book_code.R

## R code 0.1
print( "All models are wrong, but some are useful." )

## R code 0.2
x <- 1:2
x <- x*10
x <- log(x)
x <- sum(x)
x <- exp(x)
x

## R code 0.3
( log( 0.01^200 ) )
( 200 * log(0.01) )

## R code 0.4
# Load the data:
# car braking distances in feet paired with speeds in km/h
# see ?cars for details
data(cars)

# fit a linear regression of distance on speed
m <- lm( dist ~ speed , data=cars )

# estimated coefficients from the model
coef(m)

# plot residuals against speed
plot( resid(m) ~ speed , data=cars )

## R code 0.5
install.packages(c("coda","mvtnorm","devtools"))
library(devtools)
devtools::install_github("rmcelreath/rethinking",ref="Experimental")

## R code 2.1
ways <- c( 0 , 3 , 8 , 9 , 0 )
ways/sum(ways)

## R code 2.2
dbinom( 6 , size=9 , prob=0.5 )

## R code 2.3
# define grid
p_grid <- seq( from=0 , to=1 , length.out=20 )

# define prior
prior <- rep( 1 , 20 )

# compute likelihood at each value in grid
likelihood <- dbinom( 6 , size=9 , prob=p_grid )

# compute product of likelihood and prior
unstd.posterior <- likelihood * prior

# standardize the posterior, so it sums to 1
posterior <- unstd.posterior / sum(unstd.posterior)

## R code 2.4
plot( p_grid , posterior , type="b" ,
    xlab="probability of water" , ylab="posterior probability" )
mtext( "20 points" )

## R code 2.5
prior <- ifelse( p_grid < 0.5 , 0 , 1 )
prior <- exp( -5*abs( p_grid - 0.5 ) )

## R code 2.6
library(rethinking)
globe.qa <- quap(
    alist(
        W ~ dbinom( W+L ,p) ,  # binomial likelihood
        p ~ dunif(0,1)     # uniform prior
    ) ,
    data=list(W=6,L=3) )

# display summary of quadratic approximation
precis( globe.qa )

## R code 2.7
# analytical calculation
W <- 6
L <- 3
curve( dbeta( x , W+1 , L+1 ) , from=0 , to=1 )
# quadratic approximation
curve( dnorm( x , 0.67 , 0.16 ) , lty=2 , add=TRUE )

## R code 2.8
n_samples <- 1000
p <- rep( NA , n_samples )
p[1] <- 0.5
W <- 6
L <- 3
for ( i in 2:n_samples ) {
    p_new <- rnorm( 1 , p[i-1] , 0.1 )
    if ( p_new < 0 ) p_new <- abs( p_new )
    if ( p_new > 1 ) p_new <- 2 - p_new
    q0 <- dbinom( W , W+L , p[i-1] )
    q1 <- dbinom( W , W+L , p_new )
    p[i] <- ifelse( runif(1) < q1/q0 , p_new , p[i-1] )
}

## R code 2.9
dens( p , xlim=c(0,1) )
curve( dbeta( x , W+1 , L+1 ) , lty=2 , add=TRUE )

## R code 3.1
Pr_Positive_Vampire <- 0.95
Pr_Positive_Mortal <- 0.01
Pr_Vampire <- 0.001
Pr_Positive <- Pr_Positive_Vampire * Pr_Vampire +
               Pr_Positive_Mortal * ( 1 - Pr_Vampire )
( Pr_Vampire_Positive <- Pr_Positive_Vampire*Pr_Vampire / Pr_Positive )

## R code 3.2
p_grid <- seq( from=0 , to=1 , length.out=1000 )
prob_p <- rep( 1 , 1000 )
prob_data <- dbinom( 6 , size=9 , prob=p_grid )
posterior <- prob_data * prob_p
posterior <- posterior / sum(posterior)

## R code 3.3
samples <- sample( p_grid , prob=posterior , size=1e4 , replace=TRUE )

## R code 3.4
plot( samples )

## R code 3.5
library(rethinking)
dens( samples )

## R code 3.6
# add up posterior probability where p < 0.5
sum( posterior[ p_grid < 0.5 ] )

## R code 3.7
sum( samples < 0.5 ) / 1e4

## R code 3.8
sum( samples > 0.5 & samples < 0.75 ) / 1e4

## R code 3.9
quantile( samples , 0.8 )

## R code 3.10
quantile( samples , c( 0.1 , 0.9 ) )

## R code 3.11
p_grid <- seq( from=0 , to=1 , length.out=1000 )
prior <- rep(1,1000)
likelihood <- dbinom( 3 , size=3 , prob=p_grid )
posterior <- likelihood * prior
posterior <- posterior / sum(posterior)
samples <- sample( p_grid , size=1e4 , replace=TRUE , prob=posterior )

## R code 3.12
PI( samples , prob=0.5 )

## R code 3.13
HPDI( samples , prob=0.5 )

## R code 3.14
p_grid[ which.max(posterior) ]

## R code 3.15
chainmode( samples , adj=0.01 )

## R code 3.16
mean( samples )
median( samples )

## R code 3.17
sum( posterior*abs( 0.5 - p_grid ) )

## R code 3.18
loss <- sapply( p_grid , function(d) sum( posterior*abs( d - p_grid ) ) )

## R code 3.19
p_grid[ which.min(loss) ]

## R code 3.20
dbinom( 0:2 , size=2 , prob=0.7 )

## R code 3.21
rbinom( 1 , size=2 , prob=0.7 )

## R code 3.22
rbinom( 10 , size=2 , prob=0.7 )

## R code 3.23
dummy_w <- rbinom( 1e5 , size=2 , prob=0.7 )
table(dummy_w)/1e5

## R code 3.24
dummy_w <- rbinom( 1e5 , size=9 , prob=0.7 )
simplehist( dummy_w , xlab="dummy water count" )

## R code 3.25
w <- rbinom( 1e4 , size=9 , prob=0.6 )

## R code 3.26
w <- rbinom( 1e4 , size=9 , prob=samples )

## R code 3.27
p_grid <- seq( from=0 , to=1 , length.out=1000 )
prior <- rep( 1 , 1000 )
likelihood <- dbinom( 6 , size=9 , prob=p_grid )
posterior <- likelihood * prior
posterior <- posterior / sum(posterior)
set.seed(100)
samples <- sample( p_grid , prob=posterior , size=1e4 , replace=TRUE )

## R code 3.28
birth1 <- c(1,0,0,0,1,1,0,1,0,1,0,0,1,1,0,1,1,0,0,0,1,0,0,0,1,0,
0,0,0,1,1,1,0,1,0,1,1,1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0,0,0,0,0,0,
1,1,0,1,0,0,1,0,0,0,1,0,0,1,1,1,1,0,1,0,1,1,1,1,1,0,0,1,0,1,1,0,
1,0,1,1,1,0,1,1,1,1)
birth2 <- c(0,1,0,1,0,1,1,1,0,0,1,1,1,1,1,0,0,1,1,1,0,0,1,1,1,0,
1,1,1,0,1,1,1,0,1,0,0,1,1,1,1,0,0,1,0,1,1,1,1,1,1,1,1,1,1,1,1,1,
1,1,1,0,1,1,0,1,1,0,1,1,1,0,0,0,0,0,0,1,0,0,0,1,1,0,0,1,0,0,1,1,
0,0,0,1,1,1,0,0,0,0)

## R code 3.29
library(rethinking)
data(homeworkch3)

## R code 3.30
sum(birth1) + sum(birth2)

## R code 4.1
pos <- replicate( 1000 , sum( runif(16,-1,1) ) )

## R code 4.2
prod( 1 + runif(12,0,0.1) )

## R code 4.3
growth <- replicate( 10000 , prod( 1 + runif(12,0,0.1) ) )
dens( growth , norm.comp=TRUE )

## R code 4.4
big <- replicate( 10000 , prod( 1 + runif(12,0,0.5) ) )
small <- replicate( 10000 , prod( 1 + runif(12,0,0.01) ) )

## R code 4.5
log.big <- replicate( 10000 , log(prod(1 + runif(12,0,0.5))) )

## R code 4.6
w <- 6; n <- 9;
p_grid <- seq(from=0,to=1,length.out=100)
posterior <- dbinom(w,n,p_grid)*dunif(p_grid,0,1)
posterior <- posterior/sum(posterior)

## R code 4.7
library(rethinking)
data(Howell1)
d <- Howell1

## R code 4.8
str( d )

## R code 4.9
precis( d )

## R code 4.10
d$height

## R code 4.11
d2 <- d[ d$age >= 18 , ]

## R code 4.12
curve( dnorm( x , 178 , 20 ) , from=100 , to=250 )

## R code 4.13
curve( dunif( x , 0 , 50 ) , from=-10 , to=60 )

## R code 4.14
sample_mu <- rnorm( 1e4 , 178 , 20 )
sample_sigma <- runif( 1e4 , 0 , 50 )
prior_h <- rnorm( 1e4 , sample_mu , sample_sigma )
dens( prior_h )

## R code 4.15
sample_mu <- rnorm( 1e4 , 178 , 100 )
prior_h <- rnorm( 1e4 , sample_mu , sample_sigma )
dens( prior_h )

## R code 4.16
mu.list <- seq( from=150, to=160 , length.out=100 )
sigma.list <- seq( from=7 , to=9 , length.out=100 )
post <- expand.grid( mu=mu.list , sigma=sigma.list )
post$LL <- sapply( 1:nrow(post) , function(i) sum(
    dnorm( d2$height , post$mu[i] , post$sigma[i] , log=TRUE ) ) )
post$prod <- post$LL + dnorm( post$mu , 178 , 20 , TRUE ) +
    dunif( post$sigma , 0 , 50 , TRUE )
post$prob <- exp( post$prod - max(post$prod) )

## R code 4.17
contour_xyz( post$mu , post$sigma , post$prob )

## R code 4.18
image_xyz( post$mu , post$sigma , post$prob )

## R code 4.19
sample.rows <- sample( 1:nrow(post) , size=1e4 , replace=TRUE ,
    prob=post$prob )
sample.mu <- post$mu[ sample.rows ]
sample.sigma <- post$sigma[ sample.rows ]

## R code 4.20
plot( sample.mu , sample.sigma , cex=0.5 , pch=16 , col=col.alpha(rangi2,0.1) )

## R code 4.21
dens( sample.mu )
dens( sample.sigma )

## R code 4.22
PI( sample.mu )
PI( sample.sigma )

## R code 4.23
d3 <- sample( d2$height , size=20 )

## R code 4.24
mu.list <- seq( from=150, to=170 , length.out=200 )
sigma.list <- seq( from=4 , to=20 , length.out=200 )
post2 <- expand.grid( mu=mu.list , sigma=sigma.list )
post2$LL <- sapply( 1:nrow(post2) , function(i)
    sum( dnorm( d3 , mean=post2$mu[i] , sd=post2$sigma[i] ,
    log=TRUE ) ) )
post2$prod <- post2$LL + dnorm( post2$mu , 178 , 20 , TRUE ) +
    dunif( post2$sigma , 0 , 50 , TRUE )
post2$prob <- exp( post2$prod - max(post2$prod) )
sample2.rows <- sample( 1:nrow(post2) , size=1e4 , replace=TRUE ,
    prob=post2$prob )
sample2.mu <- post2$mu[ sample2.rows ]
sample2.sigma <- post2$sigma[ sample2.rows ]
plot( sample2.mu , sample2.sigma , cex=0.5 ,
    col=col.alpha(rangi2,0.1) ,
    xlab="mu" , ylab="sigma" , pch=16 )

## R code 4.25
dens( sample2.sigma , norm.comp=TRUE )

## R code 4.26
library(rethinking)
data(Howell1)
d <- Howell1
d2 <- d[ d$age >= 18 , ]

## R code 4.27
flist <- alist(
    height ~ dnorm( mu , sigma ) ,
    mu ~ dnorm( 178 , 20 ) ,
    sigma ~ dunif( 0 , 50 )
)

## R code 4.28
m4.1 <- quap( flist , data=d2 )

## R code 4.29
precis( m4.1 )

## R code 4.30
start <- list(
    mu=mean(d2$height),
    sigma=sd(d2$height)
)
m4.1 <- quap( flist , data=d2 , start=start )

## R code 4.31
m4.2 <- quap(
    alist(
        height ~ dnorm( mu , sigma ) ,
        mu ~ dnorm( 178 , 0.1 ) ,
        sigma ~ dunif( 0 , 50 )
    ) , data=d2 )
precis( m4.2 )

## R code 4.32
vcov( m4.1 )

## R code 4.33
diag( vcov( m4.1 ) )
cov2cor( vcov( m4.1 ) )

## R code 4.34
library(rethinking)
post <- extract.samples( m4.1 , n=1e4 )
head(post)

## R code 4.35
precis(post)

## R code 4.36
library(MASS)
post <- mvrnorm( n=1e4 , mu=coef(m4.1) , Sigma=vcov(m4.1) )

## R code 4.37
plot( d2$height ~ d2$weight )

## R code 4.38
set.seed(2971)
N <- 100                   # 100 lines
a <- rnorm( N , 178 , 20 )
b <- rnorm( N , 0 , 10 )

## R code 4.39
plot( NULL , xlim=range(d2$weight) , ylim=c(-100,400) ,
    xlab="weight" , ylab="height" )
abline( h=0 , lty=2 )
abline( h=272 , lty=1 , lwd=0.5 )
mtext( "b ~ dnorm(0,10)" )
xbar <- mean(d2$weight)
for ( i in 1:N ) curve( a[i] + b[i]*(x - xbar) ,
    from=min(d2$weight) , to=max(d2$weight) , add=TRUE ,
    col=col.alpha("black",0.2) )

## R code 4.40
b <- rlnorm( 1e4 , 0 , 1 )
dens( b , xlim=c(0,5) , adj=0.1 )

## R code 4.41
set.seed(2971)
N <- 100                   # 100 lines
a <- rnorm( N , 178 , 20 )
b <- rlnorm( N , 0 , 1 )

## R code 4.42
# load data again, since it's a long way back
library(rethinking)
data(Howell1)
d <- Howell1
d2 <- d[ d$age >= 18 , ]

# define the average weight, x-bar
xbar <- mean(d2$weight)

# fit model
m4.3 <- quap(
    alist(
        height ~ dnorm( mu , sigma ) ,
        mu <- a + b*( weight - xbar ) ,
        a ~ dnorm( 178 , 20 ) ,
        b ~ dlnorm( 0 , 1 ) ,
        sigma ~ dunif( 0 , 50 )
    ) ,
    data=d2 )

## R code 4.43
m4.3b <- quap(
    alist(
        height ~ dnorm( mu , sigma ) ,
        mu <- a + exp(log_b)*( weight - xbar ),
        a ~ dnorm( 178 , 20 ) ,
        log_b ~ dnorm( 0 , 1 ) ,
        sigma ~ dunif( 0 , 50 )
    ) ,
    data=d2 )

## R code 4.44
precis( m4.3 )

## R code 4.45
round( vcov( m4.3 ) , 3 )

## R code 4.46
plot( height ~ weight , data=d2 , col=rangi2 )
post <- extract.samples( m4.3 )
a_map <- mean(post$a)
b_map <- mean(post$b)
curve( a_map + b_map*(x - xbar) , add=TRUE )

## R code 4.47
post <- extract.samples( m4.3 )
post[1:5,]

## R code 4.48
N <- 10
dN <- d2[ 1:N , ]
mN <- quap(
    alist(
        height ~ dnorm( mu , sigma ) ,
        mu <- a + b*( weight - mean(weight) ) ,
        a ~ dnorm( 178 , 20 ) ,
        b ~ dlnorm( 0 , 1 ) ,
        sigma ~ dunif( 0 , 50 )
    ) , data=dN )

## R code 4.49
# extract 20 samples from the posterior
post <- extract.samples( mN , n=20 )

# display raw data and sample size
plot( dN$weight , dN$height ,
    xlim=range(d2$weight) , ylim=range(d2$height) ,
    col=rangi2 , xlab="weight" , ylab="height" )
mtext(concat("N = ",N))

# plot the lines, with transparency
for ( i in 1:20 )
    curve( post$a[i] + post$b[i]*(x-mean(dN$weight)) ,
        col=col.alpha("black",0.3) , add=TRUE )

## R code 4.50
post <- extract.samples( m4.3 )
mu_at_50 <- post$a + post$b * ( 50 - xbar )

## R code 4.51
dens( mu_at_50 , col=rangi2 , lwd=2 , xlab="mu|weight=50" )

## R code 4.52
PI( mu_at_50 , prob=0.89 )

## R code 4.53
mu <- link( m4.3 )
str(mu)

## R code 4.54
# define sequence of weights to compute predictions for
# these values will be on the horizontal axis
weight.seq <- seq( from=25 , to=70 , by=1 )

# use link to compute mu
# for each sample from posterior
# and for each weight in weight.seq
mu <- link( m4.3 , data=data.frame(weight=weight.seq) )
str(mu)

## R code 4.55
# use type="n" to hide raw data
plot( height ~ weight , d2 , type="n" )

# loop over samples and plot each mu value
for ( i in 1:100 )
    points( weight.seq , mu[i,] , pch=16 , col=col.alpha(rangi2,0.1) )

## R code 4.56
# summarize the distribution of mu
mu.mean <- apply( mu , 2 , mean )
mu.PI <- apply( mu , 2 , PI , prob=0.89 )

## R code 4.57
# plot raw data
# fading out points to make line and interval more visible
plot( height ~ weight , data=d2 , col=col.alpha(rangi2,0.5) )

# plot the MAP line, aka the mean mu for each weight
lines( weight.seq , mu.mean )

# plot a shaded region for 89% PI
shade( mu.PI , weight.seq )

## R code 4.58
post <- extract.samples(m4.3)
mu.link <- function(weight) post$a + post$b*( weight - xbar )
weight.seq <- seq( from=25 , to=70 , by=1 )
mu <- sapply( weight.seq , mu.link )
mu.mean <- apply( mu , 2 , mean )
mu.CI <- apply( mu , 2 , PI , prob=0.89 )

## R code 4.59
sim.height <- sim( m4.3 , data=list(weight=weight.seq) )
str(sim.height)

## R code 4.60
height.PI <- apply( sim.height , 2 , PI , prob=0.89 )

## R code 4.61
# plot raw data
plot( height ~ weight , d2 , col=col.alpha(rangi2,0.5) )

# draw MAP line
lines( weight.seq , mu.mean )

# draw HPDI region for line
shade( mu.HPDI , weight.seq )

# draw PI region for simulated heights
shade( height.PI , weight.seq )

## R code 4.62
sim.height <- sim( m4.3 , data=list(weight=weight.seq) , n=1e4 )
height.PI <- apply( sim.height , 2 , PI , prob=0.89 )

## R code 4.63
post <- extract.samples(m4.3)
weight.seq <- 25:70
sim.height <- sapply( weight.seq , function(weight)
    rnorm(
        n=nrow(post) ,
        mean=post$a + post$b*( weight - xbar ) ,
        sd=post$sigma ) )
height.PI <- apply( sim.height , 2 , PI , prob=0.89 )

## R code 4.64
library(rethinking)
data(Howell1)
d <- Howell1
str(d)

## R code 4.65
d$weight_s <- ( d$weight - mean(d$weight) )/sd(d$weight)
d$weight_s2 <- d$weight_s^2
m4.5 <- quap(
    alist(
        height ~ dnorm( mu , sigma ) ,
        mu <- a + b1*weight_s + b2*weight_s2 ,
        a ~ dnorm( 178 , 20 ) ,
        b1 ~ dlnorm( 0 , 1 ) ,
        b2 ~ dnorm( 0 , 1 ) ,
        sigma ~ dunif( 0 , 50 )
    ) ,
    data=d )

## R code 4.66
precis( m4.5 )

## R code 4.67
weight.seq <- seq( from=-2.2 , to=2 , length.out=30 )
pred_dat <- list( weight_s=weight.seq , weight_s2=weight.seq^2 )
mu <- link( m4.5 , data=pred_dat )
mu.mean <- apply( mu , 2 , mean )
mu.PI <- apply( mu , 2 , PI , prob=0.89 )
sim.height <- sim( m4.5 , data=pred_dat )
height.PI <- apply( sim.height , 2 , PI , prob=0.89 )

## R code 4.68
plot( height ~ weight_s , d , col=col.alpha(rangi2,0.5) )
lines( weight.seq , mu.mean )
shade( mu.PI , weight.seq )
shade( height.PI , weight.seq )

## R code 4.69
d$weight_s3 <- d$weight_s^3
m4.6 <- quap(
    alist(
        height ~ dnorm( mu , sigma ) ,
        mu <- a + b1*weight_s + b2*weight_s2 + b3*weight_s3 ,
        a ~ dnorm( 178 , 20 ) ,
        b1 ~ dlnorm( 0 , 1 ) ,
        b2 ~ dnorm( 0 , 10 ) ,
        b3 ~ dnorm( 0 , 10 ) ,
        sigma ~ dunif( 0 , 50 )
    ) ,
    data=d )

