########################################################################################################################################################################################################################
#Load Packages
########################################################################################################################################################################################################################
library(truncnorm)
library(lattice)
library(nlme)
library(MCMCpack)
library(combinat)

########################################################################################################################################################################################################################
#Load Data
########################################################################################################################################################################################################################
data=read.csv("Training_data.csv")
data_validation=read.csv("Validation_data.csv")

set.seed(1) #set seed so each run matches reported results 

########################################################################################################################################################################################################################
#Functions for model fitting
########################################################################################################################################################################################################################
posterior_mu_theta=function(Theta,sigma_theta_2,mu_0,sigma_0_2)
{
  mu_0_star=(sigma_theta_2*mu_0)/(sigma_theta_2 + N*sigma_0_2) + (sigma_0_2*sum(Theta))/(sigma_theta_2 + N*sigma_0_2)
  sigma_0_star_2=(sigma_theta_2*sigma_0_2)/(sigma_theta_2 + N*sigma_0_2)
  list(mu_0_star=mu_0_star,sigma_0_star_2=sigma_0_star_2)
}

posterior_sigma_theta_2=function(Theta,mu_theta,a_theta,b_theta)
{
  a_theta_star=a_theta + N/2
  b_theta_star=b_theta + sum((Theta-mu_theta)^2)/2
  list(a_theta_star=a_theta_star,b_theta_star=b_theta_star)
}

posterior_sigma_2=function(Y,t,D,J,Theta,I,gamma,Tau)
{
  a_sigma=sum(J)/2
  
  temp_I=rep(c(rep(0,n_0),I),J)
  temp_gamma=rep(c(rep(0,n_0),gamma),J)
  temp_gamma[is.na(temp_gamma)]=0
  temp_Tau=rep(c(rep(0,n_0),Tau),J)
  temp_Tau[is.na(temp_Tau)]=0
  
  Theta_star=rep(Theta,J)+temp_gamma*(t-temp_Tau)*as.numeric(t>temp_Tau)
  b_sigma=sum((Y-Theta_star)^2)/2
  
  list(a_sigma=a_sigma,b_sigma=b_sigma)
}

posterior_mu_gamma=function(I,gamma,sigma_gamma_2,mu_1,sigma_1_2)
{
  n_I=sum(I)
  mu_1_star=(sigma_gamma_2*mu_1)/(sigma_gamma_2 + n_I*sigma_1_2) + (sigma_1_2*sum(log(gamma),na.rm=T))/(sigma_gamma_2 + n_I*sigma_1_2)
  sigma_1_star_2=(sigma_gamma_2*sigma_1_2)/(sigma_gamma_2 + n_I*sigma_1_2)
  list(mu_1_star=mu_1_star,sigma_1_star_2=sigma_1_star_2)
}

posterior_sigma_gamma_2=function(I,gamma,mu_gamma,a_gamma,b_gamma)
{
  n_I=sum(I)
  a_gamma_star=a_gamma+n_I/2
  b_gamma_star=b_gamma+sum((log(gamma)-mu_gamma)^2,na.rm=T)/2
  list(a_gamma_star=a_gamma_star,b_gamma_star=b_gamma_star)
}

posterior_theta=function(Y_i,t_i,D_i,J_i,mu_theta,sigma_theta_2,sigma_2,I_i,gamma_i,tau_i)
{
  Y_star_i=Y_i
  if(is.na(I_i)==FALSE & I_i==1)
  {
    Y_star_i=Y_i-gamma_i*(t_i-tau_i)*as.numeric(t_i>tau_i)
  } 
  mu_theta_star=(sigma_2*mu_theta)/(sigma_2 + J_i*sigma_theta_2) + (sigma_theta_2*sum(Y_star_i))/(sigma_2 + J_i*sigma_theta_2)
  sigma_theta_star_2=(sigma_2*sigma_theta_2)/(sigma_2 + J_i*sigma_theta_2)
  list(mu_theta_star=mu_theta_star,sigma_theta_star_2=sigma_theta_star_2)
}

log_likelihood_cases=function(Y_i,t_i,J_i,sigma_2,Theta_i,I_i,gamma_i,tau_i)
{
  log_pr=NA
  if(I_i==0)
  {
    log_pr=log((2*pi*sigma_2)^(-J_i/2)) + (-sum((Y_i-Theta_i)^2)/(2*sigma_2))
  }
  if(I_i==1)
  {
    log_pr=log((2*pi*sigma_2)^(-J_i/2)) + (-sum((Y_i-Theta_i-gamma_i*(t_i-tau_i)*as.numeric(t_i>tau_i))^2)/(2*sigma_2))
  }
  log_pr
}

update_I=function(Y_i,t_i,J_i,d_i,mu_gamma,sigma_gamma_2,mu_tau,sigma_tau_2,Tau_star,sigma_2,Theta_i,I_i,gamma_i,tau_i,Pi)
{
  #new_I=NA
  new_gamma=NA
  new_tau=NA
  if(I_i==0)
  {
    gamma_star=exp(rnorm(1,mean=mu_gamma,sd=sqrt(sigma_gamma_2)))
    tau_star=rtruncnorm(1,a=d_i-Tau_star,b=d_i,mean=d_i-mu_tau,sd=sqrt(sigma_tau_2))
    temp_log_r=log_likelihood_cases(Y_i,t_i,J_i,sigma_2,Theta_i,I_i=1,gamma_i=gamma_star,tau_i=tau_star)-log_likelihood_cases(Y_i,t_i,J_i,sigma_2,Theta_i,I_i=0,gamma_i=NA,tau_i=NA)
    log_r=min(temp_log_r+log(Pi)-log(1-Pi),log(1))
    u=runif(1,min=0,max=1)
    new_I=as.numeric(log(u)<log_r)
    if(new_I==1)
    {
      new_gamma=gamma_star
      new_tau=tau_star
    }
  }
  if(I_i==1)
  {
    temp_log_r=log_likelihood_cases(Y_i,t_i,J_i,sigma_2,Theta_i,I_i=0,gamma_i=NA,tau_i=NA)-log_likelihood_cases(Y_i,t_i,J_i,sigma_2,Theta_i,I_i=1,gamma_i=gamma_i,tau_i=tau_i)
    log_r=min(temp_log_r+log(1-Pi)-log(Pi),log(1))
    u=runif(1,min=0,max=1)
    new_I=1-as.numeric(log(u)<log_r)
    if(new_I==1)
    {
      new_gamma=gamma_i
      new_tau=tau_i
    }
  }
  list(new_I=new_I,new_gamma=new_gamma,new_tau=new_tau)
}

update_gamma=function(Y_i,t_i,J_i,mu_gamma,sigma_gamma_2,sigma_2,Theta_i,I_i,gamma_i,tau_i,delta_gamma)
{
  new_gamma=NA
  gamma_accepted=NA
  new_I=0
  new_tau=NA
  if(I_i==1)
  {
    gamma_star=exp(rnorm(1,mean=log(gamma_i),sd=delta_gamma))
    log_temp_r1=log_likelihood_cases(Y_i,t_i,J_i,sigma_2,Theta_i,I_i=1,gamma_i=gamma_star,tau_i=tau_i)-log_likelihood_cases(Y_i,t_i,J_i,sigma_2,Theta_i,I_i=1,gamma_i=gamma_i,tau_i=tau_i)
    log_temp_r2=log(dlnorm(x=gamma_star,meanlog=mu_gamma,sdlog=sqrt(sigma_gamma_2)))-log(dlnorm(x=gamma_i,meanlog=mu_gamma,sdlog=sqrt(sigma_gamma_2)))
    log_temp_r3=log(dlnorm(x=gamma_i,meanlog=log(gamma_star),sdlog=delta_gamma))-log(dlnorm(x=gamma_star,meanlog=log(gamma_i),sdlog=delta_gamma))
    log_r=min(log_temp_r1+log_temp_r2+log_temp_r3,log(1))
    u=runif(1,min=0,max=1)
    new_gamma=as.numeric(log(u)<log_r)*gamma_star + (1-as.numeric(log(u)<log_r))*gamma_i
    gamma_accepted=as.numeric(log(u)<log_r)
    new_I=I_i
    new_tau=tau_i
  }
  list(new_gamma=new_gamma,gamma_accepted=gamma_accepted,new_I=new_I,new_tau=new_tau)
}

update_tau=function(Y_i,t_i,J_i,d_i,sigma_2,mu_tau,sigma_tau_2,Theta_i,I_i,gamma_i,tau_i,delta_tau,Tau_star)
{
  new_gamma=NA
  new_I=0
  new_tau=NA
  tau_accepted=NA
  if(I_i==1)
  {
    tau_star=rtruncnorm(1,a=d_i-Tau_star,b=d_i,mean=tau_i,sd=delta_tau)
    log_temp_r1=log_likelihood_cases(Y_i,t_i,J_i,sigma_2,Theta_i,I_i=1,gamma_i=gamma_i,tau_i=tau_star)-log_likelihood_cases(Y_i,t_i,J_i,sigma_2,Theta_i,I_i=1,gamma_i=gamma_i,tau_i=tau_i)
    log_temp_r2=log(dtruncnorm(x=tau_star,a=d_i-Tau_star,b=d_i,mean=d_i-mu_tau,sd=sqrt(sigma_tau_2)))-log(dtruncnorm(x=tau_i,a=d_i-Tau_star,b=d_i,mean=d_i-mu_tau,sd=sqrt(sigma_tau_2)))
    log_temp_r3=log(dtruncnorm(x=tau_i,a=d_i-Tau_star,b=d_i,mean=tau_star,sd=delta_tau))-log(dtruncnorm(x=tau_star,a=d_i-Tau_star,b=d_i,mean=tau_i,sd=delta_tau))
    log_r=min(log_temp_r1+log_temp_r2+log_temp_r3,log(1))
    u=runif(1,min=0,max=1)
    new_tau=as.numeric(log(u)<log_r)*tau_star + (1-as.numeric(log(u)<log_r))*tau_i
    tau_accepted=as.numeric(log(u)<log_r)
    new_I=I_i
    new_gamma=gamma_i
  }
  list(new_gamma=new_gamma,tau_accepted=tau_accepted,new_I=new_I,new_tau=new_tau)
}

update_mu_tau=function(Tau,I,d,D,mu_tau,sigma_tau_2,delta_mu_tau,mu_2,sigma_2_2,Tau_star)
{
  n_I=sum(I)
  mu_tau_star=rnorm(1,mean=mu_tau,sd=delta_mu_tau)
  log_temp1=sum(log(dtruncnorm(x=Tau[I==1],a=d[D==1][I==1]-Tau_star,b=d[D==1][I==1],mean=d[D==1][I==1]-mu_tau_star,sd=sqrt(sigma_tau_2)))) + log(dnorm(x=mu_tau_star,mean=mu_2,sd=sqrt(sigma_2_2)))
  log_temp2=sum(log(dtruncnorm(x=Tau[I==1],a=d[D==1][I==1]-Tau_star,b=d[D==1][I==1],mean=d[D==1][I==1]-mu_tau,sd=sqrt(sigma_tau_2)))) + log(dnorm(x=mu_tau,mean=mu_2,sd=sqrt(sigma_2_2)))
  log_r=min(log_temp1 - log_temp2,log(1))
  u=runif(1,min=0,max=1)
  new_mu_tau=as.numeric(log(u)<log_r)*mu_tau_star + (1-as.numeric(log(u)<log_r))*mu_tau
  mu_tau_accepted=as.numeric(log(u)<log_r)
  list(new_mu_tau=new_mu_tau,mu_tau_accepted=mu_tau_accepted)
}

update_sigma_tau=function(Tau,I,d,D,mu_tau,sigma_tau_2,delta_sigma_tau,a_tau,b_tau,Tau_star)
{
  n_I=sum(I)
  sigma_tau_2_star=rtruncnorm(1,a=0,b=Inf,mean=sigma_tau_2,sd=delta_sigma_tau)
  log_temp1=sum(log(dtruncnorm(x=Tau[I==1],a=d[D==1][I==1]-Tau_star,b=d[D==1][I==1],mean=d[D==1][I==1]-mu_tau,sd=sqrt(sigma_tau_2_star)))) + log(dinvgamma(x=sigma_tau_2_star,shape=a_tau,scale=b_tau))
  log_temp2=sum(log(dtruncnorm(x=Tau[I==1],a=d[D==1][I==1]-Tau_star,b=d[D==1][I==1],mean=d[D==1][I==1]-mu_tau,sd=sqrt(sigma_tau_2)))) + log(dinvgamma(x=sigma_tau_2,shape=a_tau,scale=b_tau))
  log_temp3=log(dtruncnorm(x=sigma_tau_2,a=0,b=Inf,mean=sigma_tau_2_star,sd=delta_sigma_tau))-log(dtruncnorm(x=sigma_tau_2_star,a=0,b=Inf,mean=sigma_tau_2,sd=delta_sigma_tau))
  log_r=min(log_temp1 - log_temp2 + log_temp3,log(1))
  u=runif(1,min=0,max=1)
  new_sigma_tau_2=as.numeric(log(u)<log_r)*sigma_tau_2_star + (1-as.numeric(log(u)<log_r))*sigma_tau_2
  sigma_tau_accepted=as.numeric(log(u)<log_r)
  list(new_sigma_tau_2=new_sigma_tau_2,sigma_tau_accepted=sigma_tau_accepted)
}

posterior_Pi=function(I,p_1,p_2)
{
  p_1_star=p_1 + sum(I)
  p_2_star=p_2 + sum(1-I)
  list(p_1_star=p_1_star,p_2_star=p_2_star)
}

