ABC in Roma [R lab #1]

Here are the R codes of the first R lab written and ran by Serena Arima in supplement of my lectures. This is quite impressive and helpful to the students, as illustrated by the first example below (using the abc software).

### Example 1: Conjugate model (Normal-Inverse Gamma)
### y1,y2,...,yn ~N(mu,sigma2)
### mu|sigma2 ~ N(0,sigma2), sigma2 ~IG(1/2,1/2)

library(abc)

### Iris data: sepal width of Iris Setosa
data(iris3)
y=iris3[,2,1]

### We want to obtain the following quantities
### "par.sim" "post.mu" "post.sigma2" "stat.obs" "stat.sim"

## STAT.OBS: stat.obs are mean and variance (log scale) of the data
mean(y)
log(var(y))
stat.obs
stat.oss=c(mean(y),log(var(y)))

### PAR.SIM: par.sim simulated values from the prior distribution
n.sim=10000
gdl=1
invsigma.sim=rchisq(n.sim,df=gdl)
sigma.sim=gdl/invsigma.sim

m=3
mu.sim=c()
for(i in 1:length(sigma.sim)){
 mu.sim=c(mu.sim,rnorm(1,mean=m,sd=sqrt(((sigma.sim[i])))))}
prior.sim=data.frame(mu.sim,sigma.sim)

### STAT.SIM: for mu and sigma simulated from the prior,
### generate data from the model y ~ N(mu,sigma^2)
stat.simulated=matrix(NA,nrow=length(mu.sim),ncol=2)

for(i in 1:length(mu.sim)){
 y.new=rnorm(length(y),mu.sim[i],sqrt(sigma.sim[i]))
 stat.simulated[i,1]=mean(y.new)
 stat.simulated[i,2]=log(var(y.new))}

### Obtain posterior distribution using ABC
post.value=abc(target=stat.oss, param=prior.sim,
sumstat=data.frame(stat.simulated),tol=0.001,method="rejection")
summary(post.value)

posterior.values=post.value$unadj.values
mu.post=posterior.values[,1]
sigma.post=posterior.values[,2]

### True values, thanks to conjugancy
post.mean.mu=(length(y)/(length(y)+1))*mean(y)
post.a.sigma=length(y)/2
post.b.sigma=0.5+0.5*sum((y-mean(y))^2)

hist(mu.post,main="Posterior distribution of mu")
abline(v=post.mean.mu,col=2,lwd=2)

hist(sigma.post,main="Posterior distribution of sigma2")
abline(v=post.b.sigma/(post.a.sigma-1),col=2,lwd=2)

I am having a great time teaching this “ABC in Roma” course, in particular because of the level of interaction and exchange with the participants (after, if not during, the classes).