#1.)
data<-c(rep(0,4),rep(1,6),rep(2,6),rep(3,8),rep(4,3),6,6,10)
table(data)

verohodnost<-function(x){
verohodnost<-prod(dpois(data,x))}

#a) apriorni hustota - rovnomerna

apost<-function(x){
verohodnost(x)*1
}

K<-integrate(Vectorize(apost),0,Inf)$value

apost2<-function(x){
apost2<-apost(x)/K
return(apost2)}

#stale ale integral neni roven 1
KK=integrate(Vectorize(apost2),0,Inf)$value
KK

apost3<-function(x){
apost3<-apost2(x)/KK
return(apost3)}

#ted uz konecne ano
integrate(Vectorize(apost3),0,Inf)


ind<-seq(1,4, by=0.01) #graf vykreslime na mrizce od 1 do 4
plot(Vectorize(apost3)(ind)~ind, type="l",xlab=expression(lambda),
ylab=expression(paste(pi,"(",lambda,"|x)" )),main='Graf aposteriorni hustoty',
ylim=c(0,1.5))

curve(dgamma(x,shape=77,scale=1/30),add=TRUE,col=7) #skutecna aposteriorni hustota


#b) apriorni hustota - exponencialni

apost_E<-function(x){
verohodnost(x)*dexp(x,1/5)}

K2<-integrate(Vectorize(apost_E),0,Inf)$value

apost2_E<-function(x){
apost2_E<-apost_E(x)/K2
return(apost2_E)}

#stale ale integral neni roven 1
KK2=integrate(Vectorize(apost2_E),0,Inf)$value
KK2

apost3_E<-function(x){
apost3_E<-apost2_E(x)/KK2
return(apost3_E)}

#ted uz konecne ano
integrate(Vectorize(apost3_E),0,Inf)



par(new=TRUE)
plot(Vectorize(apost3_E)(ind)~ind, type="l",col=2,ylim=c(0,1.5),
xlab=expression(lambda), ylab=expression(paste(pi,"(",lambda,"|x)" )))

#c) apriorni hustota - Jeffreysova

apost_J<-function(x){
verohodnost(x)*1/sqrt(x)}

K3<-integrate(Vectorize(apost_J),0,Inf)$value

apost2_J<-function(x){
apost2_J<-apost_J(x)/K3
return(apost2_J)}

#stale ale integral neni roven 1
KK3=integrate(Vectorize(apost2_J),0,Inf)$value
KK3

apost3_J<-function(x){
apost3_J<-apost2_J(x)/KK3
return(apost3_J)}

#ted uz konecne ano
integrate(Vectorize(apost3_J),0,Inf)

par(new=TRUE)
plot(Vectorize(apost3_J)(ind)~ind, type="l",col=3,ylim=c(0,1.5),
xlab=expression(lambda), ylab=expression(paste(pi,"(",lambda,"|x)" )))


#2.)

#a) graf pro pozitivne diagnostikovane
aposterP<-function(x){
aposterP<-0.7*x/(0.7*x+0.02*(1-x))
return(aposterP)}

curve(aposterP,0,1,xlab='Prevalence',
ylab='Aposteriorni pravdepodobnost',main='Graf aposteriorni pravdepodobnosti',col=2)

#b) graf pro negativne diagnostikovane
aposterN<-function(x){
aposterN<-0.98*(1-x)/(0.3*x+0.98*(1-x))
return(aposterN)}

curve(aposterN,col=4,add=TRUE)
abline(v=0.03,lty=3)