########################################################################################################################################################################################################################
#Analyze Training Data (pg 6 Supplementary Materials)
########################################################################################################################################################################################################################
#Get sample size 
N=length(unique(data$ID)) #total number of patients
n_0=length(unique(data$ID[(data$D==0)])) #number of control patients

#Extract vectors
D=data$D[(data$obs_number==1)] #case/control status
d=data$d[(data$obs_number==1)] #diagnosis/follow-up times
J=data$J[(data$obs_number==1)] #number of screening visits
K=3 #Number of biomarkers in the study

#Specify fixed parameters
p_1=30
p_2=30

#Biomarker 1
mu_0_1=2
sigma_0_1_2=0.1
a_theta_1=2
b_theta_1=0.05
mu_1_1=log(16)
sigma_1_1_2=0.1 
a_gamma_1=4
b_gamma_1=0.1
Tau_star_1=2
mu_2_1=1
sigma_2_1_2=0.1
a_tau_1=10
b_tau_1=(0.75^2)*(a_tau_1+1)

#Biomarker 2
mu_0_2=3
sigma_0_2_2=0.1
a_theta_2=2
b_theta_2=0.05
mu_1_2=log(16)
sigma_1_2_2=0.05
a_gamma_2=2*2
b_gamma_2=0.05*2
Tau_star_2=2
mu_2_2=1
sigma_2_2_2=0.1
a_tau_2=10
b_tau_2=(0.75^2)*(a_tau_2+1)

#Biomarker 3
mu_0_3=2
sigma_0_3_2=0.1
a_theta_3=2
b_theta_3=0.05
mu_1_3=log(16)
sigma_1_3_2=0.05
a_gamma_3=2*2
b_gamma_3=0.05*2
Tau_star_3=2
mu_2_3=1
sigma_2_3_2=0.1
a_tau_3=10
b_tau_3=(0.75^2)*(a_tau_1+1)

S=5000
delta_gamma_set=0.1
delta_tau_set=0.5
delta_mu_tau_set=0.75
delta_sigma_tau_set=0.3

###########################Initialize parameters from priors for first chain

#biomarker 1
Pi_1_out=rbeta(1,p_1,p_2)
mu_theta_1_out=rnorm(1,mean=mu_0_1,sd=sqrt(sigma_0_1_2))
sigma_theta_1_2_out=rinvgamma(1,shape=a_theta_1,scale=b_theta_1)
Theta_1_out=array(rnorm(N,mean=mu_theta_1_out,sd=sigma_theta_1_2_out),c(N,1))
I_1_out=array(rbinom(3*(N-n_0),size=1,prob=0.5),c(N-n_0,3))
mu_gamma_1_out=rnorm(1,mean=mu_1_1,sd=sqrt(sigma_1_1_2))
sigma_gamma_1_2_out=rinvgamma(1,shape=a_gamma_1,scale=b_gamma_1)
mu_tau_1_out=1
sigma_tau_1_2_out=(0.75)^2
gamma_1_out=array(exp(rnorm(3*(N-n_0),mean=mu_gamma_1_out,sd=sqrt(sigma_gamma_1_2_out))),c(N-n_0,3))
gamma_1_out[I_1_out[,1]==0,1]=NA
gamma_1_out[I_1_out[,2]==0,2]=NA
gamma_1_out[I_1_out[,3]==0,3]=NA
Tau_1_out=array(rtruncnorm(n=3*(N-n_0),a=d[D==1]-Tau_star_1,b=d[D==1],mean=d[D==1]-mu_tau_1_out,sd=sqrt(sigma_tau_1_2_out)),c(N-n_0,3))
Tau_1_out[I_1_out[,1]==0,1]=NA
Tau_1_out[I_1_out[,2]==0,2]=NA
Tau_1_out[I_1_out[,3]==0,3]=NA
sigma_1_2_out=0.1

#biomarker 2
Pi_2_out=rbeta(1,p_1,p_2)
mu_theta_2_out=rnorm(1,mean=mu_0_2,sd=sqrt(sigma_0_2_2))
sigma_theta_2_2_out=rinvgamma(1,shape=a_theta_2,scale=b_theta_2)
Theta_2_out=array(rnorm(N,mean=mu_theta_2_out,sd=sigma_theta_2_2_out),c(N,1))
I_2_out=array(rbinom(3*(N-n_0),size=1,prob=0.5),c(N-n_0,3))
mu_gamma_2_out=rnorm(1,mean=mu_1_2,sd=sqrt(sigma_1_2_2))
sigma_gamma_2_2_out=rinvgamma(1,shape=a_gamma_2,scale=b_gamma_2)
mu_tau_2_out=1
sigma_tau_2_2_out=(0.75)^2
gamma_2_out=array(exp(rnorm(3*(N-n_0),mean=mu_gamma_2_out,sd=sqrt(sigma_gamma_2_2_out))),c(N-n_0,3))
gamma_2_out[I_2_out[,1]==0,1]=NA
gamma_2_out[I_2_out[,2]==0,2]=NA
gamma_2_out[I_2_out[,3]==0,3]=NA
Tau_2_out=array(rtruncnorm(n=3*(N-n_0),a=d[D==1]-Tau_star_1,b=d[D==1],mean=d[D==1]-mu_tau_2_out,sd=sqrt(sigma_tau_2_2_out)),c(N-n_0,3))
Tau_2_out[I_2_out[,1]==0,1]=NA
Tau_2_out[I_2_out[,2]==0,2]=NA
Tau_2_out[I_2_out[,3]==0,3]=NA
sigma_2_2_out=0.1

#biomarker 3
Pi_3_out=rbeta(1,p_1,p_2)
mu_theta_3_out=rnorm(1,mean=mu_0_3,sd=sqrt(sigma_0_3_2))
sigma_theta_3_2_out=rinvgamma(1,shape=a_theta_3,scale=b_theta_3)
Theta_3_out=array(rnorm(N,mean=mu_theta_3_out,sd=sigma_theta_3_2_out),c(N,1))
I_3_out=array(rbinom(3*(N-n_0),size=1,prob=0.5),c(N-n_0,3))
mu_gamma_3_out=rnorm(1,mean=mu_1_3,sd=sqrt(sigma_1_3_2))
sigma_gamma_3_2_out=rinvgamma(1,shape=a_gamma_3,scale=b_gamma_3)
mu_tau_3_out=1
sigma_tau_3_2_out=(0.75)^2
gamma_3_out=array(exp(rnorm(3*(N-n_0),mean=mu_gamma_3_out,sd=sqrt(sigma_gamma_3_2_out))),c(N-n_0,3))
gamma_3_out[I_3_out[,1]==0,1]=NA
gamma_3_out[I_3_out[,2]==0,2]=NA
gamma_3_out[I_3_out[,3]==0,3]=NA
Tau_3_out=array(rtruncnorm(n=3*(N-n_0),a=d[D==1]-Tau_star_3,b=d[D==1],mean=d[D==1]-mu_tau_3_out,sd=sqrt(sigma_tau_3_2_out)),c(N-n_0,3))
Tau_3_out[I_3_out[,1]==0,1]=NA
Tau_3_out[I_3_out[,2]==0,2]=NA
Tau_3_out[I_3_out[,3]==0,3]=NA
sigma_3_2_out=0.1

#Update parameters
for(i in 1:S)
{
  #MU_THETA_1
  mu_theta_parameters=posterior_mu_theta(Theta=Theta_1_out[,i],sigma_theta_2=sigma_theta_1_2_out[i],mu_0=mu_0_1,sigma_0_2=sigma_0_1_2)
  mu_theta_1_out=c(mu_theta_1_out,rnorm(1,mean=mu_theta_parameters$mu_0_star,sd=sqrt(mu_theta_parameters$sigma_0_star_2)))
  
  #MU_THETA_2
  mu_theta_parameters=posterior_mu_theta(Theta=Theta_2_out[,i],sigma_theta_2=sigma_theta_2_2_out[i],mu_0=mu_0_2,sigma_0_2=sigma_0_2_2)
  mu_theta_2_out=c(mu_theta_2_out,rnorm(1,mean=mu_theta_parameters$mu_0_star,sd=sqrt(mu_theta_parameters$sigma_0_star_2)))
  
  #MU_THETA_3
  mu_theta_parameters=posterior_mu_theta(Theta=Theta_3_out[,i],sigma_theta_2=sigma_theta_3_2_out[i],mu_0=mu_0_3,sigma_0_2=sigma_0_3_2)
  mu_theta_3_out=c(mu_theta_3_out,rnorm(1,mean=mu_theta_parameters$mu_0_star,sd=sqrt(mu_theta_parameters$sigma_0_star_2)))
  
  #SIGMA_THETA_1
  sigma_theta_parameters=posterior_sigma_theta_2(Theta=Theta_1_out[,i],mu_theta=mu_theta_1_out[i+1],a_theta=a_theta_1,b_theta=b_theta_1)
  sigma_theta_1_2_out=c(sigma_theta_1_2_out,rinvgamma(1,shape=sigma_theta_parameters$a_theta_star,scale=sigma_theta_parameters$b_theta_star))
  
  #SIGMA_THETA_2
  sigma_theta_parameters=posterior_sigma_theta_2(Theta=Theta_2_out[,i],mu_theta=mu_theta_2_out[i+1],a_theta=a_theta_2,b_theta=b_theta_2)
  sigma_theta_2_2_out=c(sigma_theta_2_2_out,rinvgamma(1,shape=sigma_theta_parameters$a_theta_star,scale=sigma_theta_parameters$b_theta_star))
  
  #SIGMA_THETA_3
  sigma_theta_parameters=posterior_sigma_theta_2(Theta=Theta_3_out[,i],mu_theta=mu_theta_3_out[i+1],a_theta=a_theta_3,b_theta=b_theta_3)
  sigma_theta_3_2_out=c(sigma_theta_3_2_out,rinvgamma(1,shape=sigma_theta_parameters$a_theta_star,scale=sigma_theta_parameters$b_theta_star))
  
  #SIGMA_1
  sigma_parameters=posterior_sigma_2(Y=data$Y1,t=data$t,D=D,J=J,Theta=Theta_1_out[,i],I=I_1_out[,3*i],gamma=gamma_1_out[,3*i],Tau=Tau_1_out[,3*i])
  sigma_1_2_out=c(sigma_1_2_out,rinvgamma(1,shape=sigma_parameters$a_sigma,scale=sigma_parameters$b_sigma))
  
  #SIGMA_2
  sigma_parameters=posterior_sigma_2(Y=data$Y2,t=data$t,D=D,J=J,Theta=Theta_2_out[,i],I=I_2_out[,3*i],gamma=gamma_2_out[,3*i],Tau=Tau_2_out[,3*i])
  sigma_2_2_out=c(sigma_2_2_out,rinvgamma(1,shape=sigma_parameters$a_sigma,scale=sigma_parameters$b_sigma))
  
  #SIGMA_3
  sigma_parameters=posterior_sigma_2(Y=data$Y3,t=data$t,D=D,J=J,Theta=Theta_3_out[,i],I=I_3_out[,3*i],gamma=gamma_3_out[,3*i],Tau=Tau_3_out[,3*i])
  sigma_3_2_out=c(sigma_3_2_out,rinvgamma(1,shape=sigma_parameters$a_sigma,scale=sigma_parameters$b_sigma))
  
  #PI_1
  Pi_parameters=posterior_Pi(I=I_1_out[,3*i],p_1=p_1,p_2=p_2)
  Pi_1_out=c(Pi_1_out,rbeta(1,shape1=Pi_parameters$p_1_star,shape2=Pi_parameters$p_2_star))
  
  #PI_2
  Pi_parameters=posterior_Pi(I=I_2_out[,3*i],p_1=p_1,p_2=p_2)
  Pi_2_out=c(Pi_2_out,rbeta(1,shape1=Pi_parameters$p_1_star,shape2=Pi_parameters$p_2_star))
  
  #PI_3
  Pi_parameters=posterior_Pi(I=I_3_out[,3*i],p_1=p_1,p_2=p_2)
  Pi_3_out=c(Pi_3_out,rbeta(1,shape1=Pi_parameters$p_1_star,shape2=Pi_parameters$p_2_star))
  
  #MU_GAMMA_1
  mu_gamma_parameters=posterior_mu_gamma(I=I_1_out[,3*i],gamma=gamma_1_out[,3*i],sigma_gamma_2=sigma_gamma_1_2_out[i],mu_1=mu_1_1,sigma_1_2=sigma_1_1_2)
  mu_gamma_1_out=c(mu_gamma_1_out,rnorm(1,mean=mu_gamma_parameters$mu_1_star,sd=sqrt(mu_gamma_parameters$sigma_1_star_2)))
  