## R code 4.70
plot( height ~ weight_s , d , col=col.alpha(rangi2,0.5) , xaxt="n" )

## R code 4.71
at <- c(-2,-1,0,1,2)
labels <- at*sd(d$weight) + mean(d$weight)
axis( side=1 , at=at , labels=round(labels,1) )

## R code 4.72
library(rethinking)
data(cherry_blossoms)
d <- cherry_blossoms
precis(d)

## R code 4.73
d2 <- d[ complete.cases(d$temp) , ] # complete cases on temp
num_knots <- 15
knot_list <- quantile( d2$year , probs=seq(0,1,length.out=num_knots) )

## R code 4.74
library(splines)
B <- bs(d2$year,
    knots=knot_list[-c(1,num_knots)] ,
    degree=3 , intercept=TRUE )

## R code 4.75
plot( NULL , xlim=range(d2$year) , ylim=c(0,1) , xlab="year" , ylab="basis value" )
for ( i in 1:ncol(B) ) lines( d2$year , B[,i] )

## R code 4.76
m4.7 <- quap(
    alist(
        T ~ dnorm( mu , sigma ) ,
        mu <- a + B %*% w ,
        a ~ dnorm(6,10),
        w ~ dnorm(0,1),
        sigma ~ dexp(1)
    ),
    data=list( T=d2$temp , B=B ) ,
    start=list( w=rep( 0 , ncol(B) ) ) )

## R code 4.77
post <- extract.samples(m4.7)
w <- apply( post$w , 2 , mean )
plot( NULL , xlim=range(d2$year) , ylim=c(-2,2) ,
    xlab="year" , ylab="basis * weight" )
for ( i in 1:ncol(B) ) lines( d2$year , w[i]*B[,i] )

## R code 4.78
mu <- link( m4.7 )
mu_PI <- apply(mu,2,PI,0.97)
plot( d2$year , d2$temp , col=col.alpha(rangi2,0.3) , pch=16 )
shade( mu_PI , d2$year , col=col.alpha("black",0.5) )

## R code 4.79
m4.7alt <- quap(
    alist(
        T ~ dnorm( mu , sigma ) ,
        mu <- a + sapply( 1:1124 , function(i) sum( B[i,]*w ) ) ,
        a ~ dnorm(6,10),
        w ~ dnorm(0,1),
        sigma ~ dexp(1)
    ),
    data=list( T=d2$temp , B=B ) ,
    start=list( w=rep( 0 , ncol(B) ) ) )

## R code 4.80
plot( height ~ weight , data=Howell1 ,
    col=col.alpha(rangi2,0.4) )

## R code 5.1
# load data and copy
library(rethinking)
data(WaffleDivorce)
d <- WaffleDivorce

# standardize variables
d$A <- scale( d$MedianAgeMarriage )
d$D <- scale( d$Divorce )

## R code 5.2
sd( d$MedianAgeMarriage )

## R code 5.3
m5.1 <- quap(
    alist(
        D ~ dnorm( mu , sigma ) ,
        mu <- a + bA * A ,
        a ~ dnorm( 0 , 0.2 ) ,
        bA ~ dnorm( 0 , 0.5 ) ,
        sigma ~ dexp( 1 )
    ) , data = d )

## R code 5.4
set.seed(10)
prior <- extract.prior( m5.1 )
mu <- link( m5.1 , post=prior , data=list( A=c(-2,2) ) )
plot( NULL , xlim=c(-2,2) , ylim=c(-2,2) )
for ( i in 1:50 ) lines( c(-2,2) , mu[i,] , col=col.alpha("black",0.4) )

## R code 5.5
# compute percentile interval of mean
A_seq <- seq( from=-3 , to=3.2 , length.out=30 )
mu <- link( m5.1 , data=list(A=A_seq) )
mu.mean <- apply( mu , 2, mean )
mu.PI <- apply( mu , 2 , PI )

# plot it all
plot( D ~ A , data=d , col=rangi2 )
lines( A_seq , mu.mean , lwd=2 )
shade( mu.PI , A_seq )

## R code 5.6
d$M <- scale( d$Marriage )
m5.2 <- quap(
    alist(
        D ~ dnorm( mu , sigma ) ,
        mu <- a + bM * M ,
        a ~ dnorm( 0 , 0.2 ) ,
        bM ~ dnorm( 0 , 0.5 ) ,
        sigma ~ dexp( 1 )
    ) , data = d )

## R code 5.7
library(dagitty)
dag5.1 <- dagitty( "dag {
    A -> D
    A -> M
    M -> D
}")
coordinates(dag5.1) <- list( x=c(A=0,D=1,M=2) , y=c(A=0,D=1,M=0) )
drawdag( dag5.1 )

## R code 5.8
DMA_dag2 <- dagitty('dag{ D <- A -> M }')
impliedConditionalIndependencies( DMA_dag2 )

## R code 5.9
DMA_dag1 <- dagitty('dag{ D <- A -> M -> D }')
impliedConditionalIndependencies( DMA_dag1 )

## R code 5.10
m5.3 <- quap(
    alist(
        D ~ dnorm( mu , sigma ) ,
        mu <- a + bM*M + bA*A ,
        a ~ dnorm( 0 , 0.2 ) ,
        bM ~ dnorm( 0 , 0.5 ) ,
        bA ~ dnorm( 0 , 0.5 ) ,
        sigma ~ dexp( 1 )
    ) , data = d )
precis( m5.3 )

## R code 5.11
plot( coeftab(m5.1,m5.2,m5.3), par=c("bA","bM") )

## R code 5.12
N <- 50 # number of simulated States
age <- rnorm( N )        # sim A
mar <- rnorm( N , -age )  # sim A -> M
div <- rnorm( N , age )  # sim A -> D

## R code 5.13
m5.4 <- quap(
    alist(
        M ~ dnorm( mu , sigma ) ,
        mu <- a + bAM * A ,
        a ~ dnorm( 0 , 0.2 ) ,
        bAM ~ dnorm( 0 , 0.5 ) ,
        sigma ~ dexp( 1 )
    ) , data = d )

## R code 5.14
mu <- link(m5.4)
mu_mean <- apply( mu , 2 , mean )
mu_resid <- d$M - mu_mean

## R code 5.15
# call link without specifying new data
# so it uses original data
mu <- link( m5.3 )

# summarize samples across cases
mu_mean <- apply( mu , 2 , mean )
mu_PI <- apply( mu , 2 , PI )

# simulate observations
# again no new data, so uses original data
D_sim <- sim( m5.3 , n=1e4 )
D_PI <- apply( D_sim , 2 , PI )

## R code 5.16
plot( mu_mean ~ d$D , col=rangi2 , ylim=range(mu_PI) ,
    xlab="Observed divorce" , ylab="Predicted divorce" )
abline( a=0 , b=1 , lty=2 )
for ( i in 1:nrow(d) ) lines( rep(d$D[i],2) , mu_PI[,i] , col=rangi2 )

## R code 5.17
identify( x=d$D , y=mu_mean , labels=d$Loc )

## R code 5.18
N <- 100                         # number of cases
x_real <- rnorm( N )             # x_real as Gaussian with mean 0 and stddev 1
x_spur <- rnorm( N , x_real )    # x_spur as Gaussian with mean=x_real
y <- rnorm( N , x_real )         # y as Gaussian with mean=x_real
d <- data.frame(y,x_real,x_spur) # bind all together in data frame

## R code 5.19
data(WaffleDivorce)
d <- list()
d$A <- standardize( WaffleDivorce$MedianAgeMarriage )
d$D <- standardize( WaffleDivorce$Divorce )
d$M <- standardize( WaffleDivorce$Marriage )

m5.3_A <- quap(
    alist(
      ## A -> D <- M
        D ~ dnorm( mu , sigma ) ,
        mu <- a + bM*M + bA*A ,
        a ~ dnorm( 0 , 0.2 ) ,
        bM ~ dnorm( 0 , 0.5 ) ,
        bA ~ dnorm( 0 , 0.5 ) ,
        sigma ~ dexp( 1 ),
      ## A -> M
        M ~ dnorm( mu_M , sigma_M ),
        mu_M <- aM + bAM*A,
        aM ~ dnorm( 0 , 0.2 ),
        bAM ~ dnorm( 0 , 0.5 ),
        sigma_M ~ dexp( 1 )
    ) , data = d )

## R code 5.20
A_seq <- seq( from=-2 , to=2 , length.out=30 )

## R code 5.21
# prep data
sim_dat <- data.frame( A=A_seq )

# simulate M and then D, using A_seq
s <- sim( m5.3_A , data=sim_dat , vars=c("M","D") )

## R code 5.22
# display counterfactual predictions
plot( sim_dat$A , colMeans(s$D) , ylim=c(-2,2) , type="l" ,
    xlab="manipulated A" , ylab="counterfactual D"  )
shade( apply(s$D,2,PI) , sim_dat$A )
mtext( "Total counterfactual effect of A on D" )

## R code 5.23
sim_dat <- data.frame( M=seq(from=-2,to=2,length.out=30) , A=0 )
s <- sim( m5.3_A , data=sim_dat , vars="D" )

plot( sim_dat$M , colMeans(s) , ylim=c(-2,2) , type="l" ,
    xlab="manipulated M" , ylab="counterfactual D"  )
shade( apply(s,2,PI) , sim_dat$M )
mtext( "Total counterfactual effect of M on D" )

## R code 5.24
A_seq <- seq( from=-2 , to=2 , length.out=30 )

## R code 5.25
post <- extract.samples( m5.3\_A )
M_sim <- with( post , sapply( 1:30 ,
    function(i) rnorm( 1e3 , aM + bAM*A_seq[i] , sigma_M ) ) )

## R code 5.26
D_sim <- with( post , sapply( 1:30 ,
    function(i) rnorm( 1e3 , a + bA*A_seq[i] + bM*M_sim[,i] , sigma ) ) )

## R code 5.27
library(rethinking)
data(milk)
d <- milk
str(d)

## R code 5.28
d$K <- scale( d$kcal.per.g )
d$N <- scale( d$neocortex.perc )
d$M <- scale( log(d$mass) )

## R code 5.29
m5.5_draft <- quap(
    alist(
        K ~ dnorm( mu , sigma ) ,
        mu <- a + bN*N ,
        a ~ dnorm( 0 , 1 ) ,
        bN ~ dnorm( 0 , 1 ) ,
        sigma ~ dexp( 1 )
    ) , data=d )

## R code 5.30
d$neocortex.perc

## R code 5.31
dcc <- d[ complete.cases(d$K,d$N,d$M) , ]

## R code 5.32
m5.5_draft <- quap(
    alist(
        K ~ dnorm( mu , sigma ) ,
        mu <- a + bN*N ,
        a ~ dnorm( 0 , 1 ) ,
        bN ~ dnorm( 0 , 1 ) ,
        sigma ~ dexp( 1 )
    ) , data=dcc )

## R code 5.33
prior <- extract.prior( m5.5_draft )
xseq <- c(-2,2)
mu <- link( m5.5_draft , post=prior , data=list(N=xseq) )
plot( NULL , xlim=xseq , ylim=xseq )
for ( i in 1:50 ) lines( xseq , mu[i,] , col=col.alpha("black",0.3) )

## R code 5.34
m5.5 <- quap(
    alist(
        K ~ dnorm( mu , sigma ) ,
        mu <- a + bN*N ,
        a ~ dnorm( 0 , 0.2 ) ,
        bN ~ dnorm( 0 , 0.5 ) ,
        sigma ~ dexp( 1 )
    ) , data=dcc )

## R code 5.35
precis( m5.5 )

## R code 5.36
xseq <- seq( from=min(dcc$N)-0.15 , to=max(dcc$N)+0.15 , length.out=30 )
mu <- link( m5.5 , data=list(N=xseq) )
mu_mean <- apply(mu,2,mean)
mu_PI <- apply(mu,2,PI)
plot( K ~ N , data=dcc )
lines( xseq , mu_mean , lwd=2 )
shade( mu_PI , xseq )

## R code 5.37
m5.6 <- quap(
    alist(
        K ~ dnorm( mu , sigma ) ,
        mu <- a + bM*M ,
        a ~ dnorm( 0 , 0.2 ) ,
        bM ~ dnorm( 0 , 0.5 ) ,
        sigma ~ dexp( 1 )
    ) , data=dcc )
precis(m5.6)

## R code 5.38
m5.7 <- quap(
    alist(
        K ~ dnorm( mu , sigma ) ,
        mu <- a + bN*N + bM*M ,
        a ~ dnorm( 0 , 0.2 ) ,
        bN ~ dnorm( 0 , 0.5 ) ,
        bM ~ dnorm( 0 , 0.5 ) ,
        sigma ~ dexp( 1 )
    ) , data=dcc )
precis(m5.7)

## R code 5.39
plot( coeftab( m5.5 , m5.6 , m5.7 ) , pars=c("bM","bN") )

## R code 5.40
xseq <- seq( from=min(dcc$M)-0.15 , to=max(dcc$M)+0.15 , length.out=30 )
mu <- link( m5.7 , data=data.frame( M=xseq , N=0 ) )
mu_mean <- apply(mu,2,mean)
mu_PI <- apply(mu,2,PI)
plot( NULL , xlim=range(dcc$M) , ylim=range(dcc$K) )
lines( xseq , mu_mean , lwd=2 )
shade( mu_PI , xseq )

## R code 5.41
# M -> K <- N
# M -> N
n <- 100
M <- rnorm( n )
N <- rnorm( n , M )
K <- rnorm( n , N - M )
d_sim <- data.frame(K=K,N=N,M=M)

## R code 5.42
# M -> K <- N
# N -> M
n <- 100
N <- rnorm( n )
M <- rnorm( n , N )
K <- rnorm( n , N - M )
d_sim2 <- data.frame(K=K,N=N,M=M)

# M -> K <- N
# M <- U -> N
n <- 100
U <- rnorm( n )
N <- rnorm( n , U )
M <- rnorm( n , U )
K <- rnorm( n , N - M )
d_sim3 <- data.frame(K=K,N=N,M=M)

## R code 5.43
dag5.7 <- dagitty( "dag{
    M -> K <- N
    M -> N }" )
coordinates(dag5.7) <- list( x=c(M=0,K=1,N=2) , y=c(M=0.5,K=1,N=0.5) )
MElist <- equivalentDAGs(dag5.7)

## R code 5.44
data(Howell1)
d <- Howell1
str(d)

## R code 5.45
mu_female <- rnorm(1e4,178,20)
mu_male <- rnorm(1e4,178,20) + rnorm(1e4,0,10)
precis( data.frame( mu_female , mu_male ) )

## R code 5.46
d$sex <- ifelse( d$male==1 , 2 , 1 )
str( d$sex )

## R code 5.47
m5.8 <- quap(
    alist(
        height ~ dnorm( mu , sigma ) ,
        mu <- a[sex] ,
        a[sex] ~ dnorm( 178 , 20 ) ,
        sigma ~ dunif( 0 , 50 )
    ) , data=d )
precis( m5.8 , depth=2 )

## R code 5.48
post <- extract.samples(m5.8)
post$diff_fm <- post$a[,1] - post$a[,2]
precis( post , depth=2 )

## R code 5.49
data(milk)
d <- milk
unique(d$clade)

## R code 5.50
d$clade_id <- as.integer( d$clade )

## R code 5.51
d$K <- scale( d$kcal.per.g )
m5.9 <- quap(
    alist(
        K ~ dnorm( mu , sigma ),
        mu <- a[clade_id],
        a[clade_id] ~ dnorm( 0 , 0.5 ),
        sigma ~ dexp( 1 )
    ) , data=d )
labels <- paste( "a[" , 1:4 , "]:" , levels(d$clade) , sep="" )
plot( precis( m5.9 , depth=2 , pars="a" ) , labels=labels ,
    xlab="expected kcal (std)" )

## R code 5.52
set.seed(63)
d$house <- sample( rep(1:4,each=8) , size=nrow(d) )

## R code 5.53
m5.10 <- quap(
    alist(
        K ~ dnorm( mu , sigma ),
        mu <- a[clade_id] + h[house],
        a[clade_id] ~ dnorm( 0 , 0.5 ),
        h[house] ~ dnorm( 0 , 0.5 ),
        sigma ~ dexp( 1 )
    ) , data=d )

## R code 6.1
set.seed(1914)
N <- 200 # num grant proposals
p <- 0.1 # proportion to select
# uncorrelated newsworthiness and trustworthiness
nw <- rnorm(N)
tw <- rnorm(N)
# select top 10% of combined scores
s <- nw + tw  # total score
q <- quantile( s , 1-p ) # top 10% threshold
selected <- ifelse( s >= q , TRUE , FALSE )
cor( tw[selected] , nw[selected] )

## R code 6.2
N <- 100                          # number of individuals
set.seed(909)
height <- rnorm(N,10,2)           # sim total height of each
leg_prop <- runif(N,0.4,0.5)      # leg as proportion of height
leg_left <- leg_prop*height +     # sim left leg as proportion + error
    rnorm( N , 0 , 0.02 )
leg_right <- leg_prop*height +    # sim right leg as proportion + error
    rnorm( N , 0 , 0.02 )
                                  # combine into data frame
d <- data.frame(height,leg_left,leg_right)

## R code 6.3
m6.1 <- quap(
    alist(
        height ~ dnorm( mu , sigma ) ,
        mu <- a + bl*leg_left + br*leg_right ,
        a ~ dnorm( 10 , 100 ) ,
        bl ~ dnorm( 2 , 10 ) ,
        br ~ dnorm( 2 , 10 ) ,
        sigma ~ dexp( 1 )
    ) ,
    data=d )
precis(m6.1)

## R code 6.4
plot(precis(m6.1))

## R code 6.5
post <- extract.samples(m6.1)
plot( bl ~ br , post , col=col.alpha(rangi2,0.1) , pch=16 )

## R code 6.6
sum_blbr <- post$bl + post$br
dens( sum_blbr , col=rangi2 , lwd=2 , xlab="sum of bl and br" )

## R code 6.7
m6.2 <- quap(
    alist(
        height ~ dnorm( mu , sigma ) ,
        mu <- a + bl*leg_left,
        a ~ dnorm( 10 , 100 ) ,
        bl ~ dnorm( 2 , 10 ) ,
        sigma ~ dexp( 1 )
    ) ,
    data=d )
precis(m6.2)

## R code 6.8
library(rethinking)
data(milk)
d <- milk
d$K <- scale( d$kcal.per.g )
d$F <- scale( d$perc.fat )
d$L <- scale( d$perc.lactose )

## R code 6.9
# kcal.per.g regressed on perc.fat
m6.3 <- quap(
    alist(
        K ~ dnorm( mu , sigma ) ,
        mu <- a + bF*F ,
        a ~ dnorm( 0 , 0.2 ) ,
        bF ~ dnorm( 0 , 0.5 ) ,
        sigma ~ dexp( 1 )
    ) , data=d )

# kcal.per.g regressed on perc.lactose
m6.4 <- quap(
    alist(
        K ~ dnorm( mu , sigma ) ,
        mu <- a + bL*L ,
        a ~ dnorm( 0 , 0.2 ) ,
        bL ~ dnorm( 0 , 0.5 ) ,
        sigma ~ dexp( 1 )
    ) , data=d )

precis( m6.3 )
precis( m6.4 )

## R code 6.10
m6.5 <- quap(
    alist(
        K ~ dnorm( mu , sigma ) ,
        mu <- a + bF*F + bL*L ,
        a ~ dnorm( 0 , 0.2 ) ,
        bF ~ dnorm( 0 , 0.5 ) ,
        bL ~ dnorm( 0 , 0.5 ) ,
        sigma ~ dexp( 1 )
    ) ,
    data=d )
precis( m6.5 )

## R code 6.11
pairs( ~ kcal.per.g + perc.fat + perc.lactose , data=d , col=rangi2 )

## R code 6.12
library(rethinking)
data(milk)
d <- milk
sim.coll <- function( r=0.9 ) {
    d$x <- rnorm( nrow(d) , mean=r*d$perc.fat ,
        sd=sqrt( (1-r^2)*var(d$perc.fat) ) )
    m <- lm( kcal.per.g ~ perc.fat + x , data=d )
    sqrt( diag( vcov(m) ) )[2] # stddev of parameter
}
rep.sim.coll <- function( r=0.9 , n=100 ) {
    stddev <- replicate( n , sim.coll(r) )
    mean(stddev)
}
r.seq <- seq(from=0,to=0.99,by=0.01)
stddev <- sapply( r.seq , function(z) rep.sim.coll(r=z,n=100) )
plot( stddev ~ r.seq , type="l" , col=rangi2, lwd=2 , xlab="correlation" )

## R code 6.13
set.seed(71)
# number of plants
N <- 100

# simulate initial heights
h0 <- rnorm(N,10,2)

# assign treatments and simulate fungus and growth
treatment <- rep( 0:1 , each=N/2 )
fungus <- rbinom( N , size=1 , prob=0.5 - treatment*0.4 )
h1 <- h0 + rnorm(N, 5 - 3*fungus)

