How to use more than one cluster type at the same time? For example stories and participants (2) How to calculate predictions (3) How to sample chains efficiently 2 4 6 8 10 12 1 2 3 4 5 6 7 Story Response 0 10 20 30 40 50 1 2 3 4 5 6 7 Participant Response
5 6 7 Story Response 0 10 20 30 40 50 1 2 3 4 5 6 7 Participant Response Add clusters: More index variables, more population priors (this lecture) Add features: More parameters, more dimensions in each population prior (next lecture) Cluster tanks stories individuals departments Features survival treatment effect average response admission rate, bias
i ) α j ∼ Normal( ¯ α, σ A ) log-odds of left lever Prior treatment/block effects Prior for actor effects (handedness) logit(p i ) = β T[i],B[i] + α A[i] β j,k ∼ Normal(0,σ B )
i ) α j ∼ Normal( ¯ α, σ A ) σ A , σ B ∼ Exponential(1) log-odds of left lever Prior treatment/block effects Prior for actor effects (handedness) Prior for “variance components” logit(p i ) = β T[i],B[i] + α A[i] β j,k ∼ Normal(0,σ B )
treatment effects? Because treatments are not completely different Because there are many possible treatments, you used a few Because it results in better estimates If parameters get the same prior, usually better to learn the prior from the sample β j,k ∼ Normal(0,σ B )
additive Total variation in outcome is sum of the components Not true for generalized linear models Link function breaks additivity Variation in one component moderates variation in the others 0 1 2 3 4 5 0.0 0.4 0.8 sigma_A Density 0 1 2 3 4 5 0.0 1.5 sigma_B Density actor variation treatment/block variation
(causal effects)? Predict for same groups: Use varying effect estimates for each group Predict for new groups: Ignore varying effect estimates, marginalize over population distribution 0 1 2 3 4 5 0.0 0.4 0.8 sigma_A Density 0 1 2 3 4 5 0.0 1.5 sigma_B Density actor variation treatment/block variation
1000 n_samples <- 2000 S1 <- matrix(0,nrow=n_samples,ncol=n_groups) for ( s in 1:n_groups ) { # sim a tank from posterior population aT <- rnorm(n_samples,0,post$sigma) # sample P and G for this group P <- sample( 1:2 , size=1 , prob=c(0.5,0.5) ) # 50% pred G <- sample( 1:2 , size=1 , prob=c(0.75,0.25) ) # 25% large # sim survival p <- inv_logit( aT + post$b[,P,G] ) S1[,s] <- rbinom(2000,35,p) }
1000 n_samples <- 2000 S1 <- matrix(0,nrow=n_samples,ncol=n_groups) for ( s in 1:n_groups ) { # sim a tank from posterior population aT <- rnorm(n_samples,0,post$sigma) # sample P and G for this group P <- sample( 1:2 , size=1 , prob=c(0.5,0.5) ) # 50% pred G <- sample( 1:2 , size=1 , prob=c(0.75,0.25) ) # 25% large # sim survival p <- inv_logit( aT + post$b[,P,G] ) S1[,s] <- rbinom(2000,35,p) }
1000 n_samples <- 2000 S1 <- matrix(0,nrow=n_samples,ncol=n_groups) for ( s in 1:n_groups ) { # sim a tank from posterior population aT <- rnorm(n_samples,0,post$sigma) # sample P and G for this group P <- sample( 1:2 , size=1 , prob=c(0.5,0.5) ) # 50% pred G <- sample( 1:2 , size=1 , prob=c(0.75,0.25) ) # 25% large # sim survival p <- inv_logit( aT + post$b[,P,G] ) S1[,s] <- rbinom(2000,35,p) }
1000 n_samples <- 2000 S1 <- matrix(0,nrow=n_samples,ncol=n_groups) for ( s in 1:n_groups ) { # sim a tank from posterior population aT <- rnorm(n_samples,0,post$sigma) # sample P and G for this group P <- sample( 1:2 , size=1 , prob=c(0.5,0.5) ) # 50% pred G <- sample( 1:2 , size=1 , prob=c(0.75,0.25) ) # 25% large # sim survival p <- inv_logit( aT + post$b[,P,G] ) S1[,s] <- rbinom(2000,35,p) }
1000 n_samples <- 2000 S1 <- matrix(0,nrow=n_samples,ncol=n_groups) for ( s in 1:n_groups ) { # sim a tank from posterior population aT <- rnorm(n_samples,0,post$sigma) # sample P and G for this group P <- sample( 1:2 , size=1 , prob=c(0.5,0.5) ) # 50% pred G <- sample( 1:2 , size=1 , prob=c(0.75,0.25) ) # 25% large # sim survival p <- inv_logit( aT + post$b[,P,G] ) S1[,s] <- rbinom(2000,35,p) } 0 50000 100000 150000 number surviving Frequency 0 3 6 9 13 18 23 28 33 Reedfrog status quo
s in 1:n_groups ) { # sim a tank from posterior population aT <- rnorm(n_samples,0,post$sigma) # sample P and G for this group P <- sample( 1:2 , size=1 , prob=c(0.5,0.5) ) # 50% pred G <- sample( 1:2 , size=1 , prob=c(0.25,0.75) ) # 75% large # sim survival p <- inv_logit( aT + post$b[,P,G] ) S2[,s] <- rbinom(n_samples,35,p) } 0 50000 100000 150000 number surviving Frequency 0 3 6 9 13 18 23 28 33 Reedfrog intervention
variation means simulating groups (or using estimates from observed groups as appropriate) If you have a generative model, you can simulate interventions for new targets 0 50000 100000 150000 number surviving Frequency 0 3 6 9 13 18 23 28 33 0 50000 100000 150000 number surviving Frequency 0 3 6 9 13 18 23 28 33
∼ Normal( ¯ α, σ A ) σ A , σ B ∼ Exponential(1) logit(p i ) = β T[i],B[i] + α A[i] β j,k ∼ Normal(0,σ B ) P i ∼ Bernoulli(p i ) logit(p i ) = ¯ α + (z α,A[i] )σ A + (z β,T[i],B[i] )σ B z α,j ∼ Normal(0,1) z β,j ∼ Normal(0,1) σ A , σ B ∼ Exponential(1) ¯ α ∼ Normal(0,1.5) ¯ α ∼ Normal(0,1.5) “Centered” “Non-centered”
∼ Normal( ¯ α, σ A ) σ A , σ B ∼ Exponential(1) logit(p i ) = β T[i],B[i] + α A[i] β j,k ∼ Normal(0,σ B ) P i ∼ Bernoulli(p i ) logit(p i ) = ¯ α + (z α,A[i] )σ A + (z β,T[i],B[i] )σ B z α,j ∼ Normal(0,1) z β,j ∼ Normal(0,1) σ A , σ B ∼ Exponential(1) ¯ α ∼ Normal(0,1.5) ¯ α ∼ Normal(0,1.5) “Centered” “Non-centered”
∼ Normal( ¯ α, σ A ) σ A , σ B ∼ Exponential(1) logit(p i ) = β T[i],B[i] + α A[i] β j,k ∼ Normal(0,σ B ) P i ∼ Bernoulli(p i ) logit(p i ) = ¯ α + (z α,A[i] )σ A + (z β,T[i],B[i] )σ B z α,j ∼ Normal(0,1) z β,j ∼ Normal(0,1) σ A , σ B ∼ Exponential(1) ¯ α ∼ Normal(0,1.5) ¯ α ∼ Normal(0,1.5) “Centered” “Non-centered”
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