  #MU_GAMMA_2
  mu_gamma_parameters=posterior_mu_gamma(I=I_2_out[,3*i],gamma=gamma_2_out[,3*i],sigma_gamma_2=sigma_gamma_2_2_out[i],mu_1=mu_1_2,sigma_1_2=sigma_1_2_2)
  mu_gamma_2_out=c(mu_gamma_2_out,rnorm(1,mean=mu_gamma_parameters$mu_1_star,sd=sqrt(mu_gamma_parameters$sigma_1_star_2)))
  
  #MU_GAMMA_3
  mu_gamma_parameters=posterior_mu_gamma(I=I_3_out[,3*i],gamma=gamma_3_out[,3*i],sigma_gamma_2=sigma_gamma_3_2_out[i],mu_1=mu_1_3,sigma_1_2=sigma_1_3_2)
  mu_gamma_3_out=c(mu_gamma_3_out,rnorm(1,mean=mu_gamma_parameters$mu_1_star,sd=sqrt(mu_gamma_parameters$sigma_1_star_2)))
  
  #SIGMA_GAMMA_1
  sigma_gamma_parameters=posterior_sigma_gamma_2(I=I_1_out[,3*i],gamma=gamma_1_out[,3*i],mu_gamma=mu_gamma_1_out[i+1],a_gamma=a_gamma_1,b_gamma=b_gamma_1)
  sigma_gamma_1_2_out=c(sigma_gamma_1_2_out,rinvgamma(1,shape=sigma_gamma_parameters$a_gamma_star,scale=sigma_gamma_parameters$b_gamma_star))
  
  #SIGMA_GAMMA_2
  sigma_gamma_parameters=posterior_sigma_gamma_2(I=I_2_out[,3*i],gamma=gamma_2_out[,3*i],mu_gamma=mu_gamma_2_out[i+1],a_gamma=a_gamma_2,b_gamma=b_gamma_2)
  sigma_gamma_2_2_out=c(sigma_gamma_2_2_out,rinvgamma(1,shape=sigma_gamma_parameters$a_gamma_star,scale=sigma_gamma_parameters$b_gamma_star))
  
  #SIGMA_GAMMA_3
  sigma_gamma_parameters=posterior_sigma_gamma_2(I=I_3_out[,3*i],gamma=gamma_3_out[,3*i],mu_gamma=mu_gamma_3_out[i+1],a_gamma=a_gamma_3,b_gamma=b_gamma_3)
  sigma_gamma_3_2_out=c(sigma_gamma_3_2_out,rinvgamma(1,shape=sigma_gamma_parameters$a_gamma_star,scale=sigma_gamma_parameters$b_gamma_star))
  
  #MU_TAU_1
  mu_tau_output=update_mu_tau(Tau=Tau_1_out[,3*i],I=I_1_out[,3*i],d=d,D=D,mu_tau=mu_tau_1_out[i],sigma_tau_2=sigma_tau_1_2_out[i],delta_mu_tau=delta_mu_tau_set,mu_2=mu_2_1,sigma_2_2=sigma_2_1_2,Tau_star=Tau_star_1)
  mu_tau_1_out=c(mu_tau_1_out,mu_tau_output$new_mu_tau)
  
  #MU_TAU_2
  mu_tau_output=update_mu_tau(Tau=Tau_2_out[,3*i],I=I_2_out[,3*i],d=d,D=D,mu_tau=mu_tau_2_out[i],sigma_tau_2=sigma_tau_2_2_out[i],delta_mu_tau=delta_mu_tau_set,mu_2=mu_2_2,sigma_2_2=sigma_2_2_2,Tau_star=Tau_star_2)
  mu_tau_2_out=c(mu_tau_2_out,mu_tau_output$new_mu_tau)
  
  #MU_TAU_3
  mu_tau_output=update_mu_tau(Tau=Tau_3_out[,3*i],I=I_3_out[,3*i],d=d,D=D,mu_tau=mu_tau_3_out[i],sigma_tau_2=sigma_tau_3_2_out[i],delta_mu_tau=delta_mu_tau_set,mu_2=mu_2_3,sigma_2_2=sigma_2_3_2,Tau_star=Tau_star_3)
  mu_tau_3_out=c(mu_tau_3_out,mu_tau_output$new_mu_tau)
  
  #SIGMA_TAU_1
  sigma_tau_output=update_sigma_tau(Tau=Tau_1_out[,3*i],I=I_1_out[,3*i],d=d,D=D,mu_tau=mu_tau_1_out[i+1],sigma_tau_2=sigma_tau_1_2_out[i],delta_sigma_tau=delta_sigma_tau_set,a_tau=a_tau_1,b_tau=b_tau_1,Tau_star=Tau_star_1)
  sigma_tau_1_2_out=c(sigma_tau_1_2_out,sigma_tau_output$new_sigma_tau_2)
  
  #SIGMA_TAU_2
  sigma_tau_output=update_sigma_tau(Tau=Tau_2_out[,3*i],I=I_2_out[,3*i],d=d,D=D,mu_tau=mu_tau_2_out[i+1],sigma_tau_2=sigma_tau_2_2_out[i],delta_sigma_tau=delta_sigma_tau_set,a_tau=a_tau_2,b_tau=b_tau_2,Tau_star=Tau_star_2)
  sigma_tau_2_2_out=c(sigma_tau_2_2_out,sigma_tau_output$new_sigma_tau_2)
  
  #SIGMA_TAU_3
  sigma_tau_output=update_sigma_tau(Tau=Tau_3_out[,3*i],I=I_3_out[,3*i],d=d,D=D,mu_tau=mu_tau_3_out[i+1],sigma_tau_2=sigma_tau_3_2_out[i],delta_sigma_tau=delta_sigma_tau_set,a_tau=a_tau_3,b_tau=b_tau_3,Tau_star=Tau_star_3)
  sigma_tau_3_2_out=c(sigma_tau_3_2_out,sigma_tau_output$new_sigma_tau_2)
  
  Theta_1_new=rep(NA,N)
  Theta_2_new=rep(NA,N)
  Theta_3_new=rep(NA,N)
  
  I_1_new_1=rep(NA,N-n_0)
  I_1_new_2=rep(NA,N-n_0)
  I_1_new_3=rep(NA,N-n_0)
  
  I_2_new_1=rep(NA,N-n_0)
  I_2_new_2=rep(NA,N-n_0)
  I_2_new_3=rep(NA,N-n_0)
  
  I_3_new_1=rep(NA,N-n_0)
  I_3_new_2=rep(NA,N-n_0)
  I_3_new_3=rep(NA,N-n_0)
  
  gamma_1_new_1=rep(NA,N-n_0)
  gamma_1_new_2=rep(NA,N-n_0)
  gamma_1_new_3=rep(NA,N-n_0)
  
  gamma_2_new_1=rep(NA,N-n_0)
  gamma_2_new_2=rep(NA,N-n_0)
  gamma_2_new_3=rep(NA,N-n_0)
  
  gamma_3_new_1=rep(NA,N-n_0)
  gamma_3_new_2=rep(NA,N-n_0)
  gamma_3_new_3=rep(NA,N-n_0)
  
  Tau_1_new_1=rep(NA,N-n_0)
  Tau_1_new_2=rep(NA,N-n_0)
  Tau_1_new_3=rep(NA,N-n_0)
  
  Tau_2_new_1=rep(NA,N-n_0)
  Tau_2_new_2=rep(NA,N-n_0)
  Tau_2_new_3=rep(NA,N-n_0)
  
  Tau_3_new_1=rep(NA,N-n_0)
  Tau_3_new_2=rep(NA,N-n_0)
  Tau_3_new_3=rep(NA,N-n_0)
  
  for(j in 1:n_0)
  {
    #THETA_1_j
    theta_j_parameters=posterior_theta(Y_i=data$Y1[(data$ID==j)],t_i=data$t[(data$ID==j)],D_i=D[j],J_i=J[j],mu_theta=mu_theta_1_out[i+1],sigma_theta_2=sigma_theta_1_2_out[i+1],sigma_2=sigma_1_2_out[i+1],I_i=NA,gamma_i=NA,tau_i=NA)
    Theta_1_new[j]=rnorm(1,mean=theta_j_parameters$mu_theta_star,sd=sqrt(theta_j_parameters$sigma_theta_star_2))
    
    #THETA_2_j
    theta_j_parameters=posterior_theta(Y_i=data$Y2[(data$ID==j)],t_i=data$t[(data$ID==j)],D_i=D[j],J_i=J[j],mu_theta=mu_theta_2_out[i+1],sigma_theta_2=sigma_theta_2_2_out[i+1],sigma_2=sigma_2_2_out[i+1],I_i=NA,gamma_i=NA,tau_i=NA)
    Theta_2_new[j]=rnorm(1,mean=theta_j_parameters$mu_theta_star,sd=sqrt(theta_j_parameters$sigma_theta_star_2))
    
    #THETA_3_j
    theta_j_parameters=posterior_theta(Y_i=data$Y3[(data$ID==j)],t_i=data$t[(data$ID==j)],D_i=D[j],J_i=J[j],mu_theta=mu_theta_3_out[i+1],sigma_theta_2=sigma_theta_3_2_out[i+1],sigma_2=sigma_3_2_out[i+1],I_i=NA,gamma_i=NA,tau_i=NA)
    Theta_3_new[j]=rnorm(1,mean=theta_j_parameters$mu_theta_star,sd=sqrt(theta_j_parameters$sigma_theta_star_2))
  }
  for(j in (n_0+1):N)
  {
    #THETA_1_j
    theta_j_parameters=posterior_theta(Y_i=data$Y1[(data$ID==j)],t_i=data$t[(data$ID==j)],D_i=D[j],J_i=J[j],mu_theta=mu_theta_1_out[i+1],sigma_theta_2=sigma_theta_1_2_out[i+1],sigma_2=sigma_1_2_out[i+1],I_i=I_1_out[j-n_0,3*i],gamma_i=gamma_1_out[j-n_0,3*i],tau_i=Tau_1_out[j-n_0,3*i])
    Theta_1_new[j]=rnorm(1,mean=theta_j_parameters$mu_theta_star,sd=sqrt(theta_j_parameters$sigma_theta_star_2))
    
    #THETA_2_j
    theta_j_parameters=posterior_theta(Y_i=data$Y2[(data$ID==j)],t_i=data$t[(data$ID==j)],D_i=D[j],J_i=J[j],mu_theta=mu_theta_2_out[i+1],sigma_theta_2=sigma_theta_2_2_out[i+1],sigma_2=sigma_2_2_out[i+1],I_i=I_2_out[j-n_0,3*i],gamma_i=gamma_2_out[j-n_0,3*i],tau_i=Tau_2_out[j-n_0,3*i])
    Theta_2_new[j]=rnorm(1,mean=theta_j_parameters$mu_theta_star,sd=sqrt(theta_j_parameters$sigma_theta_star_2))
    
    #THETA_3_j
    theta_j_parameters=posterior_theta(Y_i=data$Y3[(data$ID==j)],t_i=data$t[(data$ID==j)],D_i=D[j],J_i=J[j],mu_theta=mu_theta_3_out[i+1],sigma_theta_2=sigma_theta_3_2_out[i+1],sigma_2=sigma_3_2_out[i+1],I_i=I_3_out[j-n_0,3*i],gamma_i=gamma_3_out[j-n_0,3*i],tau_i=Tau_3_out[j-n_0,3*i])
    Theta_3_new[j]=rnorm(1,mean=theta_j_parameters$mu_theta_star,sd=sqrt(theta_j_parameters$sigma_theta_star_2))
    
    #I_1
    Pi_1_input=Pi_1_out[i+1]
    I_output=update_I(Y_i=data$Y1[(data$ID==j)],t_i=data$t[(data$ID==j)],J_i=J[j],d_i=d[j],mu_gamma=mu_gamma_1_out[i+1],sigma_gamma_2=sigma_gamma_1_2_out[i+1],mu_tau=mu_tau_1_out[i+1],sigma_tau_2=sigma_tau_1_2_out[i+1],Tau_star=Tau_star_1,sigma_2=sigma_1_2_out[i+1],Theta_i=Theta_1_new[j],I_i=I_1_out[j-n_0,3*i],gamma_i=gamma_1_out[j-n_0,3*i],tau_i=Tau_1_out[j-n_0,3*i],Pi=Pi_1_input)
    I_1_new_1[j-n_0]=I_output$new_I
    Tau_1_new_1[j-n_0]=I_output$new_tau
    gamma_1_new_1[j-n_0]=I_output$new_gamma
    
    #I_2
    Pi_2_input=Pi_2_out[i+1]
    I_output=update_I(Y_i=data$Y2[(data$ID==j)],t_i=data$t[(data$ID==j)],J_i=J[j],d_i=d[j],mu_gamma=mu_gamma_2_out[i+1],sigma_gamma_2=sigma_gamma_2_2_out[i+1],mu_tau=mu_tau_2_out[i+1],sigma_tau_2=sigma_tau_2_2_out[i+1],Tau_star=Tau_star_2,sigma_2=sigma_2_2_out[i+1],Theta_i=Theta_2_new[j],I_i=I_2_out[j-n_0,3*i],gamma_i=gamma_2_out[j-n_0,3*i],tau_i=Tau_2_out[j-n_0,3*i],Pi=Pi_2_input)
    I_2_new_1[j-n_0]=I_output$new_I
    Tau_2_new_1[j-n_0]=I_output$new_tau
    gamma_2_new_1[j-n_0]=I_output$new_gamma
    