# compose a clean data frame
d <- data.frame( h0=h0 , h1=h1 , treatment=treatment , fungus=fungus )
precis(d)

## R code 6.14
sim_p <- rlnorm( 1e4 , 0 , 0.25 )
precis( data.frame(sim_p) )

## R code 6.15
m6.6 <- quap(
    alist(
        h1 ~ dnorm( mu , sigma ),
        mu <- h0*p,
        p ~ dlnorm( 0 , 0.25 ),
        sigma ~ dexp( 1 )
    ), data=d )
precis(m6.6)

## R code 6.16
m6.7 <- quap(
    alist(
        h1 ~ dnorm( mu , sigma ),
        mu <- h0 * p,
        p <- a + bt*treatment + bf*fungus,
        a ~ dlnorm( 0 , 0.2 ) ,
        bt ~ dnorm( 0 , 0.5 ),
        bf ~ dnorm( 0 , 0.5 ),
        sigma ~ dexp( 1 )
    ), data=d )
precis(m6.7)

## R code 6.17
m6.8 <- quap(
    alist(
        h1 ~ dnorm( mu , sigma ),
        mu <- h0 * p,
        p <- a + bt*treatment,
        a ~ dlnorm( 0 , 0.2 ),
        bt ~ dnorm( 0 , 0.5 ),
        sigma ~ dexp( 1 )
    ), data=d )
precis(m6.8)

## R code 6.18
library(dagitty)
plant_dag <- dagitty( "dag {
    H_0 -> H_1
    F -> H_1
    T -> F
}")
coordinates( plant_dag ) <- list( x=c(H_0=0,T=2,F=1.5,H_1=1) ,
                                  y=c(H_0=0,T=0,F=0,H_1=0) )
drawdag( plant_dag )

## R code 6.19
impliedConditionalIndependencies(plant_dag)

## R code 6.20
set.seed(71)
N <- 1000
h0 <- rnorm(N,10,2)
treatment <- rep( 0:1 , each=N/2 )
M <- rbern(N)
fungus <- rbinom( N , size=1 , prob=0.5 - treatment*0.4 + 0.4*M )
h1 <- h0 + rnorm( N , 5 + 3*M )
d2 <- data.frame( h0=h0 , h1=h1 , treatment=treatment , fungus=fungus )

## R code 6.21
library(rethinking)
d <- sim_happiness( seed=1977 , N_years=1000 )
precis(d)

## R code 6.22
d2 <- d[ d$age>17 , ] # only adults
d2$A <- ( d2$age - 18 ) / ( 65 - 18 )

## R code 6.23
d2$mid <- d2$married + 1
m6.9 <- quap(
    alist(
        happiness ~ dnorm( mu , sigma ),
        mu <- a[mid] + bA*A,
        a[mid] ~ dnorm( 0 , 1 ),
        bA ~ dnorm( 0 , 2 ),
        sigma ~ dexp(1)
    ) , data=d2 )
precis(m6.9,depth=2)

## R code 6.24
m6.10 <- quap(
    alist(
        happiness ~ dnorm( mu , sigma ),
        mu <- a + bA*A,
        a ~ dnorm( 0 , 1 ),
        bA ~ dnorm( 0 , 2 ),
        sigma ~ dexp(1)
    ) , data=d2 )
precis(m6.10)

## R code 6.25
N <- 200  # number of grandparent-parent-child triads
b_GP <- 1 # direct effect of G on P
b_GC <- 0 # direct effect of G on C
b_PC <- 1 # direct effect of P on C
b_U <- 2  # direct effect of U on P and C

## R code 6.26
set.seed(1)
U <- 2*rbern( N , 0.5 ) - 1
G <- rnorm( N )
P <- rnorm( N , b_GP*G + b_U*U )
C <- rnorm( N , b_PC*P + b_GC*G + b_U*U )
d <- data.frame( C=C , P=P , G=G , U=U )

## R code 6.27
m6.11 <- quap(
    alist(
        C ~ dnorm( mu , sigma ),
        mu <- a + b_PC*P + b_GC*G,
        a ~ dnorm( 0 , 1 ),
        c(b_PC,b_GC) ~ dnorm( 0 , 1 ),
        sigma ~ dexp( 1 )
    ), data=d )
precis(m6.11)

## R code 6.28
m6.12 <- quap(
    alist(
        C ~ dnorm( mu , sigma ),
        mu <- a + b_PC*P + b_GC*G + b_U*U,
        a ~ dnorm( 0 , 1 ),
        c(b_PC,b_GC,b_U) ~ dnorm( 0 , 1 ),
        sigma ~ dexp( 1 )
    ), data=d )
precis(m6.12)

## R code 6.29
library(dagitty)
dag_6.1 <- dagitty( "dag {
    U [unobserved]
    X -> Y
    X <- U <- A -> C -> Y
    U -> B <- C
}")
adjustmentSets( dag_6.1 , exposure="X" , outcome="Y" )

## R code 6.30
library(dagitty)
dag_6.2 <- dagitty( "dag {
    A -> D
    A -> M -> D
    A <- S -> M
    S -> W -> D
}")
adjustmentSets( dag_6.2 , exposure="W" , outcome="D" )

## R code 6.31
impliedConditionalIndependencies( dag_6.2 )

## R code 7.1
sppnames <- c( "afarensis","africanus","habilis","boisei",
    "rudolfensis","ergaster","sapiens")
brainvolcc <- c( 438 , 452 , 612, 521, 752, 871, 1350 )
masskg <- c( 37.0 , 35.5 , 34.5 , 41.5 , 55.5 , 61.0 , 53.5 )
d <- data.frame( species=sppnames , brain=brainvolcc , mass=masskg )

## R code 7.2
d$mass_std <- (d$mass - mean(d$mass))/sd(d$mass)
d$brain_std <- d$brain / max(d$brain)

## R code 7.3
m7.1 <- quap(
    alist(
        brain_std ~ dnorm( mu , exp(log_sigma) ),
        mu <- a + b*mass_std,
        a ~ dnorm( 0.5 , 1 ),
        b ~ dnorm( 0 , 10 ),
        log_sigma ~ dnorm( 0 , 1 )
    ), data=d )

## R code 7.4
m7.1_OLS <- lm( brain_std ~ mass_std , data=d )
post <- extract.samples( m7.1_OLS )

## R code 7.5
set.seed(12)
s <- sim( m7.1 )
r <- apply(s,2,mean) - d$brain_std
resid_var <- var2(r)
outcome_var <- var2( d$brain_std )
1 - resid_var/outcome_var

## R code 7.6
R2_is_bad <- function( quap_fit ) {
    s <- sim( quap_fit , refresh=0 )
    r <- apply(s,2,mean) - d$brain_std
    1 - var2(r)/var2(d$brain_std)
}

## R code 7.7
m7.2 <- quap(
    alist(
        brain_std ~ dnorm( mu , exp(log_sigma) ),
        mu <- a + b[1]*mass_std + b[2]*mass_std^2,
        a ~ dnorm( 0.5 , 1 ),
        b ~ dnorm( 0 , 10 ),
        log_sigma ~ dnorm( 0 , 1 )
    ), data=d , start=list(b=rep(0,2)) )

## R code 7.8
m7.3 <- quap(
    alist(
        brain_std ~ dnorm( mu , exp(log_sigma) ),
        mu <- a + b[1]*mass_std + b[2]*mass_std^2 +
                  b[3]*mass_std^3,
        a ~ dnorm( 0.5 , 1 ),
        b ~ dnorm( 0 , 10 ),
        log_sigma ~ dnorm( 0 , 1 )
    ), data=d , start=list(b=rep(0,3)) )

m7.4 <- quap(
    alist(
        brain_std ~ dnorm( mu , exp(log_sigma) ),
        mu <- a + b[1]*mass_std + b[2]*mass_std^2 +
                  b[3]*mass_std^3 + b[4]*mass_std^4,
        a ~ dnorm( 0.5 , 1 ),
        b ~ dnorm( 0 , 10 ),
        log_sigma ~ dnorm( 0 , 1 )
    ), data=d , start=list(b=rep(0,4)) )

m7.5 <- quap(
    alist(
        brain_std ~ dnorm( mu , exp(log_sigma) ),
        mu <- a + b[1]*mass_std + b[2]*mass_std^2 +
                  b[3]*mass_std^3 + b[4]*mass_std^4 +
                  b[5]*mass_std^5,
        a ~ dnorm( 0.5 , 1 ),
        b ~ dnorm( 0 , 10 ),
        log_sigma ~ dnorm( 0 , 1 )
    ), data=d , start=list(b=rep(0,5)) )

## R code 7.9
m7.6 <- quap(
    alist(
        brain_std ~ dnorm( mu , 0.001 ),
        mu <- a + b[1]*mass_std + b[2]*mass_std^2 +
                  b[3]*mass_std^3 + b[4]*mass_std^4 +
                  b[5]*mass_std^5 + b[6]*mass_std^6,
        a ~ dnorm( 0.5 , 1 ),
        b ~ dnorm( 0 , 10 )
    ), data=d , start=list(b=rep(0,6)) )

## R code 7.10
post <- extract.samples(m7.1)
mass_seq <- seq( from=min(d$mass_std) , to=max(d$mass_std) , length.out=100 )
l <- link( m7.1 , data=list( mass_std=mass_seq ) )
mu <- apply( l , 2 , mean )
ci <- apply( l , 2 , PI )
plot( brain_std ~ mass_std , data=d )
lines( mass_seq , mu )
shade( ci , mass_seq )

## R code 7.11
d_minus_i <- d[ -i , ]

## R code 7.12
p <- c( 0.3 , 0.7 )
-sum( p*log(p) )

## R code 7.13
set.seed(1)
lppd( m7.1 , n=1e4 )

## R code 7.14
set.seed(1)
logprob <- sim( m7.1 , ll=TRUE , n=1e4 )
n <- ncol(logprob)
ns <- nrow(logprob)
f <- function( i ) log_sum_exp( logprob[,i] ) - log(ns)
( lppd <- sapply( 1:n , f ) )

## R code 7.15
set.seed(1)
sapply( list(m7.1,m7.2,m7.3,m7.4,m7.5,m7.6) , function(m) sum(lppd(m)) )

## R code 7.16
N <- 20
kseq <- 1:5
dev <- sapply( kseq , function(k) {
        print(k);
        r <- replicate( 1e4 , sim_train_test( N=N, k=k ) );
        c( mean(r[1,]) , mean(r[2,]) , sd(r[1,]) , sd(r[2,]) )
    } )

## R code 7.17
        r <- mcreplicate( 1e4 , sim_train_test( N=N, k=k ) , mc.cores=4 )

## R code 7.18
plot( 1:5 , dev[1,] , ylim=c( min(dev[1:2,])-5 , max(dev[1:2,])+10 ) ,
    xlim=c(1,5.1) , xlab="number of parameters" , ylab="deviance" ,
    pch=16 , col=rangi2 )
mtext( concat( "N = ",N ) )
points( (1:5)+0.1 , dev[2,] )
for ( i in kseq ) {
    pts_in <- dev[1,i] + c(-1,+1)*dev[3,i]
    pts_out <- dev[2,i] + c(-1,+1)*dev[4,i]
    lines( c(i,i) , pts_in , col=rangi2 )
    lines( c(i,i)+0.1 , pts_out )
}

## R code 7.19
data(cars)
m <- quap(
    alist(
        dist ~ dnorm(mu,sigma),
        mu <- a + b*speed,
        a ~ dnorm(0,100),
        b ~ dnorm(0,10),
        sigma ~ dexp(1)
    ) , data=cars )
set.seed(94)
post <- extract.samples(m,n=1000)

## R code 7.20
n_samples <- 1000
logprob <- sapply( 1:n_samples ,
    function(s) {
        mu <- post$a[s] + post$b[s]*cars$speed
        dnorm( cars$dist , mu , post$sigma[s] , log=TRUE )
    } )

## R code 7.21
n_cases <- nrow(cars)
lppd <- sapply( 1:n_cases , function(i) log_sum_exp(logprob[i,]) - log(n_samples) )

## R code 7.22
pWAIC <- sapply( 1:n_cases , function(i) var(logprob[i,]) )

## R code 7.23
-2*( sum(lppd) - sum(pWAIC) )

## R code 7.24
waic_vec <- -2*( lppd - pWAIC )
sqrt( n_cases*var(waic_vec) )

## R code 7.25
set.seed(11)
WAIC( m6.7 )

## R code 7.26
set.seed(77)
compare( m6.6 , m6.7 , m6.8 )

## R code 7.27
set.seed(91)
waic_m6.7 <- WAIC( m6.7 , pointwise=TRUE )$WAIC
waic_m6.8 <- WAIC( m6.8 , pointwise=TRUE )$WAIC
n <- length(waic_m6.7)
diff_m6.7_m6.8 <- waic_m6.7 - waic_m6.8
sqrt( n*var( diff_m6.7_m6.8 ) )

## R code 7.28
40.0 + c(-1,1)*10.4*2.6

## R code 7.29
plot( compare( m6.6 , m6.7 , m6.8 ) )

## R code 7.30
set.seed(92)
waic_m6.6 <- WAIC( m6.6 , pointwise=TRUE )$WAIC
diff_m6.6_m6.8 <- waic_m6.6 - waic_m6.8
sqrt( n*var( diff_m6.6_m6.8 ) )

## R code 7.31
set.seed(93)
compare( m6.6 , m6.7 , m6.8 )@dSE

## R code 7.32
library(rethinking)
data(WaffleDivorce)
d <- WaffleDivorce
d$A <- standardize( d$MedianAgeMarriage )
d$D <- standardize( d$Divorce )
d$M <- standardize( d$Marriage )

m5.1 <- quap(
    alist(
        D ~ dnorm( mu , sigma ) ,
        mu <- a + bA * A ,
        a ~ dnorm( 0 , 0.2 ) ,
        bA ~ dnorm( 0 , 0.5 ) ,
        sigma ~ dexp( 1 )
    ) , data = d )

m5.2 <- quap(
    alist(
        D ~ dnorm( mu , sigma ) ,
        mu <- a + bM * M ,
        a ~ dnorm( 0 , 0.2 ) ,
        bM ~ dnorm( 0 , 0.5 ) ,
        sigma ~ dexp( 1 )
    ) , data = d )

m5.3 <- quap(
    alist(
        D ~ dnorm( mu , sigma ) ,
        mu <- a + bM*M + bA*A ,
        a ~ dnorm( 0 , 0.2 ) ,
        bM ~ dnorm( 0 , 0.5 ) ,
        bA ~ dnorm( 0 , 0.5 ) ,
        sigma ~ dexp( 1 )
    ) , data = d )

## R code 7.33
set.seed(24071847)
compare( m5.1 , m5.2 , m5.3 , func=PSIS )

## R code 7.34
set.seed(24071847)
PSIS_m5.3 <- PSIS(m5.3,pointwise=TRUE)
set.seed(24071847)
WAIC_m5.3 <- WAIC(m5.3,pointwise=TRUE)
plot( PSIS_m5.3$k , WAIC_m5.3$penalty , xlab="PSIS Pareto k" ,
    ylab="WAIC penalty" , col=rangi2 , lwd=2 )

## R code 7.35
m5.3t <- quap(
    alist(
        D ~ dstudent( 2 , mu , sigma ) ,
        mu <- a + bM*M + bA*A ,
        a ~ dnorm( 0 , 0.2 ) ,
        bM ~ dnorm( 0 , 0.5 ) ,
        bA ~ dnorm( 0 , 0.5 ) ,
        sigma ~ dexp( 1 )
    ) , data = d )

## R code 8.1
library(rethinking)
data(rugged)
d <- rugged

# make log version of outcome
d$log_gdp <- log( d$rgdppc_2000 )

# extract countries with GDP data
dd <- d[ complete.cases(d$rgdppc_2000) , ]

# rescale variables
dd$log_gdp_std <- dd$log_gdp / mean(dd$log_gdp)
dd$rugged_std <- dd$rugged / max(dd$rugged)

## R code 8.2
m8.1 <- quap(
    alist(
        log_gdp_std ~ dnorm( mu , sigma ) ,
        mu <- a + b*( rugged_std - 0.215 ) ,
        a ~ dnorm( 1 , 1 ) ,
        b ~ dnorm( 0 , 1 ) ,
        sigma ~ dexp( 1 )
    ) , data=dd )

## R code 8.3
set.seed(7)
prior <- extract.prior( m8.1 )

# set up the plot dimensions
plot( NULL , xlim=c(0,1) , ylim=c(0.5,1.5) ,
    xlab="ruggedness" , ylab="log GDP" )
abline( h=min(dd$log_gdp_std) , lty=2 )
abline( h=max(dd$log_gdp_std) , lty=2 )

# draw 50 lines from the prior
rugged_seq <- seq( from=-0.1 , to=1.1 , length.out=30 )
mu <- link( m8.1 , post=prior , data=data.frame(rugged_std=rugged_seq) )
for ( i in 1:50 ) lines( rugged_seq , mu[i,] , col=col.alpha("black",0.3) )

## R code 8.4
sum( abs(prior$b) > 0.6 ) / length(prior$b)

## R code 8.5
m8.1 <- quap(
    alist(
        log_gdp_std ~ dnorm( mu , sigma ) ,
        mu <- a + b*( rugged_std - 0.215 ) ,
        a ~ dnorm( 1 , 0.1 ) ,
        b ~ dnorm( 0 , 0.3 ) ,
        sigma ~ dexp(1)
    ) , data=dd )

## R code 8.6
precis( m8.1 )

## R code 8.7
# make variable to index Africa (1) or not (2)
dd$cid <- ifelse( dd$cont_africa==1 , 1 , 2 )

## R code 8.8
m8.2 <- quap(
    alist(
        log_gdp_std ~ dnorm( mu , sigma ) ,
        mu <- a[cid] + b*( rugged_std - 0.215 ) ,
        a[cid] ~ dnorm( 1 , 0.1 ) ,
        b ~ dnorm( 0 , 0.3 ) ,
        sigma ~ dexp( 1 )
    ) , data=dd )

## R code 8.9
compare( m8.1 , m8.2 )

## R code 8.10
precis( m8.2 , depth=2 )

## R code 8.11
post <- extract.samples(m8.2)
diff_a1_a2 <- post$a[,1] - post$a[,2]
PI( diff_a1_a2 )

## R code 8.12
rugged.seq <- seq( from=-0.1 , to=1.1 , length.out=30 )

# compute mu over samples, fixing cid=2
mu.NotAfrica <- link( m8.2 ,
    data=data.frame( cid=2 , rugged_std=rugged.seq ) )

# compute mu over samples, fixing cid=1
mu.Africa <- link( m8.2 ,
    data=data.frame( cid=1 , rugged_std=rugged.seq ) )

# summarize to means and intervals
mu.NotAfrica_mu <- apply( mu.NotAfrica , 2 , mean )
mu.NotAfrica_ci <- apply( mu.NotAfrica , 2 , PI , prob=0.97 )
mu.Africa_mu <- apply( mu.Africa , 2 , mean )
mu.Africa_ci <- apply( mu.Africa , 2 , PI , prob=0.97 )

## R code 8.13
m8.3 <- quap(
    alist(
        log_gdp_std ~ dnorm( mu , sigma ) ,
        mu <- a[cid] + b[cid]*( rugged_std - 0.215 ) ,
        a[cid] ~ dnorm( 1 , 0.1 ) ,
        b[cid] ~ dnorm( 0 , 0.3 ) ,
        sigma ~ dexp( 1 )
    ) , data=dd )

## R code 8.14
precis( m8.5 , depth=2 )

## R code 8.15
compare( m8.1 , m8.2 , m8.3 , func=PSIS )

## R code 8.16
plot( PSIS( m8.3 , pointwise=TRUE )$k )

## R code 8.17
# plot Africa - cid=1
d.A1 <- dd[ dd$cid==1 , ]
plot( d.A1$rugged_std , d.A1$log_gdp_std , pch=16 , col=rangi2 ,
    xlab="ruggedness (standardized)" , ylab="log GDP (as proportion of mean)" ,
    xlim=c(0,1) )
mu <- link( m8.3 , data=data.frame( cid=1 , rugged_std=rugged_seq ) )
mu_mean <- apply( mu , 2 , mean )
mu_ci <- apply( mu , 2 , PI , prob=0.97 )
lines( rugged_seq , mu_mean , lwd=2 )
shade( mu_ci , rugged_seq , col=col.alpha(rangi2,0.3) )
mtext("African nations")

# plot non-Africa - cid=2
d.A0 <- dd[ dd$cid==2 , ]
plot( d.A0$rugged_std , d.A0$log_gdp_std , pch=1 , col="black" ,
    xlab="ruggedness (standardized)" , ylab="log GDP (as proportion of mean)" ,
    xlim=c(0,1) )
mu <- link( m8.3 , data=data.frame( cid=2 , rugged_std=rugged_seq ) )
mu_mean <- apply( mu , 2 , mean )
mu_ci <- apply( mu , 2 , PI , prob=0.97 )
lines( rugged_seq , mu_mean , lwd=2 )
shade( mu_ci , rugged_seq )
mtext("Non-African nations")

## R code 8.18
rugged_seq <- seq(from=-0.2,to=1.2,length.out=30)
muA <- link( m8.3 , data=data.frame(cid=1,rugged_std=rugged_seq) )
muN <- link( m8.3 , data=data.frame(cid=2,rugged_std=rugged_seq) )
delta <- muA - muN

