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