems: Huge uncertainty in best case, process not stationary, no one phylogeny correct for all traits Cultural/linguistic phylogenies remain incredible Basic truth: Phylogenies do not exist
  64. Phylogenetic regression (2) How to use it to model causes?

    Suppose we have a phylogeny. Now what? No universally correct approach Default approach is a Gaussian process regression Brain Group size B M G Mass u h history
  65. B M G B i ∼ Normal(μ i , σ)

    μ i = α + β G G i + β M M i α ∼ Normal(0,1) β G , β M ∼ Normal(0,0.5) σ ∼ Exponential(1)
  66. B M G B ∼ MVNormal(μ, K) μ i =

    α + β G G i + β M M i α ∼ Normal(0,1) β G , β M ∼ Normal(0,0.5) σ ∼ Exponential(1) K = Iσ2
  67. B ∼ MVNormal(μ, K) μ i = α + β

    G G i + β M M i α ∼ Normal(0,1) β G , β M ∼ Normal(0,0.5) σ ∼ Exponential(1) K = Iσ2 I = 1 0 0 … 0 0 1 0 … 0 0 0 1 … 0 ⋮ ⋮ ⋮ ⋱ 0 0 0 0 … 1
  68. B ∼ MVNormal(μ, K) μ i = α + β

    G G i + β M M i α ∼ Normal(0,1) β G , β M ∼ Normal(0,0.5) σ ∼ Exponential(1) K = Iσ2 I = 1 0 0 … 0 0 1 0 … 0 0 0 1 … 0 ⋮ ⋮ ⋮ ⋱ 0 0 0 0 … 1 K = σ2 0 0 … 0 0 σ2 0 … 0 0 0 σ2 … 0 ⋮ ⋮ ⋮ ⋱ 0 0 0 0 … σ2
  69. dat_list <- list( N_spp = nrow(dstan), M = standardize(log(dstan$body)), B

    = standardize(log(dstan$brain)), G = standardize(log(dstan$group_size)), Imat = diag(nrow(dstan)) ) # classical regression form mBMG0 <- ulam( alist( B ~ normal( mu , sigma ), mu <- a + bM*M + bG*G, a ~ normal( 0 , 1 ), c(bM,bG) ~ normal( 0 , 0.5 ), sigma ~ exponential( 1 ) ), data=dat_list , chains=4 , cores=4 ) B i ∼ Normal(μ i , σ) # multivariate form mBMG <- ulam( alist( B ~ multi_normal( mu , K ), mu <- a + bM*M + bG*G, matrix[N_spp,N_spp]:K <- Imat*(sigma^2), a ~ normal( 0 , 1 ), c(bM,bG) ~ normal( 0 , 0.5 ), sigma ~ exponential( 1 ) ), data=dat_list , chains=4 , cores=4 ) B ∼ MVNormal(μ, K)
  70. # multivariate form mBMG <- ulam( alist( B ~ multi_normal(

    mu , K ), mu <- a + bM*M + bG*G, matrix[N_spp,N_spp]:K <- Imat*(sigma^2), a ~ normal( 0 , 1 ), c(bM,bG) ~ normal( 0 , 0.5 ), sigma ~ exponential( 1 ) ), data=dat_list , chains=4 , cores=4 ) dat_list <- list( N_spp = nrow(dstan), M = standardize(log(dstan$body)), B = standardize(log(dstan$brain)), G = standardize(log(dstan$group_size)), Imat = diag(nrow(dstan)) ) # classical regression form mBMG0 <- ulam( alist( B ~ normal( mu , sigma ), mu <- a + bM*M + bG*G, a ~ normal( 0 , 1 ), c(bM,bG) ~ normal( 0 , 0.5 ), sigma ~ exponential( 1 ) ), data=dat_list , chains=4 , cores=4 ) B i ∼ Normal(μ i , σ) B ∼ MVNormal(μ, K) > precis( mBMG ) mean sd 5.5% 94.5% n_eff Rhat4 a 0.00 0.02 -0.03 0.03 1880 1 bG 0.12 0.02 0.09 0.16 1384 1 bM 0.89 0.02 0.86 0.93 1395 1 sigma 0.22 0.01 0.20 0.24 1433 1 > precis( mBMG0 ) mean sd 5.5% 94.5% n_eff Rhat4 a 0.00 0.02 -0.03 0.03 1740 1 bG 0.12 0.02 0.09 0.16 1491 1 bM 0.89 0.02 0.86 0.93 1439 1 sigma 0.22 0.01 0.20 0.24 1706 1
  71. B ∼ MVNormal(μ, K) μ i = α + β

    G G i + β M M i + u i α ∼ Normal(0,1) β G , β M ∼ Normal(0,0.5) σ ∼ Exponential(1) K = Iσ2 B M G u h history
  72. None
  73. None
  74. None
  75. None
  76. From Model to Kernel Evolutionary model + tree structure =

    pattern of covariation at tips Covariance declines with phylogenetic distance Phylogenetic distance: Branch length from one species to another
  77. From Model to Kernel 0 50 100 150 0 20

    40 60 phylogenetic distance covariance Evolutionary model + tree structure = pattern of covariation at tips Common simple models: Brownian motion Ornstein-Uhlenbeck (damped Brownian motion) Brownian m otion O rnstein-U hlenbeck
  78. B ∼ MVNormal(μ, K) μ i = α + β

    G G i + β M M i α ∼ Normal(0,1) β G , β M ∼ Normal(0,0.5) σ ∼ Exponential(1) K = Iσ2
  79. B ∼ MVNormal(μ, K) μ i = α + β