## R code 8.19
library(rethinking)
data(tulips)
d <- tulips
str(d)

## R code 8.20
d$blooms_std <- d$blooms / max(d$blooms)
d$water_cent <- d$water - mean(d$water)
d$shade_cent <- d$shade - mean(d$shade)

## R code 8.21
a <- rnorm( 1e4 , 0.5 , 1 )
sum( a < 0 | a > 1 ) / length( a )

## R code 8.22
a <- rnorm( 1e4 , 0.5 , 0.25 )
sum( a < 0 | a > 1 ) / length( a )

## R code 8.23
m8.4 <- quap(
    alist(
        blooms_std ~ dnorm( mu , sigma ) ,
        mu <- a + bw*water_cent + bs*shade_cent ,
        a ~ dnorm( 0.5 , 0.25 ) ,
        bw ~ dnorm( 0 , 0.25 ) ,
        bs ~ dnorm( 0 , 0.25 ) ,
        sigma ~ dexp( 1 )
    ) , data=d )

## R code 8.24
m8.5 <- quap(
    alist(
        blooms_std ~ dnorm( mu , sigma ) ,
        mu <- a + bw*water_cent + bs*shade_cent + bws*water_cent*shade_cent ,
        a ~ dnorm( 0.5 , 0.25 ) ,
        bw ~ dnorm( 0 , 0.25 ) ,
        bs ~ dnorm( 0 , 0.25 ) ,
        bws ~ dnorm( 0 , 0.25 ) ,
        sigma ~ dexp( 1 )
    ) , data=d )

## R code 8.25
par(mfrow=c(1,3)) # 3 plots in 1 row
for ( s in -1:1 ) {
    idx <- which( d$shade_cent==s )
    plot( d$water_cent[idx] , d$blooms_std[idx] , xlim=c(-1,1) , ylim=c(0,1) ,
        xlab="water" , ylab="blooms" , pch=16 , col=rangi2 )
    mu <- link( m8.4 , data=data.frame( shade_cent=s , water_cent=-1:1 ) )
    for ( i in 1:20 ) lines( -1:1 , mu[i,] , col=col.alpha("black",0.3) )
}

## R code 8.26
set.seed(7)
prior <- extract.prior(m8.5)

## R code 8.27
d$lang.per.cap <- d$num.lang / d$k.pop

## R code 9.1
num_weeks <- 1e5
positions <- rep(0,num_weeks)
current <- 10
for ( i in 1:num_weeks ) {
    # record current position
    positions[i] <- current

    # flip coin to generate proposal
    proposal <- current + sample( c(-1,1) , size=1 )
    # now make sure he loops around the archipelago
    if ( proposal < 1 ) proposal <- 10
    if ( proposal > 10 ) proposal <- 1

    # move?
    prob_move <- proposal/current
    current <- ifelse( runif(1) < prob_move , proposal , current )
}

## R code 9.2
plot( 1:100 , positions[1:100] )

## R code 9.3
plot( table( positions ) )

## R code 9.4
D <- 10
T <- 1e3
Y <- rmvnorm(T,rep(0,D),diag(D))
rad_dist <- function( Y ) sqrt( sum(Y^2) )
Rd <- sapply( 1:T , function(i) rad_dist( Y[i,] ) )
dens( Rd )

## R code 9.5
# U needs to return neg-log-probability
U <- function( q , a=0 , b=1 , k=0 , d=1 ) {
    muy <- q[1]
    mux <- q[2]
    U <- sum( dnorm(y,muy,1,log=TRUE) ) + sum( dnorm(x,mux,1,log=TRUE) ) +
        dnorm(muy,a,b,log=TRUE) + dnorm(mux,k,d,log=TRUE)
    return( -U )
}

## R code 9.6
# gradient function
# need vector of partial derivatives of U with respect to vector q
U_gradient <- function( q , a=0 , b=1 , k=0 , d=1 ) {
    muy <- q[1]
    mux <- q[2]
    G1 <- sum( y - muy ) + (a - muy)/b^2 #dU/dmuy
    G2 <- sum( x - mux ) + (k - mux)/d^2 #dU/dmux
    return( c( -G1 , -G2 ) ) # negative bc energy is neg-log-prob
}
# test data
set.seed(7)
y <- rnorm(50)
x <- rnorm(50)
x <- as.numeric(scale(x))
y <- as.numeric(scale(y))

## R code 9.7
library(shape) # for fancy arrows
Q <- list()
Q$q <- c(-0.1,0.2)
pr <- 0.3
plot( NULL , ylab="muy" , xlab="mux" , xlim=c(-pr,pr) , ylim=c(-pr,pr) )
step <- 0.03
L <- 11 # 0.03/28 for U-turns --- 11 for working example
n_samples <- 4
path_col <- col.alpha("black",0.5)
points( Q$q[1] , Q$q[2] , pch=4 , col="black" )
for ( i in 1:n_samples ) {
    Q <- HMC2( U , U_gradient , step , L , Q$q )
    if ( n_samples < 10 ) {
      for ( j in 1:L ) {
        K0 <- sum(Q$ptraj[j,]^2)/2 # kinetic energy
        lines( Q$traj[j:(j+1),1] , Q$traj[j:(j+1),2] , col=path_col , lwd=1+2*K0 )
      }
      points( Q$traj[1:L+1,] , pch=16 , col="white" , cex=0.35 )
      Arrows( Q$traj[L,1] , Q$traj[L,2] , Q$traj[L+1,1] , Q$traj[L+1,2] ,
          arr.length=0.35 , arr.adj = 0.7 )
      text( Q$traj[L+1,1] , Q$traj[L+1,2] , i , cex=0.8 , pos=4 , offset=0.4 )
    }
    points( Q$traj[L+1,1] , Q$traj[L+1,2] , pch=ifelse( Q$accept==1 , 16 , 1 ) ,
        col=ifelse( abs(Q$dH)>0.1 , "red" , "black" ) )
}

## R code 9.8
HMC2 <- function (U, grad_U, epsilon, L, current_q) {
  q = current_q
  p = rnorm(length(q),0,1) # random flick - p is momentum.
  current_p = p
  # Make a half step for momentum at the beginning
  p = p - epsilon * grad_U(q) / 2
  # initialize bookkeeping - saves trajectory
  qtraj <- matrix(NA,nrow=L+1,ncol=length(q))
  ptraj <- qtraj
  qtraj[1,] <- current_q
  ptraj[1,] <- p

## R code 9.9
  # Alternate full steps for position and momentum
  for ( i in 1:L ) {
    q = q + epsilon * p # Full step for the position
    # Make a full step for the momentum, except at end of trajectory
    if ( i!=L ) {
        p = p - epsilon * grad_U(q)
        ptraj[i+1,] <- p
    }
    qtraj[i+1,] <- q
  }

## R code 9.10
  # Make a half step for momentum at the end
  p = p - epsilon * grad_U(q) / 2
  ptraj[L+1,] <- p
  # Negate momentum at end of trajectory to make the proposal symmetric
  p = -p
  # Evaluate potential and kinetic energies at start and end of trajectory
  current_U = U(current_q)
  current_K = sum(current_p^2) / 2
  proposed_U = U(q)
  proposed_K = sum(p^2) / 2
  # Accept or reject the state at end of trajectory, returning either
  # the position at the end of the trajectory or the initial position
  accept <- 0
  if (runif(1) < exp(current_U-proposed_U+current_K-proposed_K)) {
    new_q <- q  # accept
    accept <- 1
  } else new_q <- current_q  # reject
  return(list( q=new_q, traj=qtraj, ptraj=ptraj, accept=accept ))
}

## R code 9.11
library(rethinking)
data(rugged)
d <- rugged
d$log_gdp <- log(d$rgdppc_2000)
dd <- d[ complete.cases(d$rgdppc_2000) , ]
dd$log_gdp_std <- dd$log_gdp / mean(dd$log_gdp)
dd$rugged_std <- dd$rugged / max(dd$rugged)
dd$cid <- ifelse( dd$cont_africa==1 , 1 , 2 )

## R code 9.12
m8.3 <- quap(
    alist(
        log_gdp_std ~ dnorm( mu , sigma ) ,
        mu <- a[cid] + b[cid]*( rugged_std - 0.215 ) ,
        a[cid] ~ dnorm( 1 , 0.1 ) ,
        b[cid] ~ dnorm( 0 , 0.3 ) ,
        sigma ~ dexp( 1 )
    ) , data=dd )
precis( m8.3 , depth=2 )

## R code 9.13
dat_slim <- list(
    log_gdp_std = dd$log_gdp_std,
    rugged_std = dd$rugged_std,
    cid = as.integer( dd$cid )
)
str(dat_slim)

## R code 9.14
m9.1 <- ulam(
    alist(
        log_gdp_std ~ dnorm( mu , sigma ) ,
        mu <- a[cid] + b[cid]*( rugged_std - 0.215 ) ,
        a[cid] ~ dnorm( 1 , 0.1 ) ,
        b[cid] ~ dnorm( 0 , 0.3 ) ,
        sigma ~ dexp( 1 )
    ) , data=dat_slim , chains=1 )

## R code 9.15
precis( m9.1 , depth=2 )

## R code 9.16
m9.1 <- ulam(
    alist(
        log_gdp_std ~ dnorm( mu , sigma ) ,
        mu <- a[cid] + b[cid]*( rugged_std - 0.215 ) ,
        a[cid] ~ dnorm( 1 , 0.1 ) ,
        b[cid] ~ dnorm( 0 , 0.3 ) ,
        sigma ~ dexp( 1 )
    ) , data=dat_slim , chains=4 , cores=4 )

## R code 9.17
show( m9.1 )

## R code 9.18
precis( m9.1 , 2 )

## R code 9.19
pairs( m9.1 )

## R code 9.20
traceplot( m9.1 )

## R code 9.21
trankplot( m9.1 , n_cols=2 )

## R code 9.22
y <- c(-1,1)
set.seed(11)
m9.2 <- ulam(
    alist(
        y ~ dnorm( mu , sigma ) ,
        mu <- alpha ,
        alpha ~ dnorm( 0 , 1000 ) ,
        sigma ~ dexp( 0.0001 )
    ) , data=list(y=y) , chains=3 )

## R code 9.23
precis( m9.2 )

## R code 9.24
set.seed(11)
m9.3 <- ulam(
    alist(
        y ~ dnorm( mu , sigma ) ,
        mu <- alpha ,
        alpha ~ dnorm( 1 , 10 ) ,
        sigma ~ dexp( 1 )
    ) , data=list(y=y) , chains=3 )
precis( m9.3 )

## R code 9.25
set.seed(41)
y <- rnorm( 100 , mean=0 , sd=1 )

## R code 9.26
set.seed(384)
m9.4 <- ulam(
    alist(
        y ~ dnorm( mu , sigma ) ,
        mu <- a1 + a2 ,
        a1 ~ dnorm( 0 , 1000 ),
        a2 ~ dnorm( 0 , 1000 ),
        sigma ~ dexp( 1 )
    ) , data=list(y=y) , chains=3 )
precis( m9.4 )

## R code 9.27
m9.5 <- ulam(
    alist(
        y ~ dnorm( mu , sigma ) ,
        mu <- a1 + a2 ,
        a1 ~ dnorm( 0 , 10 ),
        a2 ~ dnorm( 0 , 10 ),
        sigma ~ dexp( 1 )
    ) , data=list(y=y) , chains=3 )
precis( m9.5 )

## R code 9.28
mp <- map2stan(
    alist(
        a ~ dnorm(0,1),
        b ~ dcauchy(0,1)
    ),
    data=list(y=1),
    start=list(a=0,b=0),
    iter=1e4, warmup=100 , WAIC=FALSE )

## R code 9.29
N <- 100                          # number of individuals
height <- rnorm(N,10,2)           # sim total height of each
leg_prop <- runif(N,0.4,0.5)      # leg as proportion of height
leg_left <- leg_prop*height +     # sim left leg as proportion + error
    rnorm( N , 0 , 0.02 )
leg_right <- leg_prop*height +    # sim right leg as proportion + error
    rnorm( N , 0 , 0.02 )
                                  # combine into data frame
d <- data.frame(height,leg_left,leg_right)

## R code 9.30
m5.8s <- ulam(
    alist(
        height ~ dnorm( mu , sigma ) ,
        mu <- a + bl*leg_left + br*leg_right ,
        a ~ dnorm( 10 , 100 ) ,
        bl ~ dnorm( 2 , 10 ) ,
        br ~ dnorm( 2 , 10 ) ,
        sigma ~ dexp( 1 )
    ) ,
    data=d, chains=4,
    start=list(a=10,bl=0,br=0.1,sigma=1) )

## R code 9.31
m5.8s2 <- ulam(
    alist(
        height ~ dnorm( mu , sigma ) ,
        mu <- a + bl*leg_left + br*leg_right ,
        a ~ dnorm( 10 , 100 ) ,
        bl ~ dnorm( 2 , 10 ) ,
        br ~ dnorm( 2 , 10 ) ,
        sigma ~ dexp( 1 )
    ) ,
    data=d, chains=4,
    constraints=list(br="lower=0"),
    start=list(a=10,bl=0,br=0.1,sigma=1) )

## R code 10.1
p <- list()
p$A <- c(0,0,10,0,0)
p$B <- c(0,1,8,1,0)
p$C <- c(0,2,6,2,0)
p$D <- c(1,2,4,2,1)
p$E <- c(2,2,2,2,2)

## R code 10.2
p_norm <- lapply( p , function(q) q/sum(q))

## R code 10.3
( H <- sapply( p_norm , function(q) -sum(ifelse(q==0,0,q*log(q))) ) )

## R code 10.4
ways <- c(1,90,1260,37800,113400)
logwayspp <- log(ways)/10

## R code 10.5
# build list of the candidate distributions
p <- list()
p[[1]] <- c(1/4,1/4,1/4,1/4)
p[[2]] <- c(2/6,1/6,1/6,2/6)
p[[3]] <- c(1/6,2/6,2/6,1/6)
p[[4]] <- c(1/8,4/8,2/8,1/8)

# compute expected value of each
sapply( p , function(p) sum(p*c(0,1,1,2)) )

## R code 10.6
# compute entropy of each distribution
sapply( p , function(p) -sum( p*log(p) ) )

## R code 10.7
p <- 0.7
( A <- c( (1-p)^2 , p*(1-p) , (1-p)*p , p^2 ) )

## R code 10.8
-sum( A*log(A) )

## R code 10.9
sim.p <- function(G=1.4) {
    x123 <- runif(3)
    x4 <- ( (G)*sum(x123)-x123[2]-x123[3] )/(2-G)
    z <- sum( c(x123,x4) )
    p <- c( x123 , x4 )/z
    list( H=-sum( p*log(p) ) , p=p )
}

## R code 10.10
H <- replicate( 1e5 , sim.p(1.4) )
dens( as.numeric(H[1,]) , adj=0.1 )

## R code 10.11
entropies <- as.numeric(H[1,])
distributions <- H[2,]

## R code 10.12
max(entropies)

## R code 10.13
distributions[ which.max(entropies) ]

## R code 11.1
library(rethinking)
data(chimpanzees)
d <- chimpanzees

## R code 11.2
d$treatment <- 1 + d$prosoc_left + 2*d$condition

## R code 11.3
xtabs( ~ treatment + prosoc_left + condition , d )

## R code 11.4
m11.1 <- quap(
    alist(
        pulled_left ~ dbinom( 1 , p ) ,
        logit(p) <- a ,
        a ~ dnorm( 0 , 10 )
    ) , data=d )

## R code 11.5
set.seed(1999)
prior <- extract.prior( m11.1 , n=1e4 )

## R code 11.6
p <- inv_logit( prior$a )
dens( p , adj=0.1 )

## R code 11.7
m11.2 <- quap(
    alist(
        pulled_left ~ dbinom( 1 , p ) ,
        logit(p) <- a + b[treatment] ,
        a ~ dnorm( 0 , 1.5 ),
        b[treatment] ~ dnorm( 0 , 10 )
    ) , data=d )
set.seed(1999)
prior <- extract.prior( m11.2 , n=1e4 )
p <- sapply( 1:4 , function(k) inv_logit( prior$a + prior$b[,k] ) )

## R code 11.8
dens( abs( p[,1] - p[,2] ) , adj=0.1 )

## R code 11.9
m11.3 <- quap(
    alist(
        pulled_left ~ dbinom( 1 , p ) ,
        logit(p) <- a + b[treatment] ,
        a ~ dnorm( 0 , 1.5 ),
        b[treatment] ~ dnorm( 0 , 0.5 )
    ) , data=d )
set.seed(1999)
prior <- extract.prior( m11.3 , n=1e4 )
p <- sapply( 1:4 , function(k) inv_logit( prior$a + prior$b[,k] ) )
mean( abs( p[,1] - p[,2] ) )

## R code 11.10
# prior trimmed data list
dat_list <- list(
    pulled_left = d$pulled_left,
    actor = d$actor,
    treatment = as.integer(d$treatment) )

## R code 11.11
m11.4 <- ulam(
    alist(
        pulled_left ~ dbinom( 1 , p ) ,
        logit(p) <- a[actor] + b[treatment] ,
        a[actor] ~ dnorm( 0 , 1.5 ),
        b[treatment] ~ dnorm( 0 , 0.5 )
    ) , data=dat_list , chains=4 , log_lik=TRUE )
precis( m11.4 , depth=2 )

## R code 11.12
post <- extract.samples(m11.4)
p_left <- inv_logit( post$a )
plot( precis( as.data.frame(p_left) ) , xlim=c(0,1) )

## R code 11.13
labs <- c("R/N","L/N","R/P","L/P")
plot( precis( m11.4 , depth=2 , pars="b" ) , labels=labs )

## R code 11.14
diffs <- list(
    db13 = post$b[,1] - post$b[,3],
    db24 = post$b[,2] - post$b[,4] )
plot( precis(diffs) )

## R code 11.15
pl <- by( d$pulled_left , list( d$actor , d$treatment ) , mean )
pl[1,]

## R code 11.16
plot( NULL , xlim=c(1,28) , ylim=c(0,1) , xlab="" ,
    ylab="proportion left lever" , xaxt="n" , yaxt="n" )
axis( 2 , at=c(0,0.5,1) , labels=c(0,0.5,1) )
abline( h=0.5 , lty=2 )
for ( j in 1:7 ) abline( v=(j-1)*4+4.5 , lwd=0.5 )
for ( j in 1:7 ) text( (j-1)*4+2.5 , 1.1 , concat("actor ",j) , xpd=TRUE )
for ( j in (1:7)[-2] ) {
    lines( (j-1)*4+c(1,3) , pl[j,c(1,3)] , lwd=2 , col=rangi2 )
    lines( (j-1)*4+c(2,4) , pl[j,c(2,4)] , lwd=2 , col=rangi2 )
}
points( 1:28 , t(pl) , pch=16 , col="white" , cex=1.7 )
points( 1:28 , t(pl) , pch=c(1,1,16,16) , col=rangi2 , lwd=2 )
yoff <- 0.01
text( 1 , pl[1,1]-yoff , "R/N" , pos=1 , cex=0.8 )
text( 2 , pl[1,2]+yoff , "L/N" , pos=3 , cex=0.8 )
text( 3 , pl[1,3]-yoff , "R/P" , pos=1 , cex=0.8 )
text( 4 , pl[1,4]+yoff , "L/P" , pos=3 , cex=0.8 )
mtext( "observed proportions\n" )

## R code 11.17
dat <- list( actor=rep(1:7,each=4) , treatment=rep(1:4,times=7) )
p_post <- link( m11.4 , data=dat )
p_mu <- apply( p_post , 2 , mean )
p_ci <- apply( p_post , 2 , PI )

## R code 11.18
d$side <- d$prosoc_left + 1 # right 1, left 2
d$cond <- d$condition + 1 # no partner 1, partner 2

## R code 11.19
dat_list2 <- list(
    pulled_left = d$pulled_left,
    actor = d$actor,
    side = d$side,
    cond = d$cond )
m11.5 <- ulam(
    alist(
        pulled_left ~ dbinom( 1 , p ) ,
        logit(p) <- a[actor] + bs[side] + bc[cond] ,
        a[actor] ~ dnorm( 0 , 1.5 ),
        bs[side] ~ dnorm( 0 , 0.5 ),
        bc[cond] ~ dnorm( 0 , 0.5 )
    ) , data=dat_list2 , chains=4 , log_lik=TRUE )

## R code 11.20
compare( m11.5 , m11.4 , func=PSIS )

## R code 11.21
post <- extract.samples( m11.4 , clean=FALSE )
str(post)

## R code 11.22
m11.4_stan_code <- stancode(m11.4)
m11.4_stan <- stan( model_code=m11.4_stan_code , data=dat_list , chains=4 )
compare( m11.4_stan , m11.4 )