    #I_3
    Pi_3_input=Pi_3_out[i+1]
    I_output=update_I(Y_i=data$Y3[(data$ID==j)],t_i=data$t[(data$ID==j)],J_i=J[j],d_i=d[j],mu_gamma=mu_gamma_3_out[i+1],sigma_gamma_2=sigma_gamma_3_2_out[i+1],mu_tau=mu_tau_3_out[i+1],sigma_tau_2=sigma_tau_3_2_out[i+1],Tau_star=Tau_star_3,sigma_2=sigma_3_2_out[i+1],Theta_i=Theta_3_new[j],I_i=I_3_out[j-n_0,3*i],gamma_i=gamma_3_out[j-n_0,3*i],tau_i=Tau_3_out[j-n_0,3*i],Pi=Pi_3_input)
    I_3_new_1[j-n_0]=I_output$new_I
    Tau_3_new_1[j-n_0]=I_output$new_tau
    gamma_3_new_1[j-n_0]=I_output$new_gamma
    
    #GAMMA_1
    gamma_output=update_gamma(Y_i=data$Y1[(data$ID==j)],t_i=data$t[(data$ID==j)],J_i=J[j],mu_gamma=mu_gamma_1_out[i+1],sigma_gamma_2=sigma_gamma_1_2_out[i+1],sigma_2=sigma_1_2_out[i+1],Theta_i=Theta_1_new[j],I_i=I_1_new_1[j-n_0],gamma_i=gamma_1_new_1[j-n_0],tau_i=Tau_1_new_1[j-n_0],delta_gamma=delta_gamma_set)
    gamma_1_new_2[j-n_0]=gamma_output$new_gamma
    I_1_new_2[j-n_0]=gamma_output$new_I
    Tau_1_new_2[j-n_0]=gamma_output$new_tau
    
    #GAMMA_2
    gamma_output=update_gamma(Y_i=data$Y2[(data$ID==j)],t_i=data$t[(data$ID==j)],J_i=J[j],mu_gamma=mu_gamma_2_out[i+1],sigma_gamma_2=sigma_gamma_2_2_out[i+1],sigma_2=sigma_2_2_out[i+1],Theta_i=Theta_2_new[j],I_i=I_2_new_1[j-n_0],gamma_i=gamma_2_new_1[j-n_0],tau_i=Tau_2_new_1[j-n_0],delta_gamma=delta_gamma_set)
    gamma_2_new_2[j-n_0]=gamma_output$new_gamma
    I_2_new_2[j-n_0]=gamma_output$new_I
    Tau_2_new_2[j-n_0]=gamma_output$new_tau
    
    #GAMMA_3
    gamma_output=update_gamma(Y_i=data$Y3[(data$ID==j)],t_i=data$t[(data$ID==j)],J_i=J[j],mu_gamma=mu_gamma_3_out[i+1],sigma_gamma_2=sigma_gamma_3_2_out[i+1],sigma_2=sigma_3_2_out[i+1],Theta_i=Theta_3_new[j],I_i=I_3_new_1[j-n_0],gamma_i=gamma_3_new_1[j-n_0],tau_i=Tau_3_new_1[j-n_0],delta_gamma=delta_gamma_set)
    gamma_3_new_2[j-n_0]=gamma_output$new_gamma
    I_3_new_2[j-n_0]=gamma_output$new_I
    Tau_3_new_2[j-n_0]=gamma_output$new_tau
    
    #TAU_1
    tau_output=update_tau(Y_i=data$Y1[(data$ID==j)],t_i=data$t[(data$ID==j)],J_i=J[j],d_i=d[j],sigma_2=sigma_1_2_out[i+1],mu_tau=mu_tau_1_out[i+1],sigma_tau_2=sigma_tau_1_2_out[i+1],Theta_i=Theta_1_new[j],I_i=I_1_new_2[j-n_0],gamma_i=gamma_1_new_2[j-n_0],tau_i=Tau_1_new_2[j-n_0],delta_tau=delta_tau_set,Tau_star=Tau_star_1)
    I_1_new_3[j-n_0]=tau_output$new_I
    Tau_1_new_3[j-n_0]=tau_output$new_tau
    gamma_1_new_3[j-n_0]=tau_output$new_gamma
    
    #TAU_2
    tau_output=update_tau(Y_i=data$Y2[(data$ID==j)],t_i=data$t[(data$ID==j)],J_i=J[j],d_i=d[j],sigma_2=sigma_2_2_out[i+1],mu_tau=mu_tau_2_out[i+1],sigma_tau_2=sigma_tau_2_2_out[i+1],Theta_i=Theta_2_new[j],I_i=I_2_new_2[j-n_0],gamma_i=gamma_2_new_2[j-n_0],tau_i=Tau_2_new_2[j-n_0],delta_tau=delta_tau_set,Tau_star=Tau_star_2)
    I_2_new_3[j-n_0]=tau_output$new_I
    Tau_2_new_3[j-n_0]=tau_output$new_tau
    gamma_2_new_3[j-n_0]=tau_output$new_gamma
    
    #TAU_1
    tau_output=update_tau(Y_i=data$Y3[(data$ID==j)],t_i=data$t[(data$ID==j)],J_i=J[j],d_i=d[j],sigma_2=sigma_3_2_out[i+1],mu_tau=mu_tau_3_out[i+1],sigma_tau_2=sigma_tau_3_2_out[i+1],Theta_i=Theta_3_new[j],I_i=I_3_new_2[j-n_0],gamma_i=gamma_3_new_2[j-n_0],tau_i=Tau_3_new_2[j-n_0],delta_tau=delta_tau_set,Tau_star=Tau_star_3)
    I_3_new_3[j-n_0]=tau_output$new_I
    Tau_3_new_3[j-n_0]=tau_output$new_tau
    gamma_3_new_3[j-n_0]=tau_output$new_gamma
    
  }
  Theta_1_out=cbind(Theta_1_out,Theta_1_new)
  Theta_2_out=cbind(Theta_2_out,Theta_2_new)
  Theta_3_out=cbind(Theta_3_out,Theta_3_new)
  
  I_1_out=cbind(I_1_out,I_1_new_1,I_1_new_2,I_1_new_3)
  I_2_out=cbind(I_2_out,I_2_new_1,I_2_new_2,I_2_new_3)
  I_3_out=cbind(I_3_out,I_3_new_1,I_3_new_2,I_3_new_3)
  
  gamma_1_out=cbind(gamma_1_out,gamma_1_new_1,gamma_1_new_2,gamma_1_new_3)
  gamma_2_out=cbind(gamma_2_out,gamma_2_new_1,gamma_2_new_2,gamma_2_new_3)
  gamma_3_out=cbind(gamma_3_out,gamma_3_new_1,gamma_3_new_2,gamma_3_new_3)
  
  Tau_1_out=cbind(Tau_1_out,Tau_1_new_1,Tau_1_new_2,Tau_1_new_3)
  Tau_2_out=cbind(Tau_2_out,Tau_2_new_1,Tau_2_new_2,Tau_2_new_3)
  Tau_3_out=cbind(Tau_3_out,Tau_3_new_1,Tau_3_new_2,Tau_3_new_3)
  
  print(i)
}

###########################Initialize parameters from priors for second chain
#Biomarker 1
Pi_1_out_2=rbeta(1,p_1,p_2)
mu_theta_1_out_2=rnorm(1,mean=mu_0_1,sd=sqrt(sigma_0_1_2))
sigma_theta_1_2_out_2=rinvgamma(1,shape=a_theta_1,scale=b_theta_1)
Theta_1_out_2=array(rnorm(N,mean=mu_theta_1_out_2,sd=sigma_theta_1_2_out_2),c(N,1))
I_1_out_2=array(rbinom(3*(N-n_0),size=1,prob=0.5),c(N-n_0,3))
mu_gamma_1_out_2=rnorm(1,mean=mu_1_1,sd=sqrt(sigma_1_1_2))
sigma_gamma_1_2_out_2=rinvgamma(1,shape=a_gamma_1,scale=b_gamma_1)
mu_tau_1_out_2=2
sigma_tau_1_2_out_2=(2)^2
gamma_1_out_2=array(exp(rnorm(3*(N-n_0),mean=mu_gamma_1_out_2,sd=sqrt(sigma_gamma_1_2_out_2))),c(N-n_0,3))
gamma_1_out_2[I_1_out_2[,1]==0,1]=NA
gamma_1_out_2[I_1_out_2[,2]==0,2]=NA
gamma_1_out_2[I_1_out_2[,3]==0,3]=NA
Tau_1_out_2=array(rtruncnorm(n=3*(N-n_0),a=d[D==1]-Tau_star_1,b=d[D==1],mean=d[D==1]-mu_tau_1_out_2,sd=sqrt(sigma_tau_1_2_out_2)),c(N-n_0,3))
Tau_1_out_2[I_1_out_2[,1]==0,1]=NA
Tau_1_out_2[I_1_out_2[,2]==0,2]=NA
Tau_1_out_2[I_1_out_2[,3]==0,3]=NA
sigma_1_2_out_2=1

#Biomarker 2
Pi_2_out_2=rbeta(1,p_1,p_2)
mu_theta_2_out_2=rnorm(1,mean=mu_0_2,sd=sqrt(sigma_0_2_2))
sigma_theta_2_2_out_2=rinvgamma(1,shape=a_theta_2,scale=b_theta_2)
Theta_2_out_2=array(rnorm(N,mean=mu_theta_2_out_2,sd=sigma_theta_2_2_out_2),c(N,1))
I_2_out_2=array(rbinom(3*(N-n_0),size=1,prob=0.5),c(N-n_0,3))
mu_gamma_2_out_2=rnorm(1,mean=mu_1_2,sd=sqrt(sigma_1_2_2))
sigma_gamma_2_2_out_2=rinvgamma(1,shape=a_gamma_2,scale=b_gamma_2)
mu_tau_2_out_2=0.5
sigma_tau_2_2_out_2=1
gamma_2_out_2=array(exp(rnorm(3*(N-n_0),mean=mu_gamma_2_out_2,sd=sqrt(sigma_gamma_2_2_out_2))),c(N-n_0,3))
gamma_2_out_2[I_2_out_2[,1]==0,1]=NA
gamma_2_out_2[I_2_out_2[,2]==0,2]=NA
gamma_2_out_2[I_2_out_2[,3]==0,3]=NA
Tau_2_out_2=array(rtruncnorm(n=3*(N-n_0),a=d[D==1]-Tau_star_1,b=d[D==1],mean=d[D==1]-mu_tau_2_out_2,sd=sqrt(sigma_tau_2_2_out_2)),c(N-n_0,3))
Tau_2_out_2[I_2_out_2[,1]==0,1]=NA
Tau_2_out_2[I_2_out_2[,2]==0,2]=NA
Tau_2_out_2[I_2_out_2[,3]==0,3]=NA
sigma_2_2_out_2=3

#biomarker 3
Pi_3_out_2=rbeta(1,p_1,p_2)
mu_theta_3_out_2=rnorm(1,mean=mu_0_3,sd=sqrt(sigma_0_3_2))
sigma_theta_3_2_out_2=rinvgamma(1,shape=a_theta_3,scale=b_theta_3)
Theta_3_out_2=array(rnorm(N,mean=mu_theta_3_out_2,sd=sigma_theta_3_2_out_2),c(N,1))
I_3_out_2=array(rbinom(3*(N-n_0),size=1,prob=0.5),c(N-n_0,3))
mu_gamma_3_out_2=rnorm(1,mean=mu_1_3,sd=sqrt(sigma_1_3_2))
sigma_gamma_3_2_out_2=rinvgamma(1,shape=a_gamma_3,scale=b_gamma_3)
mu_tau_3_out_2=1
sigma_tau_3_2_out_2=(0.75)^2
gamma_3_out_2=array(exp(rnorm(3*(N-n_0),mean=mu_gamma_3_out_2,sd=sqrt(sigma_gamma_3_2_out_2))),c(N-n_0,3))
gamma_3_out_2[I_3_out_2[,1]==0,1]=NA
gamma_3_out_2[I_3_out_2[,2]==0,2]=NA
gamma_3_out_2[I_3_out_2[,3]==0,3]=NA
Tau_3_out_2=array(rtruncnorm(n=3*(N-n_0),a=d[D==1]-Tau_star_3,b=d[D==1],mean=d[D==1]-mu_tau_3_out_2,sd=sqrt(sigma_tau_3_2_out_2)),c(N-n_0,3))
Tau_3_out_2[I_3_out_2[,1]==0,1]=NA
Tau_3_out_2[I_3_out_2[,2]==0,2]=NA
Tau_3_out_2[I_3_out_2[,3]==0,3]=NA
sigma_3_2_out_2=1