    G G i + β M M i α ∼ Normal(0,1) β G , β M ∼ Normal(0,0.5) η2 ∼ HalfNormal(1,0.25) K = η2 exp(−ρd i,j ) ρ ∼ HalfNormal(3,0.25) K Ornstein-Uhlenbeck kernel Maximum covariance prior Rate prior
  80. B ∼ MVNormal(μ, K) μ i = α + β

    G G i + β M M i α ∼ Normal(0,1) β G , β M ∼ Normal(0,0.5) η2 ∼ HalfNormal(1,0.25) K = η2 exp(−ρd i,j ) 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.5 1.0 1.5 phylogenetic distance covariance ρ ∼ HalfNormal(3,0.25) K covariance function prior
  81. B ∼ MVNormal(μ, K) μ i = α α ∼

    Normal(0,1) η2 ∼ HalfNormal(1,0.25) K = η2 exp(−ρd i,j ) 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.5 1.0 1.5 phylogenetic distance covariance ρ ∼ HalfNormal(3,0.25) K covariance function prior
  82. # Ornstein-Uhlenbeck (L1 gaussian process) # add scaled and reordered

    distance matrix dat_list$Dmat <- Dmat[ spp_obs , spp_obs ] / max(Dmat) mB_OU <- ulam( alist( B ~ multi_normal( mu , K ), mu <- a + 0*M, matrix[N_spp,N_spp]:K <- cov_GPL1(Dmat,etasq,rho,0.01), a ~ normal(0,1), etasq ~ half_normal(1,0.25), rho ~ half_normal(3,0.25) ), data=dat_list , chains=4 , cores=4 ) B ∼ MVNormal(μ, K) μ i = α α ∼ Normal(0,1) η2 ∼ HalfNormal(1,0.25) K = η2 exp(−ρd i,j ) ρ ∼ HalfNormal(3,0.25)
  83. # Ornstein-Uhlenbeck (L1 gaussian process) # add scaled and reordered

    distance matrix dat_list$Dmat <- Dmat[ spp_obs , spp_obs ] / max(Dmat) mB_OU <- ulam( alist( B ~ multi_normal( mu , K ), mu <- a + 0*M, matrix[N_spp,N_spp]:K <- cov_GPL1(Dmat,etasq,rho,0.01), a ~ normal(0,1), etasq ~ half_normal(1,0.25), rho ~ half_normal(3,0.25) ), data=dat_list , chains=4 , cores=4 ) B ∼ MVNormal(μ, K) μ i = α α ∼ Normal(0,1) η2 ∼ HalfNormal(1,0.25) K = η2 exp(−ρd i,j ) ρ ∼ HalfNormal(3,0.25)
  84. B ∼ MVNormal(μ, K) μ i = α α ∼

    Normal(0,1) η2 ∼ HalfNormal(1,0.25) K = η2 exp(−ρd i,j ) ρ ∼ HalfNormal(3,0.25) 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.5 1.0 1.5 phylogenetic distance covariance prior posterior
  85. mBMG_OU <- ulam( alist( B ~ multi_normal( mu , K

    ), mu <- a + bM*M + bG*G, matrix[N_spp,N_spp]:K <- cov_GPL1(Dmat,etasq,rho,0.01), a ~ normal(0,1), c(bM,bG) ~ normal(0,0.5), etasq ~ half_normal(1,0.25), rho ~ half_normal(3,0.25) ), data=dat_list , chains=4 , cores=4 ) B ∼ MVNormal(μ, K) μ i = α + β G G i + β M M i α ∼ Normal(0,1) β G , β M ∼ Normal(0,0.5) η2 ∼ HalfNormal(1,0.25) K = η2 exp(−ρd i,j ) ρ ∼ HalfNormal(3,0.25)
  86. B ∼ MVNormal(μ, K) μ i = α + β

    G G i + β M M i α ∼ Normal(0,1) β G , β M ∼ Normal(0,0.5) η2 ∼ HalfNormal(1,0.25) K = η2 exp(−ρd i,j ) ρ ∼ HalfNormal(3,0.25) 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.5 1.0 1.5 phylogenetic distance covariance prior posterior B posterior BMG
  87. Ornstein-Uhlenbeck ordinary -0.05 0.00 0.05 0.10 0.15 0.20 0 5

    10 15 effect of group size Density B M G u h
  88. Phylogenetic regression Lingering problems: (1) What about phylogenetic uncertainty? (2)

    Don’t these traits influence one another reciprocally over time?
  89. LETTER RESEARCH González-Forero & Gardner 2018 Optimal life history approach

    Ringen, Martin, Jaeggi 2021 Drift-coevolution dynamics
  90. Gaussian Processes Partial pooling for continuous categories Very general approximation

    engine Causal theory => covariance kernel Sensitivity to kernel priors — choose wisely
  91. Gaussian Possibilities Automatic relevance determination (ARD): Multiple distance dimensions inside

    the kernel Multi-output Gaussian processes: Draw vectors from kernel Telemetry, navigation: Real-time tracking and error correction (Kálmán filter)
  92. Course Schedule Week 1 Bayesian inference Chapters 1, 2, 3

    Week 2 Linear models & Causal Inference Chapter 4 Week 3 Causes, Confounds & Colliders Chapters 5 & 6 Week 4 Overfitting / MCMC Chapters 7, 8, 9 Week 5 Generalized Linear Models Chapters 10, 11 Week 6 Ordered categories & Multilevel models Chapters 12 & 13 Week 7 More Multilevel models Chapters 13 & 14 Week 8 Social Networks & Gaussian Processes Chapter 14 Week 9 Measurement & Missingness Chapter 15 Week 10 Generalized Linear Madness Chapter 16 https://github.com/rmcelreath/stat_rethinking_2022
  93. None