## R code 11.23
post <- extract.samples(m11.4)
mean( exp(post$b[,4]-post$b[,2]) )

## R code 11.24
data(chimpanzees)
d <- chimpanzees
d$treatment <- 1 + d$prosoc_left + 2*d$condition
d$side <- d$prosoc_left + 1 # right 1, left 2
d$cond <- d$condition + 1 # no partner 1, partner 2
d_aggregated <- aggregate(
    d$pulled_left ,
    list( treatment=d$treatment , actor=d$actor ,
          side=d$side , cond=d$cond ) ,
    sum )
colnames(d_aggregated)[5] <- "left_pulls"

## R code 11.25
dat <- with( d_aggregated , list(
    left_pulls = left_pulls,
    treatment = treatment,
    actor = actor,
    side = side,
    cond = cond ) )

m11.6 <- ulam(
    alist(
        left_pulls ~ dbinom( 18 , p ) ,
        logit(p) <- a[actor] + b[treatment] ,
        a[actor] ~ dnorm( 0 , 1.5 ) ,
        b[treatment] ~ dnorm( 0 , 0.5 )
    ) , data=dat , chains=4 , log_lik=TRUE )

## R code 11.26
compare( m11.6 , m11.4 , func=PSIS )

## R code 11.27
# deviance of aggregated 6-in-9
-2*dbinom(6,9,0.2,log=TRUE)
# deviance of dis-aggregated
-2*sum(dbern(c(1,1,1,1,1,1,0,0,0),0.2,log=TRUE))

## R code 11.28
library(rethinking)
data(UCBadmit)
d <- UCBadmit

## R code 11.29
dat_list <- list(
    admit = d$admit,
    applications = d$applications,
    gid = ifelse( d$applicant.gender=="male" , 1 , 2 )
)
m11.7 <- ulam(
    alist(
        admit ~ dbinom( applications , p ) ,
        logit(p) <- a[gid] ,
        a[gid] ~ dnorm( 0 , 1.5 )
    ) , data=dat_list , chains=4 )
precis( m11.7 , depth=2 )

## R code 11.30
post <- extract.samples(m11.7)
diff_a <- post$a[,1] - post$a[,2]
diff_p <- inv_logit(post$a[,1]) - inv_logit(post$a[,2])
precis( list( diff_a=diff_a , diff_p=diff_p ) )

## R code 11.31
postcheck( m11.7 )
# draw lines connecting points from same dept
for ( i in 1:6 ) {
    x <- 1 + 2*(i-1)
    y1 <- d$admit[x]/d$applications[x]
    y2 <- d$admit[x+1]/d$applications[x+1]
    lines( c(x,x+1) , c(y1,y2) , col=rangi2 , lwd=2 )
    text( x+0.5 , (y1+y2)/2 + 0.05 , d$dept[x] , cex=0.8 , col=rangi2 )
}

## R code 11.32
dat_list$dept_id <- rep(1:6,each=2)
m11.8 <- ulam(
    alist(
        admit ~ dbinom( applications , p ) ,
        logit(p) <- a[gid] + delta[dept_id] ,
        a[gid] ~ dnorm( 0 , 1.5 ) ,
        delta[dept_id] ~ dnorm( 0 , 1.5 )
    ) , data=dat_list , chains=4 , iter=4000 )
precis( m11.8 , depth=2 )

## R code 11.33
post <- extract.samples(m11.8)
diff_a <- post$a[,1] - post$a[,2]
diff_p <- inv_logit(post$a[,1]) - inv_logit(post$a[,2])
precis( list( diff_a=diff_a , diff_p=diff_p ) )

## R code 11.34
pg <- with( dat_list , sapply( 1:6 , function(k)
    applications[dept_id==k]/sum(applications[dept_id==k]) ) )
rownames(pg) <- c("male","female")
colnames(pg) <- unique(d$dept)
round( pg , 2 )

## R code 11.35
y <- rbinom(1e5,1000,1/1000)
c( mean(y) , var(y) )

## R code 11.36
library(rethinking)
data(Kline)
d <- Kline
d

## R code 11.37
d$P <- scale( log(d$population) )
d$contact_id <- ifelse( d$contact=="high" , 2 , 1 )

## R code 11.38
curve( dlnorm( x , 0 , 10 ) , from=0 , to=100 , n=200 )

## R code 11.39
a <- rnorm(1e4,0,10)
lambda <- exp(a)
mean( lambda )

## R code 11.40
curve( dlnorm( x , 3 , 0.5 ) , from=0 , to=100 , n=200 )

## R code 11.41
N <- 100
a <- rnorm( N , 3 , 0.5 )
b <- rnorm( N , 0 , 10 )
plot( NULL , xlim=c(-2,2) , ylim=c(0,100) )
for ( i in 1:N ) curve( exp( a[i] + b[i]*x ) , add=TRUE , col=grau() )

## R code 11.42
set.seed(10)
N <- 100
a <- rnorm( N , 3 , 0.5 )
b <- rnorm( N , 0 , 0.2 )
plot( NULL , xlim=c(-2,2) , ylim=c(0,100) )
for ( i in 1:N ) curve( exp( a[i] + b[i]*x ) , add=TRUE , col=grau() )

## R code 11.43
x_seq <- seq( from=log(100) , to=log(200000) , length.out=100 )
lambda <- sapply( x_seq , function(x) exp( a + b*x ) )
plot( NULL , xlim=range(x_seq) , ylim=c(0,500) , xlab="log population" ,
    ylab="total tools" )
for ( i in 1:N ) lines( x_seq , lambda[i,] , col=grau() , lwd=1.5 )

## R code 11.44
plot( NULL , xlim=range(exp(x_seq)) , ylim=c(0,500) , xlab="population" ,
    ylab="total tools" )
for ( i in 1:N ) lines( exp(x_seq) , lambda[i,] , col=grau() , lwd=1.5 )

## R code 11.45
dat <- list(
    T = d$total_tools ,
    P = d$P ,
    cid = d$contact_id )

# intercept only
m11.9 <- ulam(
    alist(
        T ~ dpois( lambda ),
        log(lambda) <- a,
        a ~ dnorm(3,0.5)
    ), data=dat , chains=4 , log_lik=TRUE )

# interaction model
m11.10 <- ulam(
    alist(
        T ~ dpois( lambda ),
        log(lambda) <- a[cid] + b[cid]*P,
        a[cid] ~ dnorm( 3 , 0.5 ),
        b[cid] ~ dnorm( 0 , 0.2 )
    ), data=dat , chains=4 , log_lik=TRUE )

## R code 11.46
compare( m11.9 , m11.10 , func=PSIS )

## R code 11.47
k <- PSIS( m11.10 , pointwise=TRUE )$k
plot( dat$P , dat$T , xlab="log population (std)" , ylab="total tools" ,
    col=rangi2 , pch=ifelse( dat$cid==1 , 1 , 16 ) , lwd=2 ,
    ylim=c(0,75) , cex=1+normalize(k) )

# set up the horizontal axis values to compute predictions at
ns <- 100
P_seq <- seq( from=-1.4 , to=3 , length.out=ns )

# predictions for cid=1 (low contact)
lambda <- link( m11.10 , data=data.frame( P=P_seq , cid=1 ) )
lmu <- apply( lambda , 2 , mean )
lci <- apply( lambda , 2 , PI )
lines( P_seq , lmu , lty=2 , lwd=1.5 )
shade( lci , P_seq , xpd=TRUE )

# predictions for cid=2 (high contact)
lambda <- link( m11.10 , data=data.frame( P=P_seq , cid=2 ) )
lmu <- apply( lambda , 2 , mean )
lci <- apply( lambda , 2 , PI )
lines( P_seq , lmu , lty=1 , lwd=1.5 )
shade( lci , P_seq , xpd=TRUE )

## R code 11.48
plot( d$population , d$total_tools , xlab="population" , ylab="total tools" ,
    col=rangi2 , pch=ifelse( dat$cid==1 , 1 , 16 ) , lwd=2 ,
    ylim=c(0,75) , cex=1+normalize(k) )

ns <- 100
P_seq <- seq( from=-5 , to=3 , length.out=ns )
# 1.53 is sd of log(population)
# 9 is mean of log(population)
pop_seq <- exp( P_seq*1.53 + 9 )

lambda <- link( m11.10 , data=data.frame( P=P_seq , cid=1 ) )
lmu <- apply( lambda , 2 , mean )
lci <- apply( lambda , 2 , PI )
lines( pop_seq , lmu , lty=2 , lwd=1.5 )
shade( lci , pop_seq , xpd=TRUE )

lambda <- link( m11.10 , data=data.frame( P=P_seq , cid=2 ) )
lmu <- apply( lambda , 2 , mean )
lci <- apply( lambda , 2 , PI )
lines( pop_seq , lmu , lty=1 , lwd=1.5 )
shade( lci , pop_seq , xpd=TRUE )

## R code 11.49
dat2 <- list( T=d$total_tools, P=d$population, cid=d$contact_id )
m11.11 <- ulam(
    alist(
        T ~ dpois( lambda ),
        lambda <- exp(a[cid])*P^b[cid]/g,
        a[cid] ~ dnorm(1,1),
        b[cid] ~ dexp(1),
        g ~ dexp(1)
    ), data=dat2 , chains=4 , log_lik=TRUE )

## R code 11.50
num_days <- 30
y <- rpois( num_days , 1.5 )

## R code 11.51
num_weeks <- 4
y_new <- rpois( num_weeks , 0.5*7 )

## R code 11.52
y_all <- c( y , y_new )
exposure <- c( rep(1,30) , rep(7,4) )
monastery <- c( rep(0,30) , rep(1,4) )
d <- data.frame( y=y_all , days=exposure , monastery=monastery )

## R code 11.53
# compute the offset
d$log_days <- log( d$days )

# fit the model
m11.12 <- quap(
    alist(
        y ~ dpois( lambda ),
        log(lambda) <- log_days + a + b*monastery,
        a ~ dnorm( 0 , 1 ),
        b ~ dnorm( 0 , 1 )
    ), data=d )

## R code 11.54
post <- extract.samples( m10.15 )
lambda_old <- exp( post$a )
lambda_new <- exp( post$a + post$b )
precis( data.frame( lambda_old , lambda_new ) )

## R code 11.55
# simulate career choices among 500 individuals
N <- 500             # number of individuals
income <- c(1,2,5)   # expected income of each career
score <- 0.5*income  # scores for each career, based on income
# next line converts scores to probabilities
p <- softmax(score[1],score[2],score[3])

# now simulate choice
# outcome career holds event type values, not counts
career <- rep(NA,N)  # empty vector of choices for each individual
# sample chosen career for each individual
set.seed(34302)
for ( i in 1:N ) career[i] <- sample( 1:3 , size=1 , prob=p )

## R code 11.56
code_m11.13 <- "
data{
    int N; // number of individuals
    int K; // number of possible careers
    int career[N]; // outcome
    vector[K] career_income;
}
parameters{
    vector[K-1] a; // intercepts
    real<lower=0> b; // association of income with choice
}
model{
    vector[K] p;
    vector[K] s;
    a ~ normal( 0 , 1 );
    b ~ normal( 0 , 0.5 );
    s[1] = a[1] + b*career_income[1];
    s[2] = a[2] + b*career_income[2];
    s[3] = 0; // pivot
    p = softmax( s );
    career ~ categorical( p );
}
"

## R code 11.57
dat_list <- list( N=N , K=3 , career=career , career_income=income )
m11.13 <- stan( model_code=code_m11.13 , data=dat_list , chains=4 )
precis( m11.13 , 2 )

## R code 11.58
post <- extract.samples( m11.13 )

# set up logit scores
s1 <- with( post , a[,1] + b*income[1] )
s2_orig <- with( post , a[,2] + b*income[2] )
s2_new <- with( post , a[,2] + b*income[2]*2 )

# compute probabilities for original and counterfactual
p_orig <- sapply( 1:length(post$b) , function(i)
    softmax( c(s1[i],s2_orig[i],0) ) )
p_new <- sapply( 1:length(post$b) , function(i)
    softmax( c(s1[i],s2_new[i],0) ) )

# summarize
p_diff <- p_new[2,] - p_orig[2,]
precis( p_diff )

## R code 11.59
N <- 500
# simulate family incomes for each individual
family_income <- runif(N)
# assign a unique coefficient for each type of event
b <- c(-2,0,2)
career <- rep(NA,N)  # empty vector of choices for each individual
for ( i in 1:N ) {
    score <- 0.5*(1:3) + b*family_income[i]
    p <- softmax(score[1],score[2],score[3])
    career[i] <- sample( 1:3 , size=1 , prob=p )
}

code_m11.14 <- "
data{
    int N; // number of observations
    int K; // number of outcome values
    int career[N]; // outcome
    real family_income[N];
}
parameters{
    vector[K-1] a; // intercepts
    vector[K-1] b; // coefficients on family income
}
model{
    vector[K] p;
    vector[K] s;
    a ~ normal(0,1.5);
    b ~ normal(0,1);
    for ( i in 1:N ) {
        for ( j in 1:(K-1) ) s[j] = a[j] + b[j]*family_income[i];
        s[K] = 0; // the pivot
        p = softmax( s );
        career[i] ~ categorical( p );
    }
}
"

dat_list <- list( N=N , K=3 , career=career , family_income=family_income )
m11.14 <- stan( model_code=code_m11.14 , data=dat_list , chains=4 )
precis( m11.14 , 2 )

## R code 11.60
library(rethinking)
data(UCBadmit)
d <- UCBadmit

## R code 11.61
# binomial model of overall admission probability
m_binom <- quap(
    alist(
        admit ~ dbinom(applications,p),
        logit(p) <- a,
        a ~ dnorm( 0 , 1.5 )
    ), data=d )

# Poisson model of overall admission rate and rejection rate
# 'reject' is a reserved word in Stan, cannot use as variable name
dat <- list( admit=d$admit , rej=d$reject )
m_pois <- ulam(
    alist(
        admit ~ dpois(lambda1),
        rej ~ dpois(lambda2),
        log(lambda1) <- a1,
        log(lambda2) <- a2,
        c(a1,a2) ~ dnorm(0,1.5)
    ), data=dat , chains=3 , cores=3 )

## R code 11.62
inv_logit(coef(m_binom))

## R code 11.63
k <- coef(m_pois)
a1 <- k['a1']
a2 <- k['a2']
exp(a1)/(exp(a1)+exp(a2))

## R code 11.64
x2 <- replicate( 1e5 , min(runif( 2 , 1 , 100 )) )
x5 <- replicate( 1e5 , min(runif( 5 , 1 , 100 )) )

## R code 11.65
N <- 10
M <- 2
x <- replicate( 1e5 , sort(runif(N,1,100))[M] )

## R code 11.66
library(rethinking)
data(AustinCats)
d <- AustinCats

d$adopt <- ifelse( d$out_event=="Adoption" , 1L , 0L )
dat <- list(
    days_to_event = as.numeric( d$days_to_event ),
    color_id = ifelse( d$color=="Black" , 1L , 2L ) ,
    adopted = d$adopt
)

m11.15 <- ulam(
    alist(
        days_to_event|adopted==1 ~ exponential( lambda ),
        days_to_event|adopted==0 ~ custom(exponential_lccdf( !Y | lambda )),
        lambda <- 1.0/mu,
        log(mu) <- a[color_id],
        a[color_id] ~ normal(0,1)
    ), data=dat , chains=4 , cores=4 )

precis( m11.15 , 2 )

## R code 11.67
post <- extract.samples( m11.15 )
post$D <- exp(post$a)
precis( post , 2 )

## R code 11.68
model{
    vector[22356] lambda;
    b ~ normal( 0 , 1 );
    a ~ normal( 0 , 1 );
    for ( i in 1:22356 ) {
        lambda[i] = a + b * black[i];
        lambda[i] = exp(lambda[i]);
    }
    for ( i in 1:22356 )
        if ( adopted[i] == 0 ) target += exponential_lccdf(days_to_event[i] | lambda[i]);
    for ( i in 1:22356 )
        if ( adopted[i] == 1 ) days_to_event[i] ~ exponential( lambda[i] );
}

## R code 11.69
    for ( i in 1:22356 ) {
        if ( adopted[i] == 0 ) target += -lambda[i]*days_to_event[i];
        if ( adopted[i] == 1 ) target += log(lambda[i]) - lambda[i]*days_to_event[i];
    }

## R code 12.1
pbar <- 0.5
theta <- 5
curve( dbeta2(x,pbar,theta) , from=0 , to=1 ,
    xlab="probability" , ylab="Density" )

## R code 12.2
library(rethinking)
data(UCBadmit)
d <- UCBadmit
d$gid <- ifelse( d$applicant.gender=="male" , 1L , 2L )
dat <- list( A=d$admit , N=d$applications , gid=d$gid )
m12.1 <- ulam(
    alist(
        A ~ dbetabinom( N , pbar , theta ),
        logit(pbar) <- a[gid],
        a[gid] ~ dnorm( 0 , 1.5 ),
        transpars> theta <<- phi + 2.0,
        phi ~ dexp(1)
    ), data=dat , chains=4 )

## R code 12.3
post <- extract.samples( m12.1 )
post$da <- post$a[,1] - post$a[,2]
precis( post , depth=2 )

## R code 12.4
gid <- 2
# draw posterior mean beta distribution
curve( dbeta2(x,mean(logistic(post$a[,gid])),mean(post$theta)) , from=0 , to=1 ,
    ylab="Density" , xlab="probability admit", ylim=c(0,3) , lwd=2 )

# draw 50 beta distributions sampled from posterior
for ( i in 1:50 ) {
    p <- logistic( post$a[i,gid] )
    theta <- post$theta[i]
    curve( dbeta2(x,p,theta) , add=TRUE , col=col.alpha("black",0.2) )
}
mtext( "distribution of female admission rates" )

## R code 12.5
postcheck( m12.1 )

## R code 12.6
library(rethinking)
data(Kline)
d <- Kline
d$P <- standardize( log(d$population) )
d$contact_id <- ifelse( d$contact=="high" , 2L , 1L )

dat2 <- list(
    T = d$total_tools,
    P = d$population,
    cid = d$contact_id )

m12.2 <- ulam(
    alist(
        T ~ dgampois( lambda , phi ),
        lambda <- exp(a[cid])*P^b[cid] / g,
        a[cid] ~ dnorm(1,1),
        b[cid] ~ dexp(1),
        g ~ dexp(1),
        phi ~ dexp(1)
    ), data=dat2 , chains=4 , log_lik=TRUE )

## R code 12.7
# define parameters
prob_drink <- 0.2 # 20% of days
rate_work <- 1    # average 1 manuscript per day

# sample one year of production
N <- 365

# simulate days monks drink
set.seed(365)
drink <- rbinom( N , 1 , prob_drink )

# simulate manuscripts completed
y <- (1-drink)*rpois( N , rate_work )

## R code 12.8
simplehist( y , xlab="manuscripts completed" , lwd=4 )
zeros_drink <- sum(drink)
zeros_work <- sum(y==0 & drink==0)
zeros_total <- sum(y==0)
lines( c(0,0) , c(zeros_work,zeros_total) , lwd=4 , col=rangi2 )

## R code 12.9
m12.3 <- ulam(
    alist(
        y ~ dzipois( p , lambda ),
        logit(p) <- ap,
        log(lambda) <- al,
        ap ~ dnorm( -1.5 , 1 ),
        al ~ dnorm( 1 , 0.5 )
    ) , data=list(y=y) , chains=4 )
precis( m12.3 )

## R code 12.10
post <- extract.samples( m12.3 )
mean( inv_logit( post$ap ) ) # probability drink
mean( exp( post$al ) )       # rate finish manuscripts, when not drinking

## R code 12.11
m12.3_alt <- ulam(
    alist(
        y|y>0 ~ custom( log1m(p) + poisson_lpmf(y|lambda) ),
        y|y==0 ~ custom( log_mix( p , 0 , poisson_lpmf(0|lambda) ) ),
        logit(p) <- ap,
        log(lambda) <- al,
        ap ~ dnorm(-1.5,1),
        al ~ dnorm(1,0.5)
    ) , data=list(y=as.integer(y)) , chains=4 )

## R code 12.12
library(rethinking)
data(Trolley)
d <- Trolley

## R code 12.13
simplehist( d$response , xlim=c(1,7) , xlab="response" )

## R code 12.14
# discrete proportion of each response value
pr_k <- table( d$response ) / nrow(d)

# cumsum converts to cumulative proportions
cum_pr_k <- cumsum( pr_k )

# plot
plot( 1:7 , cum_pr_k , type="b" , xlab="response" ,
ylab="cumulative proportion" , ylim=c(0,1) )

## R code 12.15
logit <- function(x) log(x/(1-x)) # convenience function
round( lco <- logit( cum_pr_k ) , 2 )

## R code 12.16
m12.4 <- ulam(
    alist(
        R ~ dordlogit( 0 , cutpoints ),
        cutpoints ~ dnorm( 0 , 1.5 )
    ) , data=list( R=d$response ), chains=4 , cores=4 )

## R code 12.17
m12.4q <- quap(
    alist(
        response ~ dordlogit( 0 , c(a1,a2,a3,a4,a5,a6) ),
        c(a1,a2,a3,a4,a5,a6) ~ dnorm( 0 , 1.5 )
    ) , data=d , start=list(a1=-2,a2=-1,a3=0,a4=1,a5=2,a6=2.5) )