#Update parameters
for(i in 1:S)
{
  #MU_THETA_1
  mu_theta_parameters=posterior_mu_theta(Theta=Theta_1_out_2[,i],sigma_theta_2=sigma_theta_1_2_out_2[i],mu_0=mu_0_1,sigma_0_2=sigma_0_1_2)
  mu_theta_1_out_2=c(mu_theta_1_out_2,rnorm(1,mean=mu_theta_parameters$mu_0_star,sd=sqrt(mu_theta_parameters$sigma_0_star_2)))
  
  #MU_THETA_2
  mu_theta_parameters=posterior_mu_theta(Theta=Theta_2_out_2[,i],sigma_theta_2=sigma_theta_2_2_out_2[i],mu_0=mu_0_2,sigma_0_2=sigma_0_2_2)
  mu_theta_2_out_2=c(mu_theta_2_out_2,rnorm(1,mean=mu_theta_parameters$mu_0_star,sd=sqrt(mu_theta_parameters$sigma_0_star_2)))
  
  #MU_THETA_3
  mu_theta_parameters=posterior_mu_theta(Theta=Theta_3_out_2[,i],sigma_theta_2=sigma_theta_3_2_out_2[i],mu_0=mu_0_3,sigma_0_2=sigma_0_3_2)
  mu_theta_3_out_2=c(mu_theta_3_out_2,rnorm(1,mean=mu_theta_parameters$mu_0_star,sd=sqrt(mu_theta_parameters$sigma_0_star_2)))
  
  #SIGMA_THETA_1
  sigma_theta_parameters=posterior_sigma_theta_2(Theta=Theta_1_out_2[,i],mu_theta=mu_theta_1_out_2[i+1],a_theta=a_theta_1,b_theta=b_theta_1)
  sigma_theta_1_2_out_2=c(sigma_theta_1_2_out_2,rinvgamma(1,shape=sigma_theta_parameters$a_theta_star,scale=sigma_theta_parameters$b_theta_star))
  
  #SIGMA_THETA_2
  sigma_theta_parameters=posterior_sigma_theta_2(Theta=Theta_2_out_2[,i],mu_theta=mu_theta_2_out_2[i+1],a_theta=a_theta_2,b_theta=b_theta_2)
  sigma_theta_2_2_out_2=c(sigma_theta_2_2_out_2,rinvgamma(1,shape=sigma_theta_parameters$a_theta_star,scale=sigma_theta_parameters$b_theta_star))
  
  #SIGMA_THETA_3
  sigma_theta_parameters=posterior_sigma_theta_2(Theta=Theta_3_out_2[,i],mu_theta=mu_theta_3_out_2[i+1],a_theta=a_theta_3,b_theta=b_theta_3)
  sigma_theta_3_2_out_2=c(sigma_theta_3_2_out_2,rinvgamma(1,shape=sigma_theta_parameters$a_theta_star,scale=sigma_theta_parameters$b_theta_star))
  
  #SIGMA_1
  sigma_parameters=posterior_sigma_2(Y=data$Y1,t=data$t,D=D,J=J,Theta=Theta_1_out_2[,i],I=I_1_out_2[,3*i],gamma=gamma_1_out_2[,3*i],Tau=Tau_1_out_2[,3*i])
  sigma_1_2_out_2=c(sigma_1_2_out_2,rinvgamma(1,shape=sigma_parameters$a_sigma,scale=sigma_parameters$b_sigma))
  
  #SIGMA_2
  sigma_parameters=posterior_sigma_2(Y=data$Y2,t=data$t,D=D,J=J,Theta=Theta_2_out_2[,i],I=I_2_out_2[,3*i],gamma=gamma_2_out_2[,3*i],Tau=Tau_2_out_2[,3*i])
  sigma_2_2_out_2=c(sigma_2_2_out_2,rinvgamma(1,shape=sigma_parameters$a_sigma,scale=sigma_parameters$b_sigma))
  
  #SIGMA_3
  sigma_parameters=posterior_sigma_2(Y=data$Y3,t=data$t,D=D,J=J,Theta=Theta_3_out_2[,i],I=I_3_out_2[,3*i],gamma=gamma_3_out_2[,3*i],Tau=Tau_3_out_2[,3*i])
  sigma_3_2_out_2=c(sigma_3_2_out_2,rinvgamma(1,shape=sigma_parameters$a_sigma,scale=sigma_parameters$b_sigma))
  
  #PI_1
  Pi_parameters=posterior_Pi(I=I_1_out_2[,3*i],p_1=p_1,p_2=p_2)
  Pi_1_out_2=c(Pi_1_out_2,rbeta(1,shape1=Pi_parameters$p_1_star,shape2=Pi_parameters$p_2_star))
  
  #PI_2
  Pi_parameters=posterior_Pi(I=I_2_out_2[,3*i],p_1=p_1,p_2=p_2)
  Pi_2_out_2=c(Pi_2_out_2,rbeta(1,shape1=Pi_parameters$p_1_star,shape2=Pi_parameters$p_2_star))
  
  #PI_3
  Pi_parameters=posterior_Pi(I=I_3_out_2[,3*i],p_1=p_1,p_2=p_2)
  Pi_3_out_2=c(Pi_3_out_2,rbeta(1,shape1=Pi_parameters$p_1_star,shape2=Pi_parameters$p_2_star))
  
  #MU_GAMMA_1
  mu_gamma_parameters=posterior_mu_gamma(I=I_1_out_2[,3*i],gamma=gamma_1_out_2[,3*i],sigma_gamma_2=sigma_gamma_1_2_out_2[i],mu_1=mu_1_1,sigma_1_2=sigma_1_1_2)
  mu_gamma_1_out_2=c(mu_gamma_1_out_2,rnorm(1,mean=mu_gamma_parameters$mu_1_star,sd=sqrt(mu_gamma_parameters$sigma_1_star_2)))
  
  #MU_GAMMA_2
  mu_gamma_parameters=posterior_mu_gamma(I=I_2_out_2[,3*i],gamma=gamma_2_out_2[,3*i],sigma_gamma_2=sigma_gamma_2_2_out_2[i],mu_1=mu_1_2,sigma_1_2=sigma_1_2_2)
  mu_gamma_2_out_2=c(mu_gamma_2_out_2,rnorm(1,mean=mu_gamma_parameters$mu_1_star,sd=sqrt(mu_gamma_parameters$sigma_1_star_2)))
  
  #MU_GAMMA_3
  mu_gamma_parameters=posterior_mu_gamma(I=I_3_out_2[,3*i],gamma=gamma_3_out_2[,3*i],sigma_gamma_2=sigma_gamma_3_2_out_2[i],mu_1=mu_1_3,sigma_1_2=sigma_1_3_2)
  mu_gamma_3_out_2=c(mu_gamma_3_out_2,rnorm(1,mean=mu_gamma_parameters$mu_1_star,sd=sqrt(mu_gamma_parameters$sigma_1_star_2)))
  
  #SIGMA_GAMMA_1
  sigma_gamma_parameters=posterior_sigma_gamma_2(I=I_1_out_2[,3*i],gamma=gamma_1_out_2[,3*i],mu_gamma=mu_gamma_1_out_2[i+1],a_gamma=a_gamma_1,b_gamma=b_gamma_1)
  sigma_gamma_1_2_out_2=c(sigma_gamma_1_2_out_2,rinvgamma(1,shape=sigma_gamma_parameters$a_gamma_star,scale=sigma_gamma_parameters$b_gamma_star))
  
  #SIGMA_GAMMA_2
  sigma_gamma_parameters=posterior_sigma_gamma_2(I=I_2_out_2[,3*i],gamma=gamma_2_out_2[,3*i],mu_gamma=mu_gamma_2_out_2[i+1],a_gamma=a_gamma_2,b_gamma=b_gamma_2)
  sigma_gamma_2_2_out_2=c(sigma_gamma_2_2_out_2,rinvgamma(1,shape=sigma_gamma_parameters$a_gamma_star,scale=sigma_gamma_parameters$b_gamma_star))
  
  #SIGMA_GAMMA_3
  sigma_gamma_parameters=posterior_sigma_gamma_2(I=I_3_out_2[,3*i],gamma=gamma_3_out_2[,3*i],mu_gamma=mu_gamma_3_out_2[i+1],a_gamma=a_gamma_3,b_gamma=b_gamma_3)
  sigma_gamma_3_2_out_2=c(sigma_gamma_3_2_out_2,rinvgamma(1,shape=sigma_gamma_parameters$a_gamma_star,scale=sigma_gamma_parameters$b_gamma_star))
  
  #MU_TAU_1
  mu_tau_output=update_mu_tau(Tau=Tau_1_out_2[,3*i],I=I_1_out_2[,3*i],d=d,D=D,mu_tau=mu_tau_1_out_2[i],sigma_tau_2=sigma_tau_1_2_out_2[i],delta_mu_tau=delta_mu_tau_set,mu_2=mu_2_1,sigma_2_2=sigma_2_1_2,Tau_star=Tau_star_1)
  mu_tau_1_out_2=c(mu_tau_1_out_2,mu_tau_output$new_mu_tau)
  
  #MU_TAU_2
  mu_tau_output=update_mu_tau(Tau=Tau_2_out_2[,3*i],I=I_2_out_2[,3*i],d=d,D=D,mu_tau=mu_tau_2_out_2[i],sigma_tau_2=sigma_tau_2_2_out_2[i],delta_mu_tau=delta_mu_tau_set,mu_2=mu_2_2,sigma_2_2=sigma_2_2_2,Tau_star=Tau_star_2)
  mu_tau_2_out_2=c(mu_tau_2_out_2,mu_tau_output$new_mu_tau)
  
  #MU_TAU_3
  mu_tau_output=update_mu_tau(Tau=Tau_3_out_2[,3*i],I=I_3_out_2[,3*i],d=d,D=D,mu_tau=mu_tau_3_out_2[i],sigma_tau_2=sigma_tau_3_2_out_2[i],delta_mu_tau=delta_mu_tau_set,mu_2=mu_2_3,sigma_2_2=sigma_2_3_2,Tau_star=Tau_star_3)
  mu_tau_3_out_2=c(mu_tau_3_out_2,mu_tau_output$new_mu_tau)
  
  #SIGMA_TAU_1
  sigma_tau_output=update_sigma_tau(Tau=Tau_1_out_2[,3*i],I=I_1_out_2[,3*i],d=d,D=D,mu_tau=mu_tau_1_out_2[i+1],sigma_tau_2=sigma_tau_1_2_out_2[i],delta_sigma_tau=delta_sigma_tau_set,a_tau=a_tau_1,b_tau=b_tau_1,Tau_star=Tau_star_1)
  sigma_tau_1_2_out_2=c(sigma_tau_1_2_out_2,sigma_tau_output$new_sigma_tau_2)
  
  #SIGMA_TAU_2
  sigma_tau_output=update_sigma_tau(Tau=Tau_2_out_2[,3*i],I=I_2_out_2[,3*i],d=d,D=D,mu_tau=mu_tau_2_out_2[i+1],sigma_tau_2=sigma_tau_2_2_out_2[i],delta_sigma_tau=delta_sigma_tau_set,a_tau=a_tau_2,b_tau=b_tau_2,Tau_star=Tau_star_2)
  sigma_tau_2_2_out_2=c(sigma_tau_2_2_out_2,sigma_tau_output$new_sigma_tau_2)
  
  #SIGMA_TAU_3
  sigma_tau_output=update_sigma_tau(Tau=Tau_3_out_2[,3*i],I=I_3_out_2[,3*i],d=d,D=D,mu_tau=mu_tau_3_out_2[i+1],sigma_tau_2=sigma_tau_3_2_out_2[i],delta_sigma_tau=delta_sigma_tau_set,a_tau=a_tau_3,b_tau=b_tau_3,Tau_star=Tau_star_3)
  sigma_tau_3_2_out_2=c(sigma_tau_3_2_out_2,sigma_tau_output$new_sigma_tau_2)
  
  Theta_1_new=rep(NA,N)
  Theta_2_new=rep(NA,N)
  Theta_3_new=rep(NA,N)
  
  I_1_new_1=rep(NA,N-n_0)
  I_1_new_2=rep(NA,N-n_0)
  I_1_new_3=rep(NA,N-n_0)
  
  I_2_new_1=rep(NA,N-n_0)
  I_2_new_2=rep(NA,N-n_0)
  I_2_new_3=rep(NA,N-n_0)
  
  I_3_new_1=rep(NA,N-n_0)
  I_3_new_2=rep(NA,N-n_0)
  I_3_new_3=rep(NA,N-n_0)
  
  gamma_1_new_1=rep(NA,N-n_0)
  gamma_1_new_2=rep(NA,N-n_0)
  gamma_1_new_3=rep(NA,N-n_0)
  
  gamma_2_new_1=rep(NA,N-n_0)
  gamma_2_new_2=rep(NA,N-n_0)
  gamma_2_new_3=rep(NA,N-n_0)
  
  gamma_3_new_1=rep(NA,N-n_0)
  gamma_3_new_2=rep(NA,N-n_0)
  gamma_3_new_3=rep(NA,N-n_0)
  
