# Lab 5 - TSdD 2024

# BF


# Es. 1 (ipotesi semplici)

# th=th.0 vs th=th.1


th.0=0
th.1=0.8
n=10
sig=1

#dati=round(rnorm(n,th.0,sig),2)
dati=c(1.88,0.14,1.15,0.16,-0.39,0.47,-0.81,1.23,-0.46,1.09)

x.med=mean(dati)
x.med
sig.n=sig/sqrt(n)
sig.n

L.0=dnorm(th.0,x.med,sig.n)
L.1=dnorm(th.1,x.med,sig.n)
B.01=L.0/L.1
B.01

# Post prob
#P(H.0|z.n)=L.0*p.0/m(z.n)
#P(H.1|z.n)=L.1*p.1/m(z.n)
#m(z.n)=L.0*p.0+L.1*p.1
p.0=0.2
p.1=0.8
prior.odds=p.0/p.1
prior.odds
m=L.0*p.0+L.1*p.1
p.0.post=L.0*p.0/m
p.1.post=L.1*p.1/m
p.0.post
p.1.post
post.odds=p.0.post/p.1.post
post.odds

post.odds/prior.odds

# se invece 
# p.0=0.8
# p.1=0.2

# B.01 invariati
# post.odds cambiano



# Es. 2 (one-sided)
# theta <= th.0 vs theta>th.0
mu.0=th.0
n.0=5

sig.0=sig/sqrt(n.0)

p.0.prior=pnorm(th.0,mu.0,sig.0)
p.1.prior=1-p.0.prior

prior.odds=p.0.prior/p.1.prior
prior.odds

p.0.prior
p.1.prior
prior.odds


mu.p=(n.0*mu.0+n*x.med)/(n+n.0)
mu.p
sig.p=sig/sqrt(n+n.0)
sig.p

p.0.post=pnorm(th.0,mu.p,sig.p)
p.1.post=1-p.0.post
post.odds=p.0.post/p.1.post

p.0.post
p.1.post
post.odds

B.01=post.odds/prior.odds
B.01


# se cambio mu.0
mu.0=1
mu.0=-1



# ES. 3 (two-sided)

mu.0=1.5
th.0=1.5
n.0=5
x.med=2   
n=10
sig.2=1
sig=sqrt(sig.2)

BF.fun=function(y){
  Num=dnorm(x.med,mu.0,sig*sqrt(1/n+y))
  Den=dnorm(x.med,th.0,sig*sqrt(1/n))
  BF=Num/Den
  return(BF)
}

BF.fun(0)

curve(BF.fun(x),from=0, to=10)