## R code 12.18
precis( m12.4 , depth=2 )

## R code 12.19
round( inv_logit(coef(m12.4)) , 3 )

## R code 12.20
round( pk <- dordlogit( 1:7 , 0 , coef(m12.4) ) , 2 )

## R code 12.21
sum( pk*(1:7) )

## R code 12.22
round( pk <- dordlogit( 1:7 , 0 , coef(m12.4)-0.5 ) , 2 )

## R code 12.23
sum( pk*(1:7) )

## R code 12.24
dat <- list(
    R = d$response,
    A = d$action,
    I = d$intention,
    C = d$contact )
m12.5 <- ulam(
    alist(
        R ~ dordlogit( phi , cutpoints ),
        phi <- bA*A + bC*C + BI*I ,
        BI <- bI + bIA*A + bIC*C ,
        c(bA,bI,bC,bIA,bIC) ~ dnorm( 0 , 0.5 ),
        cutpoints ~ dnorm( 0 , 1.5 )
    ) , data=dat , chains=4 , cores=4 )
precis( m12.5 )

## R code 12.25
plot( precis(m12.5) , xlim=c(-1.4,0) )

## R code 12.26
plot( NULL , type="n" , xlab="intention" , ylab="probability" ,
    xlim=c(0,1) , ylim=c(0,1) , xaxp=c(0,1,1) , yaxp=c(0,1,2) )

## R code 12.27
kA <- 0     # value for action
kC <- 0     # value for contact
kI <- 0:1   # values of intention to calculate over
pdat <- data.frame(A=kA,C=kC,I=kI)
phi <- link( m12.5 , data=pdat )$phi

## R code 12.28
post <- extract.samples( m12.5 )
for ( s in 1:50 ) {
    pk <- pordlogit( 1:6 , phi[s,] , post$cutpoints[s,] )
    for ( i in 1:6 ) lines( kI , pk[,i] , col=grau(0.1) )
}

## R code 12.29
kA <- 0     # value for action
kC <- 1     # value for contact
kI <- 0:1   # values of intention to calculate over
pdat <- data.frame(A=kA,C=kC,I=kI)
s <- sim( m12.5 , data=pdat )
simplehist( s , xlab="response" )

## R code 12.30
library(rethinking)
data(Trolley)
d <- Trolley
levels(d$edu)

## R code 12.31
edu_levels <- c( 6 , 1 , 8 , 4 , 7 , 2 , 5 , 3 )
d$edu_new <- edu_levels[ d$edu ]

## R code 12.32
library(gtools)
set.seed(1805)
delta <- rdirichlet( 10 , alpha=rep(2,7) )
str(delta)

## R code 12.33
h <- 3
plot( NULL , xlim=c(1,7) , ylim=c(0,0.4) , xlab="index" , ylab="probability" )
for ( i in 1:nrow(delta) ) lines( 1:7 , delta[i,] , type="b" ,
    pch=ifelse(i==h,16,1) , lwd=ifelse(i==h,4,1.5) ,
    col=ifelse(i==h,"black",col.alpha("black",0.7)) )

## R code 12.34
dat <- list(
    R = d$response ,
    action = d$action,
    intention = d$intention,
    contact = d$contact,
    E = as.integer( d$edu_new ),   # edu_new as an index
    alpha = rep( 2 , 7 ) )      # delta prior

m12.6 <- ulam(
    alist(
        R ~ ordered_logistic( phi , kappa ),
        phi <- bE*sum( delta_j[1:E] ) + bA*action + bI*intention + bC*contact,
        kappa ~ normal( 0 , 1.5 ),
        c(bA,bI,bC,bE) ~ normal( 0 , 1 ),
        vector[8]: delta_j <<- append_row( 0 , delta ),
        simplex[7]: delta ~ dirichlet( alpha )
    ), data=dat , chains=4 , cores=4 , coerce_int=FALSE )

## R code 12.35
precis( m12.6 , depth=2 , omit="kappa" )

## R code 12.36
delta_labels <- c("Elem","MidSch","SHS","HSG","SCol","Bach","Mast","Grad")
pairs( m12.6 , pars="delta" , labels=delta_labels )

## R code 12.37
dat$edu_norm <- normalize( d$edu_new )
m12.7 <- ulam(
    alist(
        R ~ ordered_logistic( mu , cutpoints ),
        mu <- bE*edu_norm + bA*action + bI*intention + bC*contact,
        c(bA,bI,bC,bE) ~ normal( 0 , 1 ),
        cutpoints ~ normal( 0 , 1.5 )
    ), data=dat , chains=4 , cores=4 )
precis( m12.7 )

## R code 12.38
library(rethinking)
data(Hurricanes)

## R code 13.1
library(rethinking)
data(reedfrogs)
d <- reedfrogs
str(d)

## R code 13.2
# make the tank cluster variable
d$tank <- 1:nrow(d)

dat <- list(
    S = d$surv,
    N = d$density,
    tank = d$tank )

# approximate posterior
m13.1 <- ulam(
    alist(
        S ~ dbinom( N , p ) ,
        logit(p) <- a[tank] ,
        a[tank] ~ dnorm( 0 , 1.5 )
    ), data=dat , chains=4 , log_lik=TRUE )

## R code 13.3
m13.2 <- ulam(
    alist(
        S ~ dbinom( N , p ) ,
        logit(p) <- a[tank] ,
        a[tank] ~ dnorm( a_bar , sigma ) ,
        a_bar ~ dnorm( 0 , 1.5 ) ,
        sigma ~ dexp( 1 )
    ), data=dat , chains=4 , log_lik=TRUE )

## R code 13.4
compare( m13.1 , m13.2 )

## R code 13.5
# extract Stan samples
post <- extract.samples(m13.2)

# compute median intercept for each tank
# also transform to probability with logistic
d$propsurv.est <- logistic( apply( post$a , 2 , mean ) )

# display raw proportions surviving in each tank
plot( d$propsurv , ylim=c(0,1) , pch=16 , xaxt="n" ,
    xlab="tank" , ylab="proportion survival" , col=rangi2 )
axis( 1 , at=c(1,16,32,48) , labels=c(1,16,32,48) )

# overlay posterior means
points( d$propsurv.est )

# mark posterior mean probability across tanks
abline( h=mean(inv_logit(post$a_bar)) , lty=2 )

# draw vertical dividers between tank densities
abline( v=16.5 , lwd=0.5 )
abline( v=32.5 , lwd=0.5 )
text( 8 , 0 , "small tanks" )
text( 16+8 , 0 , "medium tanks" )
text( 32+8 , 0 , "large tanks" )

## R code 13.6
# show first 100 populations in the posterior
plot( NULL , xlim=c(-3,4) , ylim=c(0,0.35) ,
    xlab="log-odds survive" , ylab="Density" )
for ( i in 1:100 )
    curve( dnorm(x,post$a_bar[i],post$sigma[i]) , add=TRUE ,
    col=col.alpha("black",0.2) )

# sample 8000 imaginary tanks from the posterior distribution
sim_tanks <- rnorm( 8000 , post$a_bar , post$sigma )

# transform to probability and visualize
dens( inv_logit(sim_tanks) , lwd=2 , adj=0.1 )

## R code 13.7
a_bar <- 1.5
sigma <- 1.5
nponds <- 60
Ni <- as.integer( rep( c(5,10,25,35) , each=15 ) )

## R code 13.8
set.seed(5005)
a_pond <- rnorm( nponds , mean=a_bar , sd=sigma )

## R code 13.9
dsim <- data.frame( pond=1:nponds , Ni=Ni , true_a=a_pond )

## R code 13.10
class(1:3)
class(c(1,2,3))

## R code 13.11
dsim$Si <- rbinom( nponds , prob=logistic(dsim$true_a) , size=dsim$Ni )

## R code 13.12
dsim$p_nopool <- dsim$Si / dsim$Ni

## R code 13.13
dat <- list( Si=dsim$Si , Ni=dsim$Ni , pond=dsim$pond )
m13.3 <- ulam(
    alist(
        Si ~ dbinom( Ni , p ),
        logit(p) <- a_pond[pond],
        a_pond[pond] ~ dnorm( a_bar , sigma ),
        a_bar ~ dnorm( 0 , 1.5 ),
        sigma ~ dexp( 1 )
    ), data=dat , chains=4 )

## R code 13.14
precis( m13.3 , depth=2 )

## R code 13.15
post <- extract.samples( m13.3 )
dsim$p_partpool <- apply( inv_logit(post$a_pond) , 2 , mean )

## R code 13.16
dsim$p_true <- inv_logit( dsim$true_a )

## R code 13.17
nopool_error <- abs( dsim$p_nopool - dsim$p_true )
partpool_error <- abs( dsim$p_partpool - dsim$p_true )

## R code 13.18
plot( 1:60 , nopool_error , xlab="pond" , ylab="absolute error" ,
    col=rangi2 , pch=16 )
points( 1:60 , partpool_error )

## R code 13.19
nopool_avg <- aggregate(nopool_error,list(dsim$Ni),mean)
partpool_avg <- aggregate(partpool_error,list(dsim$Ni),mean)

## R code 13.20
a <- 1.5
sigma <- 1.5
nponds <- 60
Ni <- as.integer( rep( c(5,10,25,35) , each=15 ) )
a_pond <- rnorm( nponds , mean=a , sd=sigma )
dsim <- data.frame( pond=1:nponds , Ni=Ni , true_a=a_pond )
dsim$Si <- rbinom( nponds,prob=inv_logit( dsim$true_a ),size=dsim$Ni )
dsim$p_nopool <- dsim$Si / dsim$Ni
newdat <- list(Si=dsim$Si,Ni=dsim$Ni,pond=1:nponds)
m13.3new <- stan( fit=m13.3@stanfit , data=newdat , chains=4 )

post <- extract.samples( m13.3new )
dsim$p_partpool <- apply( inv_logit(post$a_pond) , 2 , mean )
dsim$p_true <- inv_logit( dsim$true_a )
nopool_error <- abs( dsim$p_nopool - dsim$p_true )
partpool_error <- abs( dsim$p_partpool - dsim$p_true )
plot( 1:60 , nopool_error , xlab="pond" , ylab="absolute error" , col=rangi2 , pch=16 )
points( 1:60 , partpool_error )

## R code 13.21
library(rethinking)
data(chimpanzees)
d <- chimpanzees
d$treatment <- 1 + d$prosoc_left + 2*d$condition

dat_list <- list(
    pulled_left = d$pulled_left,
    actor = d$actor,
    block_id = d$block,
    treatment = as.integer(d$treatment) )

set.seed(13)
m13.4 <- ulam(
    alist(
        pulled_left ~ dbinom( 1 , p ) ,
        logit(p) <- a[actor] + g[block_id] + b[treatment] ,
        b[treatment] ~ dnorm( 0 , 0.5 ),
        # adaptive priors
        a[actor] ~ dnorm( a_bar , sigma_a ),
        g[block_id] ~ dnorm( 0 , sigma_g ),
        # hyper-priors
        a_bar ~ dnorm( 0 , 1.5 ),
        sigma_a ~ dexp(1),
        sigma_g ~ dexp(1)
    ) , data=dat_list , chains=4 , cores=4 , log_lik=TRUE )

## R code 13.22
precis( m13.4 , depth=2 )
plot( precis(m13.4,depth=2) ) # also plot

## R code 13.23
set.seed(14)
m13.5 <- ulam(
    alist(
        pulled_left ~ dbinom( 1 , p ) ,
        logit(p) <- a[actor] + b[treatment] ,
        b[treatment] ~ dnorm( 0 , 0.5 ),
        a[actor] ~ dnorm( a_bar , sigma_a ),
        a_bar ~ dnorm( 0 , 1.5 ),
        sigma_a ~ dexp(1)
    ) , data=dat_list , chains=4 , cores=4 , log_lik=TRUE )

## R code 13.24
compare( m13.4 , m13.5 )

## R code 13.25
set.seed(15)
m13.6 <- ulam(
    alist(
        pulled_left ~ dbinom( 1 , p ) ,
        logit(p) <- a[actor] + g[block_id] + b[treatment] ,
        b[treatment] ~ dnorm( 0 , sigma_b ),
        a[actor] ~ dnorm( a_bar , sigma_a ),
        g[block_id] ~ dnorm( 0 , sigma_g ),
        a_bar ~ dnorm( 0 , 1.5 ),
        sigma_a ~ dexp(1),
        sigma_g ~ dexp(1),
        sigma_b ~ dexp(1)
    ) , data=dat_list , chains=4 , cores=4 , log_lik=TRUE )
coeftab(m13.4,m13.6)

## R code 13.26
m13x <- ulam(
    alist(
        v ~ normal(0,3),
        x ~ normal(0,exp(v))
    ), data=list(N=1) , chains=4 )
precis(m13x)

## R code 13.27
m13y <- ulam(
    alist(
        v ~ normal(0,3),
        z ~ normal(0,1),
        gq> real[1]:x <<- z*exp(v)
    ), data=list(N=1) , chains=4 )
precis(m13y)

## R code 13.28
set.seed(13)
m13.4b <- ulam( m13.4 , chains=4 , cores=4 , control=list(adapt_delta=0.99) )
divergent(m13.4b)

## R code 13.29
set.seed(13)
m13.4nc <- ulam(
    alist(
        pulled_left ~ dbinom( 1 , p ) ,
        logit(p) <- a_bar + z[actor]*sigma_a + # actor intercepts
                    x[block_id]*sigma_g +      # block intercepts
                    b[treatment] ,
        b[treatment] ~ dnorm( 0 , 0.5 ),
        z[actor] ~ dnorm( 0 , 1 ),
        x[block_id] ~ dnorm( 0 , 1 ),
        a_bar ~ dnorm( 0 , 1.5 ),
        sigma_a ~ dexp(1),
        sigma_g ~ dexp(1)
    ) , data=dat_list , chains=4 , cores=4 )

## R code 13.30
neff_c <- precis( m13.4 , depth=2 )[['n_eff']]
neff_nc <- precis( m13.4nc , depth=2 )[['n_eff']]
par_names <- rownames( precis( m13.4 , depth=2 ) )
neff_table <- cbind( neff_c , neff_nc )
rownames(neff_table) <- par_names
round(t(neff_table))

## R code 13.31
chimp <- 2
d_pred <- list(
    actor = rep(chimp,4),
    treatment = 1:4,
    block_id = rep(1,4)
)
p <- link( m13.4 , data=d_pred )
p_mu <- apply( p , 2 , mean )
p_ci <- apply( p , 2 , PI )

## R code 13.32
post <- extract.samples(m13.4)
str(post)

## R code 13.33
dens( post$a[,5] )

## R code 13.34
p_link <- function( treatment , actor=1 , block_id=1 ) {
    logodds <- with( post ,
        a[,actor] + g[,block_id] + b[,treatment] )
    return( inv_logit(logodds) )
}

## R code 13.35
p_raw <- sapply( 1:4 , function(i) p_link( i , actor=2 , block_id=1 ) )
p_mu <- apply( p_raw , 2 , mean )
p_ci <- apply( p_raw , 2 , PI )

## R code 13.36
p_link_abar <- function( treatment ) {
    logodds <- with( post , a_bar + b[,treatment] )
    return( inv_logit(logodds) )
}

## R code 13.37
p_raw <- sapply( 1:4 , function(i) p_link_abar( i ) )
p_mu <- apply( p_raw , 2 , mean )
p_ci <- apply( p_raw , 2 , PI )

plot( NULL , xlab="treatment" , ylab="proportion pulled left" ,
    ylim=c(0,1) , xaxt="n" , xlim=c(1,4) )
axis( 1 , at=1:4 , labels=c("R/N","L/N","R/P","L/P") )
lines( 1:4 , p_mu )
shade( p_ci , 1:4 )

## R code 13.38
a_sim <- with( post , rnorm( length(post$a_bar) , a_bar , sigma_a ) )
p_link_asim <- function( treatment ) {
    logodds <- with( post , a_sim + b[,treatment] )
    return( inv_logit(logodds) )
}
p_raw_asim <- sapply( 1:4 , function(i) p_link_asim( i ) )

## R code 13.39
plot( NULL , xlab="treatment" , ylab="proportion pulled left" ,
    ylim=c(0,1) , xaxt="n" , xlim=c(1,4) )
axis( 1 , at=1:4 , labels=c("R/N","L/N","R/P","L/P") )
for ( i in 1:100 ) lines( 1:4 , p_raw_asim[i,] , col=col.alpha("black",0.25) , lwd=2 )

## R code 13.40
sort(unique(d$district))

## R code 13.41
d$district_id <- as.integer(as.factor(d$district))
sort(unique(d$district_id))

## R code 14.1
a <- 3.5            # average morning wait time
b <- (-1)           # average difference afternoon wait time
sigma_a <- 1        # std dev in intercepts
sigma_b <- 0.5      # std dev in slopes
rho <- (-0.7)       # correlation between intercepts and slopes

## R code 14.2
Mu <- c( a , b )

## R code 14.3
cov_ab <- sigma_a*sigma_b*rho
Sigma <- matrix( c(sigma_a^2,cov_ab,cov_ab,sigma_b^2) , ncol=2 )

## R code 14.4
matrix( c(1,2,3,4) , nrow=2 , ncol=2 )

## R code 14.5
sigmas <- c(sigma_a,sigma_b) # standard deviations
Rho <- matrix( c(1,rho,rho,1) , nrow=2 ) # correlation matrix

# now matrix multiply to get covariance matrix
Sigma <- diag(sigmas) %*% Rho %*% diag(sigmas)

## R code 14.6
N_cafes <- 20

## R code 14.7
library(MASS)
set.seed(5) # used to replicate example
vary_effects <- mvrnorm( N_cafes , Mu , Sigma )

## R code 14.8
a_cafe <- vary_effects[,1]
b_cafe <- vary_effects[,2]

## R code 14.9
plot( a_cafe , b_cafe , col=rangi2 ,
    xlab="intercepts (a_cafe)" , ylab="slopes (b_cafe)" )

# overlay population distribution
library(ellipse)
for ( l in c(0.1,0.3,0.5,0.8,0.99) )
    lines(ellipse(Sigma,centre=Mu,level=l),col=col.alpha("black",0.2))

## R code 14.10
set.seed(22)
N_visits <- 10
afternoon <- rep(0:1,N_visits*N_cafes/2)
cafe_id <- rep( 1:N_cafes , each=N_visits )
mu <- a_cafe[cafe_id] + b_cafe[cafe_id]*afternoon
sigma <- 0.5  # std dev within cafes
wait <- rnorm( N_visits*N_cafes , mu , sigma )
d <- data.frame( cafe=cafe_id , afternoon=afternoon , wait=wait )

## R code 14.11
R <- rlkjcorr( 1e4 , K=2 , eta=2 )
dens( R[,1,2] , xlab="correlation" )

## R code 14.12
m14.1 <- ulam(
    alist(
        wait ~ normal( mu , sigma ),
        mu <- a_cafe[cafe] + b_cafe[cafe]*afternoon,
        c(a_cafe,b_cafe)[cafe] ~ multi_normal( c(a,b) , Rho , sigma_cafe ),
        a ~ normal(5,2),
        b ~ normal(-1,0.5),
        sigma_cafe ~ exponential(1),
        sigma ~ exponential(1),
        Rho ~ lkj_corr(2)
    ) , data=d , chains=4 , cores=4 )

## R code 14.13
post <- extract.samples(m14.1)
dens( post$Rho[,1,2] )

## R code 14.14
# compute unpooled estimates directly from data
a1 <- sapply( 1:N_cafes ,
        function(i) mean(wait[cafe_id==i & afternoon==0]) )
b1 <- sapply( 1:N_cafes ,
        function(i) mean(wait[cafe_id==i & afternoon==1]) ) - a1

# extract posterior means of partially pooled estimates
post <- extract.samples(m13.1)
a2 <- apply( post$a_cafe , 2 , mean )
b2 <- apply( post$b_cafe , 2 , mean )

# plot both and connect with lines
plot( a1 , b1 , xlab="intercept" , ylab="slope" ,
    pch=16 , col=rangi2 , ylim=c( min(b1)-0.1 , max(b1)+0.1 ) ,
    xlim=c( min(a1)-0.1 , max(a1)+0.1 ) )
points( a2 , b2 , pch=1 )
for ( i in 1:N_cafes ) lines( c(a1[i],a2[i]) , c(b1[i],b2[i]) )

## R code 14.15
# compute posterior mean bivariate Gaussian
Mu_est <- c( mean(post$a) , mean(post$b) )
rho_est <- mean( post$Rho[,1,2] )
sa_est <- mean( post$sigma_cafe[,1] )
sb_est <- mean( post$sigma_cafe[,2] )
cov_ab <- sa_est*sb_est*rho_est
Sigma_est <- matrix( c(sa_est^2,cov_ab,cov_ab,sb_est^2) , ncol=2 )

# draw contours
library(ellipse)
for ( l in c(0.1,0.3,0.5,0.8,0.99) )
    lines(ellipse(Sigma_est,centre=Mu_est,level=l),
        col=col.alpha("black",0.2))