  Tau_1_new_1=rep(NA,N-n_0)
  Tau_1_new_2=rep(NA,N-n_0)
  Tau_1_new_3=rep(NA,N-n_0)
  
  Tau_2_new_1=rep(NA,N-n_0)
  Tau_2_new_2=rep(NA,N-n_0)
  Tau_2_new_3=rep(NA,N-n_0)
  
  Tau_3_new_1=rep(NA,N-n_0)
  Tau_3_new_2=rep(NA,N-n_0)
  Tau_3_new_3=rep(NA,N-n_0)
  
  for(j in 1:n_0)
  {
    #THETA_1_j
    theta_j_parameters=posterior_theta(Y_i=data$Y1[(data$ID==j)],t_i=data$t[(data$ID==j)],D_i=D[j],J_i=J[j],mu_theta=mu_theta_1_out_2[i+1],sigma_theta_2=sigma_theta_1_2_out_2[i+1],sigma_2=sigma_1_2_out_2[i+1],I_i=NA,gamma_i=NA,tau_i=NA)
    Theta_1_new[j]=rnorm(1,mean=theta_j_parameters$mu_theta_star,sd=sqrt(theta_j_parameters$sigma_theta_star_2))
    
    #THETA_2_j
    theta_j_parameters=posterior_theta(Y_i=data$Y2[(data$ID==j)],t_i=data$t[(data$ID==j)],D_i=D[j],J_i=J[j],mu_theta=mu_theta_2_out_2[i+1],sigma_theta_2=sigma_theta_2_2_out_2[i+1],sigma_2=sigma_2_2_out_2[i+1],I_i=NA,gamma_i=NA,tau_i=NA)
    Theta_2_new[j]=rnorm(1,mean=theta_j_parameters$mu_theta_star,sd=sqrt(theta_j_parameters$sigma_theta_star_2))
    
    #THETA_3_j
    theta_j_parameters=posterior_theta(Y_i=data$Y3[(data$ID==j)],t_i=data$t[(data$ID==j)],D_i=D[j],J_i=J[j],mu_theta=mu_theta_3_out_2[i+1],sigma_theta_2=sigma_theta_3_2_out_2[i+1],sigma_2=sigma_3_2_out_2[i+1],I_i=NA,gamma_i=NA,tau_i=NA)
    Theta_3_new[j]=rnorm(1,mean=theta_j_parameters$mu_theta_star,sd=sqrt(theta_j_parameters$sigma_theta_star_2))
  }
  for(j in (n_0+1):N)
  {
    #THETA_1_j
    theta_j_parameters=posterior_theta(Y_i=data$Y1[(data$ID==j)],t_i=data$t[(data$ID==j)],D_i=D[j],J_i=J[j],mu_theta=mu_theta_1_out_2[i+1],sigma_theta_2=sigma_theta_1_2_out_2[i+1],sigma_2=sigma_1_2_out_2[i+1],I_i=I_1_out_2[j-n_0,3*i],gamma_i=gamma_1_out_2[j-n_0,3*i],tau_i=Tau_1_out_2[j-n_0,3*i])
    Theta_1_new[j]=rnorm(1,mean=theta_j_parameters$mu_theta_star,sd=sqrt(theta_j_parameters$sigma_theta_star_2))
    
    #THETA_2_j
    theta_j_parameters=posterior_theta(Y_i=data$Y2[(data$ID==j)],t_i=data$t[(data$ID==j)],D_i=D[j],J_i=J[j],mu_theta=mu_theta_2_out_2[i+1],sigma_theta_2=sigma_theta_2_2_out_2[i+1],sigma_2=sigma_2_2_out_2[i+1],I_i=I_2_out_2[j-n_0,3*i],gamma_i=gamma_2_out_2[j-n_0,3*i],tau_i=Tau_2_out_2[j-n_0,3*i])
    Theta_2_new[j]=rnorm(1,mean=theta_j_parameters$mu_theta_star,sd=sqrt(theta_j_parameters$sigma_theta_star_2))
    
    #THETA_3_j
    theta_j_parameters=posterior_theta(Y_i=data$Y3[(data$ID==j)],t_i=data$t[(data$ID==j)],D_i=D[j],J_i=J[j],mu_theta=mu_theta_3_out_2[i+1],sigma_theta_2=sigma_theta_3_2_out_2[i+1],sigma_2=sigma_3_2_out_2[i+1],I_i=I_3_out_2[j-n_0,3*i],gamma_i=gamma_3_out_2[j-n_0,3*i],tau_i=Tau_3_out_2[j-n_0,3*i])
    Theta_3_new[j]=rnorm(1,mean=theta_j_parameters$mu_theta_star,sd=sqrt(theta_j_parameters$sigma_theta_star_2))
    
    #I_1
    Pi_1_input=Pi_1_out_2[i+1]
    I_output=update_I(Y_i=data$Y1[(data$ID==j)],t_i=data$t[(data$ID==j)],J_i=J[j],d_i=d[j],mu_gamma=mu_gamma_1_out_2[i+1],sigma_gamma_2=sigma_gamma_1_2_out_2[i+1],mu_tau=mu_tau_1_out_2[i+1],sigma_tau_2=sigma_tau_1_2_out_2[i+1],Tau_star=Tau_star_1,sigma_2=sigma_1_2_out_2[i+1],Theta_i=Theta_1_new[j],I_i=I_1_out_2[j-n_0,3*i],gamma_i=gamma_1_out_2[j-n_0,3*i],tau_i=Tau_1_out_2[j-n_0,3*i],Pi=Pi_1_input)
    I_1_new_1[j-n_0]=I_output$new_I
    Tau_1_new_1[j-n_0]=I_output$new_tau
    gamma_1_new_1[j-n_0]=I_output$new_gamma
    
    #I_2
    Pi_2_input=Pi_2_out_2[i+1]
    I_output=update_I(Y_i=data$Y2[(data$ID==j)],t_i=data$t[(data$ID==j)],J_i=J[j],d_i=d[j],mu_gamma=mu_gamma_2_out_2[i+1],sigma_gamma_2=sigma_gamma_2_2_out_2[i+1],mu_tau=mu_tau_2_out_2[i+1],sigma_tau_2=sigma_tau_2_2_out_2[i+1],Tau_star=Tau_star_2,sigma_2=sigma_2_2_out_2[i+1],Theta_i=Theta_2_new[j],I_i=I_2_out_2[j-n_0,3*i],gamma_i=gamma_2_out_2[j-n_0,3*i],tau_i=Tau_2_out_2[j-n_0,3*i],Pi=Pi_2_input)
    I_2_new_1[j-n_0]=I_output$new_I
    Tau_2_new_1[j-n_0]=I_output$new_tau
    gamma_2_new_1[j-n_0]=I_output$new_gamma
    
    #I_3
    Pi_3_input=Pi_3_out_2[i+1]
    I_output=update_I(Y_i=data$Y3[(data$ID==j)],t_i=data$t[(data$ID==j)],J_i=J[j],d_i=d[j],mu_gamma=mu_gamma_3_out_2[i+1],sigma_gamma_2=sigma_gamma_3_2_out_2[i+1],mu_tau=mu_tau_3_out_2[i+1],sigma_tau_2=sigma_tau_3_2_out_2[i+1],Tau_star=Tau_star_3,sigma_2=sigma_3_2_out_2[i+1],Theta_i=Theta_3_new[j],I_i=I_3_out_2[j-n_0,3*i],gamma_i=gamma_3_out_2[j-n_0,3*i],tau_i=Tau_3_out_2[j-n_0,3*i],Pi=Pi_3_input)
    I_3_new_1[j-n_0]=I_output$new_I
    Tau_3_new_1[j-n_0]=I_output$new_tau
    gamma_3_new_1[j-n_0]=I_output$new_gamma
    
    #GAMMA_1
    gamma_output=update_gamma(Y_i=data$Y1[(data$ID==j)],t_i=data$t[(data$ID==j)],J_i=J[j],mu_gamma=mu_gamma_1_out_2[i+1],sigma_gamma_2=sigma_gamma_1_2_out_2[i+1],sigma_2=sigma_1_2_out_2[i+1],Theta_i=Theta_1_new[j],I_i=I_1_new_1[j-n_0],gamma_i=gamma_1_new_1[j-n_0],tau_i=Tau_1_new_1[j-n_0],delta_gamma=delta_gamma_set)
    gamma_1_new_2[j-n_0]=gamma_output$new_gamma
    I_1_new_2[j-n_0]=gamma_output$new_I
    Tau_1_new_2[j-n_0]=gamma_output$new_tau
    
    #GAMMA_2
    gamma_output=update_gamma(Y_i=data$Y2[(data$ID==j)],t_i=data$t[(data$ID==j)],J_i=J[j],mu_gamma=mu_gamma_2_out_2[i+1],sigma_gamma_2=sigma_gamma_2_2_out_2[i+1],sigma_2=sigma_2_2_out_2[i+1],Theta_i=Theta_2_new[j],I_i=I_2_new_1[j-n_0],gamma_i=gamma_2_new_1[j-n_0],tau_i=Tau_2_new_1[j-n_0],delta_gamma=delta_gamma_set)
    gamma_2_new_2[j-n_0]=gamma_output$new_gamma
    I_2_new_2[j-n_0]=gamma_output$new_I
    Tau_2_new_2[j-n_0]=gamma_output$new_tau
    
    #GAMMA_3
    gamma_output=update_gamma(Y_i=data$Y3[(data$ID==j)],t_i=data$t[(data$ID==j)],J_i=J[j],mu_gamma=mu_gamma_3_out_2[i+1],sigma_gamma_2=sigma_gamma_3_2_out_2[i+1],sigma_2=sigma_3_2_out_2[i+1],Theta_i=Theta_3_new[j],I_i=I_3_new_1[j-n_0],gamma_i=gamma_3_new_1[j-n_0],tau_i=Tau_3_new_1[j-n_0],delta_gamma=delta_gamma_set)
    gamma_3_new_2[j-n_0]=gamma_output$new_gamma
    I_3_new_2[j-n_0]=gamma_output$new_I
    Tau_3_new_2[j-n_0]=gamma_output$new_tau
    
    #TAU_1
    tau_output=update_tau(Y_i=data$Y1[(data$ID==j)],t_i=data$t[(data$ID==j)],J_i=J[j],d_i=d[j],sigma_2=sigma_1_2_out_2[i+1],mu_tau=mu_tau_1_out_2[i+1],sigma_tau_2=sigma_tau_1_2_out_2[i+1],Theta_i=Theta_1_new[j],I_i=I_1_new_2[j-n_0],gamma_i=gamma_1_new_2[j-n_0],tau_i=Tau_1_new_2[j-n_0],delta_tau=delta_tau_set,Tau_star=Tau_star_1)
    I_1_new_3[j-n_0]=tau_output$new_I
    Tau_1_new_3[j-n_0]=tau_output$new_tau
    gamma_1_new_3[j-n_0]=tau_output$new_gamma
    
    #TAU_2
    tau_output=update_tau(Y_i=data$Y2[(data$ID==j)],t_i=data$t[(data$ID==j)],J_i=J[j],d_i=d[j],sigma_2=sigma_2_2_out_2[i+1],mu_tau=mu_tau_2_out_2[i+1],sigma_tau_2=sigma_tau_2_2_out_2[i+1],Theta_i=Theta_2_new[j],I_i=I_2_new_2[j-n_0],gamma_i=gamma_2_new_2[j-n_0],tau_i=Tau_2_new_2[j-n_0],delta_tau=delta_tau_set,Tau_star=Tau_star_2)
    I_2_new_3[j-n_0]=tau_output$new_I
    Tau_2_new_3[j-n_0]=tau_output$new_tau
    gamma_2_new_3[j-n_0]=tau_output$new_gamma
    
    #TAU_1
    tau_output=update_tau(Y_i=data$Y3[(data$ID==j)],t_i=data$t[(data$ID==j)],J_i=J[j],d_i=d[j],sigma_2=sigma_3_2_out_2[i+1],mu_tau=mu_tau_3_out_2[i+1],sigma_tau_2=sigma_tau_3_2_out_2[i+1],Theta_i=Theta_3_new[j],I_i=I_3_new_2[j-n_0],gamma_i=gamma_3_new_2[j-n_0],tau_i=Tau_3_new_2[j-n_0],delta_tau=delta_tau_set,Tau_star=Tau_star_3)
    I_3_new_3[j-n_0]=tau_output$new_I
    Tau_3_new_3[j-n_0]=tau_output$new_tau
    gamma_3_new_3[j-n_0]=tau_output$new_gamma
    
  }
  Theta_1_out_2=cbind(Theta_1_out_2,Theta_1_new)
  Theta_2_out_2=cbind(Theta_2_out_2,Theta_2_new)
  Theta_3_out_2=cbind(Theta_3_out_2,Theta_3_new)
  