## R code 14.16
# convert varying effects to waiting times
wait_morning_1 <- (a1)
wait_afternoon_1 <- (a1 + b1)
wait_morning_2 <- (a2)
wait_afternoon_2 <- (a2 + b2)

# plot both and connect with lines
plot( wait_morning_1 , wait_afternoon_1 , xlab="morning wait" ,
    ylab="afternoon wait" , pch=16 , col=rangi2 ,
    ylim=c( min(wait_afternoon_1)-0.1 , max(wait_afternoon_1)+0.1 ) ,
    xlim=c( min(wait_morning_1)-0.1 , max(wait_morning_1)+0.1 ) )
points( wait_morning_2 , wait_afternoon_2 , pch=1 )
for ( i in 1:N_cafes )
    lines( c(wait_morning_1[i],wait_morning_2[i]) ,
    c(wait_afternoon_1[i],wait_afternoon_2[i]) )
abline( a=0 , b=1 , lty=2 )

## R code 14.17
# now shrinkage distribution by simulation
v <- mvrnorm( 1e4 , Mu_est , Sigma_est )
v[,2] <- v[,1] + v[,2] # calculate afternoon wait
Sigma_est2 <- cov(v)
Mu_est2 <- Mu_est
Mu_est2[2] <- Mu_est[1]+Mu_est[2]

# draw contours
library(ellipse)
for ( l in c(0.1,0.3,0.5,0.8,0.99) )
    lines(ellipse(Sigma_est2,centre=Mu_est2,level=l),
        col=col.alpha("black",0.5))

## R code 14.18
library(rethinking)
data(chimpanzees)
d <- chimpanzees
d$block_id <- d$block
d$treatment <- 1L + d$prosoc_left + 2L*d$condition

dat <- list(
    L = d$pulled_left,
    tid = d$treatment,
    actor = d$actor,
    block_id = as.integer(d$block_id) )

m14.2 <- ulam(
    alist(
        L ~ binomial(1,p),
        logit(p) <- g[tid] + alpha[actor,tid] + beta[block_id,tid],

        # adaptive priors
        vector[4]:alpha[actor] ~ multi_normal(0,Rho_actor,sigma_actor),
        vector[4]:beta[block_id] ~ multi_normal(0,Rho_block,sigma_block),

        # fixed priors
        g[tid] ~ dnorm(0,1),
        sigma_actor ~ dexp(1),
        Rho_actor ~ dlkjcorr(4),
        sigma_block ~ dexp(1),
        Rho_block ~ dlkjcorr(4)
    ) , data=dat , chains=4 , cores=4 )

## R code 14.19
m14.3 <- ulam(
    alist(
        L ~ binomial(1,p),
        logit(p) <- g[tid] + alpha[actor,tid] + beta[block_id,tid],

        # adaptive priors - non-centered
        transpars> matrix[actor,4]:alpha <-
                compose_noncentered( sigma_actor , L_Rho_actor , z_actor ),
        transpars> matrix[block_id,4]:beta <-
                compose_noncentered( sigma_block , L_Rho_block , z_block ),
        matrix[4,actor]:z_actor ~ normal( 0 , 1 ),
        matrix[4,block_id]:z_block ~ normal( 0 , 1 ),

        # fixed priors
        g[tid] ~ normal(0,1),
        vector[4]:sigma_actor ~ dexp(1),
        cholesky_factor_corr[4]:L_Rho_actor ~ lkj_corr_cholesky( 2 ),
        vector[4]:sigma_block ~ dexp(1),
        cholesky_factor_corr[4]:L_Rho_block ~ lkj_corr_cholesky( 2 ),

        # compute ordinary correlation matrixes from Cholesky factors
        gq> matrix[4,4]:Rho_actor <<- multiply_lower_tri_self_transpose(L_Rho_actor),
        gq> matrix[4,4]:Rho_block <<- multiply_lower_tri_self_transpose(L_Rho_block)
    ) , data=dat , chains=4 , cores=4 , log_lik=TRUE )

## R code 14.20
# extract n_eff values for each model
neff_nc <- precis(m14.3,3,pars=c("alpha","beta"))$n_eff
neff_c <- precis(m14.2,3,pars=c("alpha","beta"))$n_eff
plot( neff_c , neff_nc , xlab="centered (default)" ,
    ylab="non-centered (cholesky)" , lwd=1.5 )
abline(a=0,b=1,lty=2)

## R code 14.21
precis( m14.3 , depth=2 , pars=c("sigma_actor","sigma_block") )

## R code 14.22
# compute mean for each actor in each treatment
pl <- by( d$pulled_left , list( d$actor , d$treatment ) , mean )

# generate posterior predictions using link
datp <- list(
    actor=rep(1:7,each=4) ,
    tid=rep(1:4,times=7) ,
    block_id=rep(5,times=4*7) )
p_post <- link( m14.3 , data=datp )
p_mu <- apply( p_post , 2 , mean )
p_ci <- apply( p_post , 2 , PI )

# set up plot
plot( NULL , xlim=c(1,28) , ylim=c(0,1) , xlab="" ,
    ylab="proportion left lever" , xaxt="n" , yaxt="n" )
axis( 2 , at=c(0,0.5,1) , labels=c(0,0.5,1) )
abline( h=0.5 , lty=2 )
for ( j in 1:7 ) abline( v=(j-1)*4+4.5 , lwd=0.5 )
for ( j in 1:7 ) text( (j-1)*4+2.5 , 1.1 , concat("actor ",j) , xpd=TRUE )

xo <- 0.1 # offset distance to stagger raw data and predictions
# raw data
for ( j in (1:7)[-2] ) {
    lines( (j-1)*4+c(1,3)-xo , pl[j,c(1,3)] , lwd=2 , col=rangi2 )
    lines( (j-1)*4+c(2,4)-x0 , pl[j,c(2,4)] , lwd=2 , col=rangi2 )
}
points( 1:28-xo , t(pl) , pch=16 , col="white" , cex=1.7 )
points( 1:28-xo , t(pl) , pch=c(1,1,16,16) , col=rangi2 , lwd=2 )

yoff <- 0.175
text( 1-xo , pl[1,1]-yoff , "R/N" , pos=1 , cex=0.8 )
text( 2-xo , pl[1,2]+yoff , "L/N" , pos=3 , cex=0.8 )
text( 3-xo , pl[1,3]-yoff , "R/P" , pos=1 , cex=0.8 )
text( 4-xo , pl[1,4]+yoff , "L/P" , pos=3 , cex=0.8 )

# posterior predictions
for ( j in (1:7)[-2] ) {
    lines( (j-1)*4+c(1,3)+xo , p_mu[(j-1)*4+c(1,3)] , lwd=2 )
    lines( (j-1)*4+c(2,4)+xo , p_mu[(j-1)*4+c(2,4)] , lwd=2 )
}
for ( i in 1:28 ) lines( c(i,i)+xo , p_ci[,i] , lwd=1 )
points( 1:28+xo , p_mu , pch=16 , col="white" , cex=1.3 )
points( 1:28+xo , p_mu , pch=c(1,1,16,16) )

## R code 14.23
set.seed(73)
N <- 500
U_sim <- rnorm( N )
Q_sim <- sample( 1:4 , size=N , replace=TRUE )
E_sim <- rnorm( N , U_sim + Q_sim )
W_sim <- rnorm( N , U_sim + 0*E_sim )
dat_sim <- list(
    W=standardize(W_sim) ,
    E=standardize(E_sim) ,
    Q=standardize(Q_sim) )

## R code 14.24
m14.4 <- ulam(
    alist(
        W ~ dnorm( mu , sigma ),
        mu <- aW + bEW*E,
        aW ~ dnorm( 0 , 0.2 ),
        bEW ~ dnorm( 0 , 0.5 ),
        sigma ~ dexp( 1 )
    ) , data=dat_sim , chains=4 , cores=4 )
precis( m14.4 )

## R code 14.25
m14.5 <- ulam(
    alist(
        c(W,E) ~ multi_normal( c(muW,muE) , Rho , Sigma ),
        muW <- aW + bEW*E,
        muE <- aE + bQE*Q,
        c(aW,aE) ~ normal( 0 , 0.2 ),
        c(bEW,bQE) ~ normal( 0 , 0.5 ),
        Rho ~ lkj_corr( 2 ),
        Sigma ~ exponential( 1 )
    ), data=dat_sim , chains=4 , cores=4 )
precis( m14.5 , depth=3 )

## R code 14.26
m14.4x <- ulam( m14.4 , data=dat_sim , chains=4 , cores=4 )
m14.5x <- ulam( m14.5 , data=dat_sim , chains=4 , cores=4 )

## R code 14.27
set.seed(73)
N <- 500
U_sim <- rnorm( N )
Q_sim <- sample( 1:4 , size=N , replace=TRUE )
E_sim <- rnorm( N , U_sim + Q_sim )
W_sim <- rnorm( N , -U_sim + 0.2*E_sim )
dat_sim <- list(
    W=standardize(W_sim) ,
    E=standardize(E_sim) ,
    Q=standardize(Q_sim) )

## R code 14.28
library(dagitty)
dagIV <- dagitty( "dag{
    E -> W
    E <- U -> W
    Q -> E
}")
instrumentalVariables( dagIV , exposure="E" , outcome="W" )

## R code 14.29
library(rethinking)
data(KosterLeckie)

## R code 14.30
kl_data <- list(
    N = nrow(kl_dyads),
    N_households = max(kl_dyads$hidB),
    did = kl_dyads$did,
    hidA = kl_dyads$hidA,
    hidB = kl_dyads$hidB,
    giftsAB = kl_dyads$giftsAB,
    giftsBA = kl_dyads$giftsBA
)

m14.4 <- ulam(
    alist(
        giftsAB ~ poisson( lambdaAB ),
        giftsBA ~ poisson( lambdaBA ),
        log(lambdaAB) <- a + gr[hidA,1] + gr[hidB,2] + d[did,1] ,
        log(lambdaBA) <- a + gr[hidB,1] + gr[hidA,2] + d[did,2] ,
        a ~ normal(0,1),

       ## gr matrix of varying effects
        vector[2]:gr[N_households] ~ multi_normal(0,Rho_gr,sigma_gr),
        Rho_gr ~ lkj_corr(4),
        sigma_gr ~ exponential(1),

       ## dyad effects
        transpars> matrix[N,2]:d <-
                compose_noncentered( rep_vector(sigma_d,2) , L_Rho_d , z ),
        matrix[2,N]:z ~ normal( 0 , 1 ),
        cholesky_factor_corr[2]:L_Rho_d ~ lkj_corr_cholesky( 8 ),
        sigma_d ~ exponential(1),

       ## compute correlation matrix for dyads
        gq> matrix[2,2]:Rho_d <<- Chol_to_Corr( L_Rho_d )
    ), data=kl_data , chains=4 , cores=4 , iter=2000 )

## R code 14.31
precis( m14.4 , depth=3 , pars=c("Rho_gr","sigma_gr") )

## R code 14.32
post <- extract.samples( m14.4 )
g <- sapply( 1:25 , function(i) post$a + post$gr[,i,1] )
r <- sapply( 1:25 , function(i) post$a + post$gr[,i,2] )
Eg_mu <- apply( exp(g) , 2 , mean )
Er_mu <- apply( exp(r) , 2 , mean )

## R code 14.33
plot( NULL , xlim=c(0,8.6) , ylim=c(0,8.6) , xlab="generalized giving" ,
    ylab="generalized receiving" , lwd=1.5 )
abline(a=0,b=1,lty=2)

# ellipses
library(ellipse)
for ( i in 1:25 ) {
    Sigma <- cov( cbind( g[,i] , r[,i] ) )
    Mu <- c( mean(g[,i]) , mean(r[,i]) )
    for ( l in c(0.5) ) {
        el <- ellipse( Sigma , centre=Mu , level=l )
        lines( exp(el) , col=col.alpha("black",0.5) )
    }
}
# household means
points( Eg_mu , Er_mu , pch=21 , bg="white" , lwd=1.5 )

## R code 14.34
precis( m14.4 , depth=3 , pars=c("Rho_d","sigma_d") )

## R code 14.35
dy1 <- apply( post$d[,,1] , 2 , mean )
dy2 <- apply( post$d[,,2] , 2 , mean )
plot( dy1 , dy2 )

## R code 14.36
# load the distance matrix
library(rethinking)
data(islandsDistMatrix)

# display (measured in thousands of km)
Dmat <- islandsDistMatrix
colnames(Dmat) <- c("Ml","Ti","SC","Ya","Fi","Tr","Ch","Mn","To","Ha")
round(Dmat,1)

## R code 14.37
# linear
curve( exp(-1*x) , from=0 , to=4 , lty=2 ,
    xlab="distance" , ylab="correlation" )

# squared
curve( exp(-1*x^2) , add=TRUE )

## R code 14.38
data(Kline2) # load the ordinary data, now with coordinates
d <- Kline2
d$society <- 1:10 # index observations

dat_list <- list(
    T = d$total_tools,
    P = d$population,
    society = d$society,
    Dmat=islandsDistMatrix )

m14.7 <- ulam(
    alist(
        T ~ dpois(lambda),
        lambda <- (a*P^b/g)*exp(k[society]),
        vector[10]:k ~ multi_normal( 0 , SIGMA ),
        matrix[10,10]:SIGMA <- cov_GPL2( Dmat , etasq , rhosq , 0.01 ),
        c(a,b,g) ~ dexp( 1 ),
        etasq ~ dexp( 2 ),
        rhosq ~ dexp( 0.5 )
    ), data=dat_list , chains=4 , cores=4 , iter=2000 )

## R code 14.39
precis( m14.7 , depth=3 )

## R code 14.40
post <- extract.samples(m14.7)

# plot the posterior median covariance function
plot( NULL , xlab="distance (thousand km)" , ylab="covariance" ,
    xlim=c(0,10) , ylim=c(0,2) )

# compute posterior mean covariance
x_seq <- seq( from=0 , to=10 , length.out=100 )
pmcov <- sapply( x_seq , function(x) post$etasq*exp(-post$rhosq*x^2) )
pmcov_mu <- apply( pmcov , 2 , mean )
lines( x_seq , pmcov_mu , lwd=2 )

# plot 60 functions sampled from posterior
for ( i in 1:50 )
    curve( post$etasq[i]*exp(-post$rhosq[i]*x^2) , add=TRUE ,
        col=col.alpha("black",0.3) )

## R code 14.41
# compute posterior median covariance among societies
K <- matrix(0,nrow=10,ncol=10)
for ( i in 1:10 )
    for ( j in 1:10 )
        K[i,j] <- median(post$etasq) *
                  exp( -median(post$rhosq) * islandsDistMatrix[i,j]^2 )
diag(K) <- median(post$etasq) + 0.01

## R code 14.42
# convert to correlation matrix
Rho <- round( cov2cor(K) , 2 )
# add row/col names for convenience
colnames(Rho) <- c("Ml","Ti","SC","Ya","Fi","Tr","Ch","Mn","To","Ha")
rownames(Rho) <- colnames(Rho)
Rho

## R code 14.43
# scale point size to logpop
psize <- d$logpop / max(d$logpop)
psize <- exp(psize*1.5)-2

# plot raw data and labels
plot( d$lon2 , d$lat , xlab="longitude" , ylab="latitude" ,
    col=rangi2 , cex=psize , pch=16 , xlim=c(-50,30) )
labels <- as.character(d$culture)
text( d$lon2 , d$lat , labels=labels , cex=0.7 , pos=c(2,4,3,3,4,1,3,2,4,2) )

# overlay lines shaded by Rho
for( i in 1:10 )
    for ( j in 1:10 )
        if ( i < j )
            lines( c( d$lon2[i],d$lon2[j] ) , c( d$lat[i],d$lat[j] ) ,
                lwd=2 , col=col.alpha("black",Rho[i,j]^2) )

## R code 14.44
# compute posterior median relationship, ignoring distance
logpop.seq <- seq( from=6 , to=14 , length.out=30 )
lambda <- sapply( logpop.seq , function(lp) exp( post$a + post$bp*lp ) )
lambda.median <- apply( lambda , 2 , median )
lambda.PI80 <- apply( lambda , 2 , PI , prob=0.8 )

# plot raw data and labels
plot( d$logpop , d$total_tools , col=rangi2 , cex=psize , pch=16 ,
    xlab="log population" , ylab="total tools" )
text( d$logpop , d$total_tools , labels=labels , cex=0.7 ,
    pos=c(4,3,4,2,2,1,4,4,4,2) )

# display posterior predictions
lines( logpop.seq , lambda.median , lty=2 )
lines( logpop.seq , lambda.PI80[1,] , lty=2 )
lines( logpop.seq , lambda.PI80[2,] , lty=2 )

# overlay correlations
for( i in 1:10 )
    for ( j in 1:10 )
        if ( i < j )
            lines( c( d$logpop[i],d$logpop[j] ) ,
                   c( d$total_tools[i],d$total_tools[j] ) ,
                   lwd=2 , col=col.alpha("black",Rho[i,j]^2) )

## R code 14.45
m14.7nc <- ulam(
    alist(
        T ~ dpois(lambda),
        lambda <- (a*P^b/g)*exp(k[society]),

        # non-centered Gaussian Process prior
        transpars> vector[10]: k <<- L_SIGMA * z,
        vector[10]: z ~ normal( 0 , 1 ),
        transpars> matrix[10,10]: L_SIGMA <<- cholesky_decompose( SIGMA ),
        transpars> matrix[10,10]: SIGMA <- cov_GPL2( Dmat , etasq , rhosq , 0.01 ),

        c(a,b,g) ~ dexp( 1 ),
        etasq ~ dexp( 2 ),
        rhosq ~ dexp( 0.5 )
    ), data=dat_list , chains=4 , cores=4 , iter=2000 )

## R code 14.46
library(rethinking)
data(Primates301)
data(Primates301_nex)

# plot it using ape package - install.packages('ape') if needed
library(ape)
plot( ladderize(Primates301_nex) , type="fan" , font=1 , no.margin=TRUE ,
    label.offset=1 , cex=0.5 )

## R code 14.47
d <- Primates301
d$name <- as.character(d$name)
dstan <- d[ complete.cases( d$group_size , d$body , d$brain ) , ]
spp_obs <- dstan$name

## R code 14.48
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) )
)

m14.8 <- ulam(
    alist(
        B ~ multi_normal( mu , SIGMA ),
        mu <- a + bM*M + bG*G,
        matrix[N_spp,N_spp]: SIGMA <- Imat * sigma_sq,
        a ~ normal( 0 , 1 ),
        c(bM,bG) ~ normal( 0 , 0.5 ),
        sigma_sq ~ exponential( 1 )
    ), data=dat_list , chains=4 , cores=4 )
precis( m14.8 )

## R code 14.49
library(ape)
tree_trimmed <- keep.tip( Primates301_nex, spp_obs )
Rbm <- corBrownian( phy=tree_trimmed )
V <- vcv(Rbm)
Dmat <- cophenetic( tree_trimmed )
plot( Dmat , V , xlab="phylogenetic distance" , ylab="covariance" )

## R code 14.50
# put species in right order
dat_list$V <- V[ spp_obs , spp_obs ]
# convert to correlation matrix
dat_list$R <- dat_list$V / max(V)

# Brownian motion model
m14.9 <- ulam(
    alist(
        B ~ multi_normal( mu , SIGMA ),
        mu <- a + bM*M + bG*G,
        matrix[N_spp,N_spp]: SIGMA <- R * sigma_sq,
        a ~ normal( 0 , 1 ),
        c(bM,bG) ~ normal( 0 , 0.5 ),
        sigma_sq ~ exponential( 1 )
    ), data=dat_list , chains=4 , cores=4 )
precis( m14.9 )

## R code 14.51
# add scaled and reordered distance matrix
dat_list$Dmat <- Dmat[ spp_obs , spp_obs ] / max(Dmat)

m14.10 <- ulam(
    alist(
        B ~ multi_normal( mu , SIGMA ),
        mu <- a + bM*M + bG*G,
        matrix[N_spp,N_spp]: SIGMA <- cov_GPL1( Dmat , etasq , rhosq , 0.01 ),
        a ~ normal(0,1),
        c(bM,bG) ~ normal(0,0.5),
        etasq ~ half_normal(1,0.25),
        rhosq ~ half_normal(3,0.25)
    ), data=dat_list , chains=4 , cores=4 )
precis( m14.10 )

## R code 14.52
post <- extract.samples(m14.10)
plot( NULL , xlim=c(0,max(dat_list$Dmat)) , ylim=c(0,1.5) ,
    xlab="phylogenetic distance" , ylab="covariance" )

# posterior
for ( i in 1:30 )
    curve( post$etasq[i]*exp(-post$rhosq[i]*x) , add=TRUE , col=rangi2 )

# prior mean and 89% interval
eta <- abs(rnorm(1e3,1,0.25))
rho <- abs(rnorm(1e3,3,0.25))
d_seq <- seq(from=0,to=1,length.out=50)
K <- sapply( d_seq , function(x) eta*exp(-rho*x) )
lines( d_seq , colMeans(K) , lwd=2 )
shade( apply(K,2,PI) , d_seq )
text( 0.5 , 0.5 , "prior" )
text( 0.2 , 0.1 , "posterior" , col=rangi2 )