  I_1_out_2=cbind(I_1_out_2,I_1_new_1,I_1_new_2,I_1_new_3)
  I_2_out_2=cbind(I_2_out_2,I_2_new_1,I_2_new_2,I_2_new_3)
  I_3_out_2=cbind(I_3_out_2,I_3_new_1,I_3_new_2,I_3_new_3)
  
  gamma_1_out_2=cbind(gamma_1_out_2,gamma_1_new_1,gamma_1_new_2,gamma_1_new_3)
  gamma_2_out_2=cbind(gamma_2_out_2,gamma_2_new_1,gamma_2_new_2,gamma_2_new_3)
  gamma_3_out_2=cbind(gamma_3_out_2,gamma_3_new_1,gamma_3_new_2,gamma_3_new_3)
  
  Tau_1_out_2=cbind(Tau_1_out_2,Tau_1_new_1,Tau_1_new_2,Tau_1_new_3)
  Tau_2_out_2=cbind(Tau_2_out_2,Tau_2_new_1,Tau_2_new_2,Tau_2_new_3)
  Tau_3_out_2=cbind(Tau_3_out_2,Tau_3_new_1,Tau_3_new_2,Tau_3_new_3)
  
  print(i)
}

########################################################################################################################################################################################################################
#Post-processing results from training data
########################################################################################################################################################################################################################
#TRACE PLOTS
trace_plot=function(y1,y2,title)
{
  ymin=min(y1,y2)
  ymax=max(y1,y2)
  x=seq(from=1,to=length(y1),by=1)
  plot(x,y1,type="l",xlab="Iteration Number",ylab=title,col="red",ylim=c(ymin,ymax))
  x=seq(from=1,to=length(y2),by=1)
  lines(x,y2,lty=1,col="blue")
}

trace_plot(mu_theta_1_out,mu_theta_1_out_2,"MU_THETA_1")
trace_plot(mu_theta_2_out,mu_theta_2_out_2,"MU_THETA_2")
trace_plot(mu_theta_3_out,mu_theta_3_out_2,"MU_THETA_3")

trace_plot(sigma_theta_1_2_out,sigma_theta_1_2_out_2,"SIGMA_THETA_1_2")
trace_plot(sigma_theta_2_2_out,sigma_theta_2_2_out_2,"SIGMA_THETA_2_2")
trace_plot(sigma_theta_3_2_out,sigma_theta_3_2_out_2,"SIGMA_THETA_3_2")

trace_plot(sigma_1_2_out,sigma_1_2_out_2,"SIGMA_1_2")
trace_plot(sigma_2_2_out,sigma_2_2_out_2,"SIGMA_2_2")
trace_plot(sigma_3_2_out,sigma_3_2_out_2,"SIGMA_3_2")

trace_plot(Pi_1_out,Pi_1_out_2,"PI_1")
trace_plot(Pi_2_out,Pi_2_out_2,"PI_2")
trace_plot(Pi_3_out,Pi_3_out_2,"PI_3")

trace_plot(mu_gamma_1_out,mu_gamma_1_out_2,"MU_GAMMA_1")
trace_plot(mu_gamma_2_out,mu_gamma_2_out_2,"MU_GAMMA_2")
trace_plot(mu_gamma_3_out,mu_gamma_3_out_2,"MU_GAMMA_3")

trace_plot(sigma_gamma_1_2_out,sigma_gamma_1_2_out_2,"SIGMA_GAMMA_1_2")
trace_plot(sigma_gamma_2_2_out,sigma_gamma_2_2_out_2,"SIGMA_GAMMA_2_2")
trace_plot(sigma_gamma_3_2_out,sigma_gamma_3_2_out_2,"SIGMA_GAMMA_3_2")

trace_plot(mu_tau_1_out,mu_tau_1_out_2,"MU_TAU_1")
trace_plot(mu_tau_2_out,mu_tau_2_out_2,"MU_TAU_2")
trace_plot(mu_tau_3_out,mu_tau_3_out_2,"MU_TAU_3")

trace_plot(sigma_tau_1_2_out,sigma_tau_1_2_out_2,"SIGMA_TAU_1_2")
trace_plot(sigma_tau_2_2_out,sigma_tau_2_2_out_2,"SIGMA_TAU_2_2")
trace_plot(sigma_tau_3_2_out,sigma_tau_3_2_out_2,"SIGMA_TAU_3_2")

n_thin=10 #Thin to reduce autocorrelation
B=2000 #burn-in interations
B_star=3*B

mu_theta_1_out_thin=mu_theta_1_out[seq(from=B+1,to=S+1,by=n_thin)]
mu_theta_1_out_2_thin=mu_theta_1_out_2[seq(from=B+1,to=S+1,by=n_thin)]
mu_theta_2_out_thin=mu_theta_2_out[seq(from=B+1,to=S+1,by=n_thin)]
mu_theta_2_out_2_thin=mu_theta_2_out_2[seq(from=B+1,to=S+1,by=n_thin)]
mu_theta_3_out_thin=mu_theta_3_out[seq(from=B+1,to=S+1,by=n_thin)]
mu_theta_3_out_2_thin=mu_theta_3_out_2[seq(from=B+1,to=S+1,by=n_thin)]

sigma_theta_1_2_out_thin=sigma_theta_1_2_out[seq(from=B+1,to=S+1,by=n_thin)]
sigma_theta_1_2_out_2_thin=sigma_theta_1_2_out_2[seq(from=B+1,to=S+1,by=n_thin)]
sigma_theta_2_2_out_thin=sigma_theta_2_2_out[seq(from=B+1,to=S+1,by=n_thin)]
sigma_theta_2_2_out_2_thin=sigma_theta_2_2_out_2[seq(from=B+1,to=S+1,by=n_thin)]
sigma_theta_3_2_out_thin=sigma_theta_3_2_out[seq(from=B+1,to=S+1,by=n_thin)]
sigma_theta_3_2_out_2_thin=sigma_theta_3_2_out_2[seq(from=B+1,to=S+1,by=n_thin)]

sigma_1_2_out_thin=sigma_1_2_out[seq(from=B+1,to=S+1,by=n_thin)]
sigma_1_2_out_2_thin=sigma_1_2_out_2[seq(from=B+1,to=S+1,by=n_thin)]
sigma_2_2_out_thin=sigma_2_2_out[seq(from=B+1,to=S+1,by=n_thin)]
sigma_2_2_out_2_thin=sigma_2_2_out_2[seq(from=B+1,to=S+1,by=n_thin)]
sigma_3_2_out_thin=sigma_3_2_out[seq(from=B+1,to=S+1,by=n_thin)]
sigma_3_2_out_2_thin=sigma_3_2_out_2[seq(from=B+1,to=S+1,by=n_thin)]

Pi_1_out_thin=Pi_1_out[seq(from=B+1,to=S+1,by=n_thin)]
Pi_1_out_2_thin=Pi_1_out_2[seq(from=B+1,to=S+1,by=n_thin)]
Pi_2_out_thin=Pi_2_out[seq(from=B+1,to=S+1,by=n_thin)]
Pi_2_out_2_thin=Pi_2_out_2[seq(from=B+1,to=S+1,by=n_thin)]
Pi_3_out_thin=Pi_3_out[seq(from=B+1,to=S+1,by=n_thin)]
Pi_3_out_2_thin=Pi_3_out_2[seq(from=B+1,to=S+1,by=n_thin)]

mu_gamma_1_out_thin=mu_gamma_1_out[seq(from=B+1,to=S+1,by=n_thin)]
mu_gamma_1_out_2_thin=mu_gamma_1_out_2[seq(from=B+1,to=S+1,by=n_thin)]
mu_gamma_2_out_thin=mu_gamma_2_out[seq(from=B+1,to=S+1,by=n_thin)]
mu_gamma_2_out_2_thin=mu_gamma_2_out_2[seq(from=B+1,to=S+1,by=n_thin)]
mu_gamma_3_out_thin=mu_gamma_3_out[seq(from=B+1,to=S+1,by=n_thin)]
mu_gamma_3_out_2_thin=mu_gamma_3_out_2[seq(from=B+1,to=S+1,by=n_thin)]

sigma_gamma_1_2_out_thin=sigma_gamma_1_2_out[seq(from=B+1,to=S+1,by=n_thin)]
sigma_gamma_1_2_out_2_thin=sigma_gamma_1_2_out_2[seq(from=B+1,to=S+1,by=n_thin)]
sigma_gamma_2_2_out_thin=sigma_gamma_2_2_out[seq(from=B+1,to=S+1,by=n_thin)]
sigma_gamma_2_2_out_2_thin=sigma_gamma_2_2_out_2[seq(from=B+1,to=S+1,by=n_thin)]
sigma_gamma_3_2_out_thin=sigma_gamma_3_2_out[seq(from=B+1,to=S+1,by=n_thin)]
sigma_gamma_3_2_out_2_thin=sigma_gamma_3_2_out_2[seq(from=B+1,to=S+1,by=n_thin)]

mu_tau_1_out_thin=mu_tau_1_out[seq(from=B+1,to=S+1,by=n_thin)]
mu_tau_1_out_2_thin=mu_tau_1_out_2[seq(from=B+1,to=S+1,by=n_thin)]
mu_tau_2_out_thin=mu_tau_2_out[seq(from=B+1,to=S+1,by=n_thin)]
mu_tau_2_out_2_thin=mu_tau_2_out_2[seq(from=B+1,to=S+1,by=n_thin)]
mu_tau_3_out_thin=mu_tau_3_out[seq(from=B+1,to=S+1,by=n_thin)]
mu_tau_3_out_2_thin=mu_tau_3_out_2[seq(from=B+1,to=S+1,by=n_thin)]

sigma_tau_1_2_out_thin=sigma_tau_1_2_out[seq(from=B+1,to=S+1,by=n_thin)]
sigma_tau_1_2_out_2_thin=sigma_tau_1_2_out_2[seq(from=B+1,to=S+1,by=n_thin)]
sigma_tau_2_2_out_thin=sigma_tau_2_2_out[seq(from=B+1,to=S+1,by=n_thin)]
sigma_tau_2_2_out_2_thin=sigma_tau_2_2_out_2[seq(from=B+1,to=S+1,by=n_thin)]
sigma_tau_3_2_out_thin=sigma_tau_3_2_out[seq(from=B+1,to=S+1,by=n_thin)]
sigma_tau_3_2_out_2_thin=sigma_tau_3_2_out_2[seq(from=B+1,to=S+1,by=n_thin)]

#posterior_chain=an array with a column for each chain with burn-in removed
gelman_rubin_statistic=function(posterior_chain)
{
  m_star=ncol(posterior_chain)
  n_star=nrow(posterior_chain)
  m=m_star*2
  n=floor(n_star/2)
  posterior_chain_2=cbind(posterior_chain[1:n,],posterior_chain[(n+1):(2*n),])
  posterior_chain_2_mean=apply(posterior_chain_2,2,mean)
  B_GR=n*var(posterior_chain_2_mean)
  posterior_chain_2_var=apply(posterior_chain_2,2,var)
  W_GR=mean(posterior_chain_2_var)
  R_hat=sqrt(((n-1)*W_GR/n + B_GR/n)/W_GR)
  R_hat
}

summary_stats=NULL
#MU_THETA_1
title="MU_THETA_1"
x1=mu_theta_1_out_thin
x2=mu_theta_1_out_2_thin
acf(x1,lag.max=100,main=paste("Chain 1",title,sep="-"))
acf(x2,lag.max=100,main=paste("Chain 2",title,sep="-"))
x=c(x1,x2)
summary_stats=rbind(summary_stats,cbind(Parameter=title,Mean=mean(x),Median=median(x),Quantile_025=as.numeric(quantile(x,probs=0.025)),Quantile_975=as.numeric(quantile(x,probs=0.975)),Min=min(x),Max=max(x),R_hat=gelman_rubin_statistic(cbind(x1,x2))))

#MU_THETA_2
title="MU_THETA_2"
x1=mu_theta_2_out_thin
x2=mu_theta_2_out_2_thin
acf(x1,lag.max=100,main=paste("Chain 1",title,sep="-"))
acf(x2,lag.max=100,main=paste("Chain 2",title,sep="-"))
x=c(x1,x2)
summary_stats=rbind(summary_stats,cbind(Parameter=title,Mean=mean(x),Median=median(x),Quantile_025=as.numeric(quantile(x,probs=0.025)),Quantile_975=as.numeric(quantile(x,probs=0.975)),Min=min(x),Max=max(x),R_hat=gelman_rubin_statistic(cbind(x1,x2))))

#MU_THETA_3
title="MU_THETA_3"
x1=mu_theta_3_out_thin
x2=mu_theta_3_out_2_thin
acf(x1,lag.max=100,main=paste("Chain 1",title,sep="-"))
acf(x2,lag.max=100,main=paste("Chain 2",title,sep="-"))
x=c(x1,x2)
summary_stats=rbind(summary_stats,cbind(Parameter=title,Mean=mean(x),Median=median(x),Quantile_025=as.numeric(quantile(x,probs=0.025)),Quantile_975=as.numeric(quantile(x,probs=0.975)),Min=min(x),Max=max(x),R_hat=gelman_rubin_statistic(cbind(x1,x2))))