## R code 14.53
S <- matrix( c( sa^2 , sa*sb*rho , sa*sb*rho , sb^2 ) , nrow=2 )

## R code 15.1
# simulate a pancake and return randomly ordered sides
sim_pancake <- function() {
    pancake <- sample(1:3,1)
    sides <- matrix(c(1,1,1,0,0,0),2,3)[,pancake]
    sample(sides)
}

# sim 10,000 pancakes
pancakes <- replicate( 1e4 , sim_pancake() )
up <- pancakes[1,]
down <- pancakes[2,]

# compute proportion 1/1 (BB) out of all 1/1 and 1/0
num_11_10 <- sum( up==1 )
num_11 <- sum( up==1 & down==1 )
num_11/num_11_10

## R code 15.2
library(rethinking)
data(WaffleDivorce)
d <- WaffleDivorce

# points
plot( d$Divorce ~ d$MedianAgeMarriage , ylim=c(4,15) ,
    xlab="Median age marriage" , ylab="Divorce rate" )

# standard errors
for ( i in 1:nrow(d) ) {
    ci <- d$Divorce[i] + c(-1,1)*d$Divorce.SE[i]
    x <- d$MedianAgeMarriage[i]
    lines( c(x,x) , ci )
}

## R code 15.3
dlist <- list(
    D_obs = standardize( d$Divorce ),
    D_sd = d$Divorce.SE / sd( d$Divorce ),
    M = standardize( d$Marriage ),
    A = standardize( d$MedianAgeMarriage ),
    N = nrow(d)
)

m15.1 <- ulam(
    alist(
        D_obs ~ dnorm( D_true , D_sd ),
        vector[N]:D_true ~ dnorm( mu , sigma ),
        mu <- a + bA*A + bM*M,
        a ~ dnorm(0,0.2),
        bA ~ dnorm(0,0.5),
        bM ~ dnorm(0,0.5),
        sigma ~ dexp(1)
    ) , data=dlist , chains=4 , cores=4 )

## R code 15.4
precis( m15.1 , depth=2 )

## R code 15.5
dlist <- list(
    D_obs = standardize( d$Divorce ),
    D_sd = d$Divorce.SE / sd( d$Divorce ),
    M_obs = standardize( d$Marriage ),
    M_sd = d$Marriage.SE / sd( d$Marriage ),
    A = standardize( d$MedianAgeMarriage ),
    N = nrow(d)
)

m15.2 <- ulam(
    alist(
        D_obs ~ dnorm( D_est , D_sd ),
        vector[N]:D_est ~ dnorm( mu , sigma ),
        mu <- a + bA*A + bM*M_est[i],
        M_obs ~ dnorm( M_est , M_sd ),
        vector[N]:M_est ~ dnorm( 0 , 1 ),
        a ~ dnorm(0,0.2),
        bA ~ dnorm(0,0.5),
        bM ~ dnorm(0,0.5),
        sigma ~ dexp( 1 )
    ) , data=dlist , chains=4 , cores=4 )

## R code 15.6
post <- extract.samples( m15.2 )
D_est <- apply( post$D_est , 2 , mean )
M_est <- apply( post$M_est , 2 , mean )
plot( dlist$M_obs , dlist$D_obs , pch=16 , col=rangi2 ,
    xlab="marriage rate (std)" , ylab="divorce rate (std)" )
points( M_est , D_est )
for ( i in 1:nrow(d) )
    lines( c( dlist$M_obs[i] , M_est[i] ) , c( dlist$D_obs[i] , D_est[i] ) )

## R code 15.7
N <- 500
A <- rnorm(N)
M <- rnorm(N,-A)
D <- rnorm(N,A)
A_obs <- rnorm(N,A)

## R code 15.8
N <- 100
S <- rnorm( N )
H <- rbinom( N , size=10 , inv_logit(S) )

## R code 15.9
D <- rbern( N ) # dogs completely random
Hm <- H
Hm[D==1] <- NA

## R code 15.10
D <- ifelse( S > 0 , 1 , 0 )
Hm <- H
Hm[D==1] <- NA

## R code 15.11
set.seed(501)
N <- 1000
X <- rnorm(N)
S <- rnorm(N)
H <- rbinom( N , size=10 , inv_logit( 2 + S - 2*X ) )
D <- ifelse( X > 1 , 1 , 0 )
Hm <- H
Hm[D==1] <- NA

## R code 15.12
dat_list <- list(
    H = H,
    S = S )

m15.3 <- ulam(
    alist(
        H ~ binomial( 10 , p ),
        logit(p) <- a + bS*S,
        a ~ normal( 0 , 1 ),
        bS ~ normal( 0 , 0.5 )
    ), data=dat_list , chains=4 )
precis( m15.3 )

## R code 15.13
dat_list0 <- list(
    H = H[D==0],
    S = S[D==0] )

m15.4 <- ulam(
    alist(
        H ~ binomial( 10 , p ),
        logit(p) <- a + bS*S,
        a ~ normal( 0 , 1 ),
        bS ~ normal( 0 , 0.5 )
    ), data=dat_list0 , chains=4 )
precis( m15.4 )

## R code 15.14
D <- ifelse( abs(X) < 1 , 1 , 0 )

## R code 15.15
N <- 100
S <- rnorm(N)
H <- rbinom( N , size=10 , inv_logit(S) )
D <- ifelse( H < 5 , 1 , 0 )
Hm <- H
Hm[D==1] <- NA

## R code 15.16
library(rethinking)
data(milk)
d <- milk
d$neocortex.prop <- d$neocortex.perc / 100
d$logmass <- log(d$mass)

## R code 15.17
dat_list <- list(
    K = standardize( d$kcal.per.g ),
    B = standardize( d$neocortex.prop ),
    M = standardize( d$logmass )
)

m15.3 <- ulam(
    alist(
        K ~ dnorm( mu , sigma ),
        mu <- a + bB*B + bM*M,
        B ~ dnorm( nu , sigma_B ),
        c(a,nu) ~ dnorm( 0 , 0.5 ),
        c(bB,bM) ~ dnorm( 0, 0.5 ),
        sigma_B ~ dexp( 1 ),
        sigma ~ dexp( 1 )
    ) , data=dat_list , chains=4 , cores=4 )

## R code 15.18
precis( m15.3 , depth=2 )

## R code 15.19
obs_idx <- which( !is.na(d$neocortex.prop) )
dat_list_obs <- list(
    K = dat_list$K[obs_idx],
    B = dat_list$B[obs_idx],
    M = dat_list$M[obs_idx]
)
m15.4 <- ulam(
    alist(
        K ~ dnorm( mu , sigma ),
        mu <- a + bB*B + bM*M,
        B ~ dnorm( nu , sigma_B ),
        c(a,nu) ~ dnorm( 0 , 0.5 ),
        c(bB,bM) ~ dnorm( 0, 0.5 ),
        sigma_B ~ dexp( 1 ),
        sigma ~ dexp( 1 )
    ) , data=dat_list_obs , chains=4 , cores=4 )
precis( m15.4 )

## R code 15.20
plot( coeftab(m15.3,m15.4) , pars=c("bB","bM") )

## R code 15.21
post <- extract.samples( m15.3 )
B_impute_mu <- apply( post$B_impute , 2 , mean )
B_impute_ci <- apply( post$B_impute , 2 , PI )

# B vs K
plot( dat_list$B , dat_list$K , pch=16 , col=rangi2 ,
    xlab="neocortex percent (std)" , ylab="kcal milk (std)" )
miss_idx <- which( is.na(dat_list$B) )
Ki <- dat_list$K[miss_idx]
points( B_impute_mu , Ki )
for ( i in 1:12 ) lines( B_impute_ci[,i] , rep(Ki[i],2) )

# M vs B
plot( dat_list$M , dat_list$B , pch=16 , col=rangi2 ,
    ylab="neocortex percent (std)" , xlab="log body mass (std)" )
Mi <- dat_list$M[miss_idx]
points( Mi , B_impute_mu )
for ( i in 1:12 ) lines( rep(Mi[i],2) , B_impute_ci[,i] )

## R code 15.22
m15.5 <- ulam(
    alist(
       # K as function of B and M
        K ~ dnorm( mu , sigma ),
        mu <- a + bB*B_merge + bM*M,

       # M and B correlation
        MB ~ multi_normal( c(muM,muB) , Rho_BM , Sigma_BM ),
        matrix[29,2]:MB <<- append_col( M , B_merge ),

       # define B_merge as mix of observed and imputed values
        vector[29]:B_merge <- merge_missing( B , B_impute ),

       # priors
        c(a,muB,muM) ~ dnorm( 0 , 0.5 ),
        c(bB,bM) ~ dnorm( 0, 0.5 ),
        sigma ~ dexp( 1 ),
        Rho_BM ~ lkj_corr(2),
        Sigma_BM ~ exponential(1)
    ) , data=dat_list , chains=4 , cores=4 )
precis( m15.5 , depth=3 , pars=c("bM","bB","Rho_BM" ) )

## R code 15.23
B_missidx <- which( is.na( dat_list$B ) )

## R code 15.24
data(Moralizing_gods)
str(Moralizing_gods)

## R code 15.25
table( Moralizing_gods$moralizing_gods , useNA="always" )

## R code 15.26
symbol <- ifelse( Moralizing_gods$moralizing_gods==1 , 16 , 1 )
symbol <- ifelse( is.na(Moralizing_gods$moralizing_gods) , 4 , symbol )
color <- ifelse( is.na(Moralizing_gods$moralizing_gods) , "black" , rangi2 )
plot( Moralizing_gods$year , Moralizing_gods$population , pch=symbol ,
    col=color , xlab="Time (year)" , ylab="Population size" , lwd=1.5 )

## R code 15.27
dmg <- Moralizing_gods
table( gods=dmg$moralizing_gods , literacy=dmg$writing , useNA="always" )

## R code 15.28
dmg <- Moralizing_gods
haw <- which( dmg$polity=="Big Island Hawaii" )
t( dmg[ haw , c("year","population","writing","moralizing_gods") ] )

## R code 15.29
set.seed(9)
N_houses <- 100L
alpha <- 5
beta <- (-3)
k <- 0.5
r <- 0.2
cat <- rbern( N_houses , k )
notes <- rpois( N_houses , alpha + beta*cat )
R_C <- rbern( N_houses , r )
cat_obs <- cat
cat_obs[R_C==1] <- (-9L)

## R code 15.30
dat <- list(
    notes = notes,
    cat = cat_obs,
    RC = R_C,
    N = as.integer(N_houses) )

m15.6 <- ulam(
    alist(
        # singing bird model
        ## cat known present/absent:
        notes|RC==0 ~ poisson( lambda ),
        log(lambda) <- a + b*cat,
        ## cat NA:
        notes|RC==1 ~ custom( log_sum_exp(
                log(k) + poisson_lpmf( notes | exp(a + b) ),
                log(1-k) + poisson_lpmf( notes | exp(a) )
            ) ),

        # priors
        a ~ normal(0,1),
        b ~ normal(0,0.5),

        # sneaking cat model
        cat|RC==0 ~ bernoulli(k),
        k ~ beta(2,2)
    ), data=dat , chains=4 , cores=4 )

## R code 15.31
m15.7 <- ulam(
    alist(
        # singing bird model
        notes|RC==0 ~ poisson( lambda ),
        notes|RC==1 ~ custom( log_sum_exp(
                log(k) + poisson_lpmf( notes | exp(a + b) ),
                log(1-k) + poisson_lpmf( notes | exp(a) )
            ) ),
        log(lambda) <- a + b*cat,
        a ~ normal(0,1),
        b ~ normal(0,0.5),

        # sneaking cat model
        cat|RC==0 ~ bernoulli(k),
        k ~ beta(2,2),

        # imputed values
        gq> vector[N]:PrC1 <- exp(lpC1)/(exp(lpC1)+exp(lpC0)),
        gq> vector[N]:lpC1 <- log(k) + poisson_lpmf( notes[i] | exp(a+b) ),
        gq> vector[N]:lpC0 <- log(1-k) + poisson_lpmf( notes[i] | exp(a) )
    ), data=dat , chains=4 , cores=4 )

## R code 15.32
set.seed(100)
x <- c( rnorm(10) , NA )
y <- c( rnorm(10,x) , 100 )
d <- list(x=x,y=y)

## R code 16.1
library(rethinking)
data(Howell1)
d <- Howell1

# scale observed variables
d$w <- d$weight / mean(d$weight)
d$h <- d$height / mean(d$height)

## R code 16.2
m16.1 <- ulam(
    alist(
        w ~ dlnorm( mu , sigma ),
        exp(mu) <- 3.141593 * k * p^2 * h^3,
        p ~ beta( 2 , 18 ),
        k ~ exponential( 0.5 ),
        sigma ~ exponential( 1 )
    ), data=d , chains=4 , cores=4 )

## R code 16.3
h_seq <- seq( from=0 , to=max(d$h) , length.out=30 )
w_sim <- sim( m16.1 , data=list(h=h_seq) )
mu_mean <- apply( w_sim , 2 , mean )
w_CI <- apply( w_sim , 2 , PI )
plot( d$h , d$w , xlim=c(0,max(d$h)) , ylim=c(0,max(d$w)) , col=rangi2 ,
    lwd=2 , xlab="height (scaled)" , ylab="weight (scaled)" )
lines( h_seq , mu_mean )
shade( w_CI , h_seq )

## R code 16.4
library(rethinking)
data(Boxes)
precis(Boxes)

## R code 16.5
table( Boxes$y ) / length( Boxes$y )

## R code 16.6
set.seed(7)
N <- 30 # number of children

# half are random
# sample from 1,2,3 at random for each
y1 <- sample( 1:3 , size=N/2 , replace=TRUE )

# half follow majority
y2 <- rep( 2 , N/2 )

# combine and shuffle y1 and y2
y <- sample( c(y1,y2) )

# count the 2s
sum(y==2)/N

## R code 16.7
data(Boxes_model)
cat(Boxes_model)

## R code 16.8
# prep data
dat_list <- list(
    N = nrow(Boxes),
    y = Boxes$y,
    majority_first = Boxes$majority_first )

# run the sampler
m16.2 <- stan( model_code=Boxes_model , data=dat_list , chains=3 , cores=3 )

# show marginal posterior for p
p_labels <- c("1 Majority","2 Minority","3 Maverick","4 Random","5 Follow First")
plot( precis(m16.2,2) , labels=p_labels )

## R code 16.9
library(rethinking)
data(Panda_nuts)

## R code 16.10
N <- 1e4
phi <- rlnorm( N , log(1) , 0.1 )
k <- rlnorm( N , log(2), 0.25 )
theta <- rlnorm( N , log(5) , 0.25 )

# relative grow curve
plot( NULL , xlim=c(0,1.5) , ylim=c(0,1) , xaxt="n" , xlab="age" ,
    ylab="body mass" )
at <- c(0,0.25,0.5,0.75,1,1.25,1.5)
axis( 1 , at=at , labels=round(at*max(Panda_nuts$age)) )
for ( i in 1:20 ) curve( (1-exp(-k[i]*x)) , add=TRUE , col=grau() , lwd=1.5 )

# implied rate of nut opening curve
plot( NULL , xlim=c(0,1.5) , ylim=c(0,1.2) , xaxt="n" , xlab="age" ,
    ylab="nuts per second" )
at <- c(0,0.25,0.5,0.75,1,1.25,1.5)
axis( 1 , at=at , labels=round(at*max(Panda_nuts$age)) )
for ( i in 1:20 ) curve( phi[i]*(1-exp(-k[i]*x))^theta[i] , add=TRUE ,
    col=grau() , lwd=1.5 )

## R code 16.11
dat_list <- list(
    n = as.integer( Panda_nuts$nuts_opened ),
    age = Panda_nuts$age / max(Panda_nuts$age),
    seconds = Panda_nuts$seconds )

m16.4 <- ulam(
    alist(
        n ~ poisson( lambda ),
        lambda <- seconds*phi*(1-exp(-k*age))^theta,
        phi ~ lognormal( log(1) , 0.1 ),
        k ~ lognormal( log(2) , 0.25 ),
        theta ~ lognormal( log(5) , 0.25 )
    ), data=dat_list , chains=4 )

## R code 16.12
post <- extract.samples(m16.4)
plot( NULL , xlim=c(0,1) , ylim=c(0,1.5) , xlab="age" ,
    ylab="nuts per second" , xaxt="n" )
at <- c(0,0.25,0.5,0.75,1,1.25,1.5)
axis( 1 , at=at , labels=round(at*max(Panda_nuts$age)) )

# raw data
pts <- dat_list$n / dat_list$seconds
point_size <- normalize( dat_list$seconds )
points( jitter(dat_list$age) , pts , col=rangi2 , lwd=2 , cex=point_size*3 )

# 30 posterior curves
for ( i in 1:30 ) with( post ,
    curve( phi[i]*(1-exp(-k[i]*x))^theta[i] , add=TRUE , col=grau() ) )

## R code 16.13
library(rethinking)
data(Lynx_Hare)
plot( 1:21 , Lynx_Hare[,3] , ylim=c(0,90) , xlab="year" ,
    ylab="thousands of pelts" , xaxt="n" , type="l" , lwd=1.5 )
at <- c(1,11,21)
axis( 1 , at=at , labels=Lynx_Hare$Year[at] )
lines( 1:21 , Lynx_Hare[,2] , lwd=1.5 , col=rangi2 )
points( 1:21 , Lynx_Hare[,3] , bg="black" , col="white" , pch=21 , cex=1.4 )
points( 1:21 , Lynx_Hare[,2] , bg=rangi2 , col="white" , pch=21 , cex=1.4 )
text( 17 , 80 , "Lepus" , pos=2 )
text( 19 , 50 , "Lynx" , pos=2 , col=rangi2 )

## R code 16.14
sim_lynx_hare <- function( n_steps , init , theta , dt=0.002 ) {
    L <- rep(NA,n_steps)
    H <- rep(NA,n_steps)
    L[1] <- init[1]
    H[1] <- init[2]
    for ( i in 2:n_steps ) {
        H[i] <- H[i-1] + dt*H[i-1]*( theta[1] - theta[2]*L[i-1] )
        L[i] <- L[i-1] + dt*L[i-1]*( theta[3]*H[i-1] - theta[4] )
    }
    return( cbind(L,H) )
}

## R code 16.15
theta <- c( 0.5 , 0.05 , 0.025 , 0.5 )
z <- sim_lynx_hare( 1e4 , as.numeric(Lynx_Hare[1,2:3]) , theta )

plot( z[,2] , type="l" , ylim=c(0,max(z[,2])) , lwd=2 , xaxt="n" ,
    ylab="number (thousands)" , xlab="" )
lines( z[,1] , col=rangi2 , lwd=2 )
mtext( "time" , 1 )

## R code 16.16
N <- 1e4
Ht <- 1e4
p <- rbeta(N,2,18)
h <- rbinom( N , size=Ht , prob=p )
h <- round( h/1000 , 2 )
dens( h , xlab="thousand of pelts" , lwd=2 )

## R code 16.17
data(Lynx_Hare_model)
cat(Lynx_Hare_model)

## R code 16.18
dat_list <- list(
    N = nrow(Lynx_Hare),
    pelts = Lynx_Hare[,2:3] )

m16.5 <- stan( model_code=Lynx_Hare_model , data=dat_list , chains=3 , cores=3 ,
    control=list( adapt_delta=0.95 ) )

## R code 16.19
post <- extract.samples(m16.5)
pelts <- dat_list$pelts
plot( 1:21 , pelts[,2] , pch=16 , ylim=c(0,120) , xlab="year" ,
    ylab="thousands of pelts" , xaxt="n" )
at <- c(1,11,21)
axis( 1 , at=at , labels=Lynx_Hare$Year[at] )
points( 1:21 , pelts[,1] , col=rangi2 , pch=16 )
# 21 time series from posterior
for ( s in 1:21 ) {
    lines( 1:21 , post$pelts_pred[s,,2] , col=col.alpha("black",0.2) , lwd=2 )
    lines( 1:21 , post$pelts_pred[s,,1] , col=col.alpha(rangi2,0.3) , lwd=2 )
}
# text labels
text( 17 , 90 , "Lepus" , pos=2 )
text( 19 , 50 , "Lynx" , pos=2 , col=rangi2 )

## R code 16.20
plot( NULL , pch=16 , xlim=c(1,21) , ylim=c(0,500) , xlab="year" ,
    ylab="thousands of animals" , xaxt="n" )
at <- c(1,11,21)
axis( 1 , at=at , labels=Lynx_Hare$Year[at] )
for ( s in 1:21 ) {
    lines( 1:21 , post$pop[s,,2] , col=col.alpha("black",0.2) , lwd=2 )
    lines( 1:21 , post$pop[s,,1] , col=col.alpha(rangi2,0.4) , lwd=2 )
}

## R code 16.21
data(Lynx_Hare)
dat_ar1 <- list(
    L = Lynx_Hare$Lynx[2:21],
    L_lag1 = Lynx_Hare$Lynx[1:20],
    H = Lynx_Hare$Hare[2:21],
    H_lag1 = Lynx_Hare$Hare[1:20] )