#SIGMA_THETA_1
title="SIGMA_THETA_1_2"
x1=sigma_theta_1_2_out_thin
x2=sigma_theta_1_2_out_2_thin
acf(x1,lag.max=100,main=paste("Chain 1",title,sep="-"))
acf(x2,lag.max=100,main=paste("Chain 2",title,sep="-"))
x=c(x1,x2)
summary_stats=rbind(summary_stats,cbind(Parameter=title,Mean=mean(x),Median=median(x),Quantile_025=as.numeric(quantile(x,probs=0.025)),Quantile_975=as.numeric(quantile(x,probs=0.975)),Min=min(x),Max=max(x),R_hat=gelman_rubin_statistic(cbind(x1,x2))))

#SIGMA_THETA_2
title="SIGMA_THETA_2_2"
x1=sigma_theta_2_2_out_thin
x2=sigma_theta_2_2_out_2_thin
acf(x1,lag.max=100,main=paste("Chain 1",title,sep="-"))
acf(x2,lag.max=100,main=paste("Chain 2",title,sep="-"))
x=c(x1,x2)
summary_stats=rbind(summary_stats,cbind(Parameter=title,Mean=mean(x),Median=median(x),Quantile_025=as.numeric(quantile(x,probs=0.025)),Quantile_975=as.numeric(quantile(x,probs=0.975)),Min=min(x),Max=max(x),R_hat=gelman_rubin_statistic(cbind(x1,x2))))

#SIGMA_THETA_3
title="SIGMA_THETA_3_2"
x1=sigma_theta_3_2_out_thin
x2=sigma_theta_3_2_out_2_thin
acf(x1,lag.max=100,main=paste("Chain 1",title,sep="-"))
acf(x2,lag.max=100,main=paste("Chain 2",title,sep="-"))
x=c(x1,x2)
summary_stats=rbind(summary_stats,cbind(Parameter=title,Mean=mean(x),Median=median(x),Quantile_025=as.numeric(quantile(x,probs=0.025)),Quantile_975=as.numeric(quantile(x,probs=0.975)),Min=min(x),Max=max(x),R_hat=gelman_rubin_statistic(cbind(x1,x2))))

#SIGMA_1
title="SIGMA_1_2"
x1=sigma_1_2_out_thin
x2=sigma_1_2_out_2_thin
acf(x1,lag.max=100,main=paste("Chain 1",title,sep="-"))
acf(x2,lag.max=100,main=paste("Chain 2",title,sep="-"))
x=c(x1,x2)
summary_stats=rbind(summary_stats,cbind(Parameter=title,Mean=mean(x),Median=median(x),Quantile_025=as.numeric(quantile(x,probs=0.025)),Quantile_975=as.numeric(quantile(x,probs=0.975)),Min=min(x),Max=max(x),R_hat=gelman_rubin_statistic(cbind(x1,x2))))

#SIGMA_2
title="SIGMA_2_2"
x1=sigma_2_2_out_thin
x2=sigma_2_2_out_2_thin
acf(x1,lag.max=100,main=paste("Chain 1",title,sep="-"))
acf(x2,lag.max=100,main=paste("Chain 2",title,sep="-"))
x=c(x1,x2)
summary_stats=rbind(summary_stats,cbind(Parameter=title,Mean=mean(x),Median=median(x),Quantile_025=as.numeric(quantile(x,probs=0.025)),Quantile_975=as.numeric(quantile(x,probs=0.975)),Min=min(x),Max=max(x),R_hat=gelman_rubin_statistic(cbind(x1,x2))))

#SIGMA_3
title="SIGMA_3_2"
x1=sigma_3_2_out_thin
x2=sigma_3_2_out_2_thin
acf(x1,lag.max=100,main=paste("Chain 1",title,sep="-"))
acf(x2,lag.max=100,main=paste("Chain 2",title,sep="-"))
x=c(x1,x2)
summary_stats=rbind(summary_stats,cbind(Parameter=title,Mean=mean(x),Median=median(x),Quantile_025=as.numeric(quantile(x,probs=0.025)),Quantile_975=as.numeric(quantile(x,probs=0.975)),Min=min(x),Max=max(x),R_hat=gelman_rubin_statistic(cbind(x1,x2))))

#PI_1
title="PI_1"
x1=Pi_1_out_thin
x2=Pi_1_out_2_thin
acf(x1,lag.max=100,main=paste("Chain 1",title,sep="-"))
acf(x2,lag.max=100,main=paste("Chain 2",title,sep="-"))
x=c(x1,x2)
summary_stats=rbind(summary_stats,cbind(Parameter=title,Mean=mean(x),Median=median(x),Quantile_025=as.numeric(quantile(x,probs=0.025)),Quantile_975=as.numeric(quantile(x,probs=0.975)),Min=min(x),Max=max(x),R_hat=gelman_rubin_statistic(cbind(x1,x2))))

#PI_2
title="PI_2"
x1=Pi_2_out_thin
x2=Pi_2_out_2_thin
acf(x1,lag.max=100,main=paste("Chain 1",title,sep="-"))
acf(x2,lag.max=100,main=paste("Chain 2",title,sep="-"))
x=c(x1,x2)
summary_stats=rbind(summary_stats,cbind(Parameter=title,Mean=mean(x),Median=median(x),Quantile_025=as.numeric(quantile(x,probs=0.025)),Quantile_975=as.numeric(quantile(x,probs=0.975)),Min=min(x),Max=max(x),R_hat=gelman_rubin_statistic(cbind(x1,x2))))

#PI_3
title="PI_3"
x1=Pi_3_out_thin
x2=Pi_3_out_2_thin
acf(x1,lag.max=100,main=paste("Chain 1",title,sep="-"))
acf(x2,lag.max=100,main=paste("Chain 2",title,sep="-"))
x=c(x1,x2)
summary_stats=rbind(summary_stats,cbind(Parameter=title,Mean=mean(x),Median=median(x),Quantile_025=as.numeric(quantile(x,probs=0.025)),Quantile_975=as.numeric(quantile(x,probs=0.975)),Min=min(x),Max=max(x),R_hat=gelman_rubin_statistic(cbind(x1,x2))))

#MU_GAMMA_1
title="MU_GAMMA_1"
x1=mu_gamma_1_out_thin
x2=mu_gamma_1_out_2_thin
acf(x1,lag.max=100,main=paste("Chain 1",title,sep="-"))
acf(x2,lag.max=100,main=paste("Chain 2",title,sep="-"))
x=c(x1,x2)
summary_stats=rbind(summary_stats,cbind(Parameter=title,Mean=mean(x),Median=median(x),Quantile_025=as.numeric(quantile(x,probs=0.025)),Quantile_975=as.numeric(quantile(x,probs=0.975)),Min=min(x),Max=max(x),R_hat=gelman_rubin_statistic(cbind(x1,x2))))

#MU_GAMMA_2
title="MU_GAMMA_2"
x1=mu_gamma_2_out_thin
x2=mu_gamma_2_out_2_thin
acf(x1,lag.max=100,main=paste("Chain 1",title,sep="-"))
acf(x2,lag.max=100,main=paste("Chain 2",title,sep="-"))
x=c(x1,x2)
summary_stats=rbind(summary_stats,cbind(Parameter=title,Mean=mean(x),Median=median(x),Quantile_025=as.numeric(quantile(x,probs=0.025)),Quantile_975=as.numeric(quantile(x,probs=0.975)),Min=min(x),Max=max(x),R_hat=gelman_rubin_statistic(cbind(x1,x2))))

#MU_GAMMA_3
title="MU_GAMMA_3"
x1=mu_gamma_3_out_thin
x2=mu_gamma_3_out_2_thin
acf(x1,lag.max=100,main=paste("Chain 1",title,sep="-"))
acf(x2,lag.max=100,main=paste("Chain 2",title,sep="-"))
x=c(x1,x2)
summary_stats=rbind(summary_stats,cbind(Parameter=title,Mean=mean(x),Median=median(x),Quantile_025=as.numeric(quantile(x,probs=0.025)),Quantile_975=as.numeric(quantile(x,probs=0.975)),Min=min(x),Max=max(x),R_hat=gelman_rubin_statistic(cbind(x1,x2))))

#SIGMA_GAMMA_1
title="SIGMA_GAMMA_1_2"
x1=sigma_gamma_1_2_out_thin
x2=sigma_gamma_1_2_out_2_thin
acf(x1,lag.max=100,main=paste("Chain 1",title,sep="-"))
acf(x2,lag.max=100,main=paste("Chain 2",title,sep="-"))
x=c(x1,x2)
summary_stats=rbind(summary_stats,cbind(Parameter=title,Mean=mean(x),Median=median(x),Quantile_025=as.numeric(quantile(x,probs=0.025)),Quantile_975=as.numeric(quantile(x,probs=0.975)),Min=min(x),Max=max(x),R_hat=gelman_rubin_statistic(cbind(x1,x2))))

#SIGMA_GAMMA_2
title="SIGMA_GAMMA_2_2"
x1=sigma_gamma_2_2_out_thin
x2=sigma_gamma_2_2_out_2_thin
acf(x1,lag.max=100,main=paste("Chain 1",title,sep="-"))
acf(x2,lag.max=100,main=paste("Chain 2",title,sep="-"))
x=c(x1,x2)
summary_stats=rbind(summary_stats,cbind(Parameter=title,Mean=mean(x),Median=median(x),Quantile_025=as.numeric(quantile(x,probs=0.025)),Quantile_975=as.numeric(quantile(x,probs=0.975)),Min=min(x),Max=max(x),R_hat=gelman_rubin_statistic(cbind(x1,x2))))

#SIGMA_GAMMA_3
title="SIGMA_GAMMA_3_2"
x1=sigma_gamma_3_2_out_thin
x2=sigma_gamma_3_2_out_2_thin
acf(x1,lag.max=100,main=paste("Chain 1",title,sep="-"))
acf(x2,lag.max=100,main=paste("Chain 2",title,sep="-"))
x=c(x1,x2)
summary_stats=rbind(summary_stats,cbind(Parameter=title,Mean=mean(x),Median=median(x),Quantile_025=as.numeric(quantile(x,probs=0.025)),Quantile_975=as.numeric(quantile(x,probs=0.975)),Min=min(x),Max=max(x),R_hat=gelman_rubin_statistic(cbind(x1,x2))))

#MU_TAU_1
title="MU_TAU_1"
x1=mu_tau_1_out_thin
x2=mu_tau_1_out_2_thin
acf(x1,lag.max=100,main=paste("Chain 1",title,sep="-"))
acf(x2,lag.max=100,main=paste("Chain 2",title,sep="-"))
x=c(x1,x2)
summary_stats=rbind(summary_stats,cbind(Parameter=title,Mean=mean(x),Median=median(x),Quantile_025=as.numeric(quantile(x,probs=0.025)),Quantile_975=as.numeric(quantile(x,probs=0.975)),Min=min(x),Max=max(x),R_hat=gelman_rubin_statistic(cbind(x1,x2))))

#MU_TAU_2
title="MU_TAU_2"
x1=mu_tau_2_out_thin
x2=mu_tau_2_out_2_thin
acf(x1,lag.max=100,main=paste("Chain 1",title,sep="-"))
acf(x2,lag.max=100,main=paste("Chain 2",title,sep="-"))
x=c(x1,x2)
summary_stats=rbind(summary_stats,cbind(Parameter=title,Mean=mean(x),Median=median(x),Quantile_025=as.numeric(quantile(x,probs=0.025)),Quantile_975=as.numeric(quantile(x,probs=0.975)),Min=min(x),Max=max(x),R_hat=gelman_rubin_statistic(cbind(x1,x2))))

#MU_TAU_3
title="MU_TAU_3"
x1=mu_tau_3_out_thin
x2=mu_tau_3_out_2_thin
acf(x1,lag.max=100,main=paste("Chain 1",title,sep="-"))
acf(x2,lag.max=100,main=paste("Chain 2",title,sep="-"))
x=c(x1,x2)
summary_stats=rbind(summary_stats,cbind(Parameter=title,Mean=mean(x),Median=median(x),Quantile_025=as.numeric(quantile(x,probs=0.025)),Quantile_975=as.numeric(quantile(x,probs=0.975)),Min=min(x),Max=max(x),R_hat=gelman_rubin_statistic(cbind(x1,x2))))

#SIGMA_TAU_1
title="SIGMA_TAU_1_2"
x1=sigma_tau_1_2_out_thin
x2=sigma_tau_1_2_out_2_thin
acf(x1,lag.max=100,main=paste("Chain 1",title,sep="-"))
acf(x2,lag.max=100,main=paste("Chain 2",title,sep="-"))
x=c(x1,x2)
summary_stats=rbind(summary_stats,cbind(Parameter=title,Mean=mean(x),Median=median(x),Quantile_025=as.numeric(quantile(x,probs=0.025)),Quantile_975=as.numeric(quantile(x,probs=0.975)),Min=min(x),Max=max(x),R_hat=gelman_rubin_statistic(cbind(x1,x2))))

#SIGMA_TAU_2
title="SIGMA_TAU_2_2"
x1=sigma_tau_2_2_out_thin
x2=sigma_tau_2_2_out_2_thin
acf(x1,lag.max=100,main=paste("Chain 1",title,sep="-"))
acf(x2,lag.max=100,main=paste("Chain 2",title,sep="-"))
x=c(x1,x2)
summary_stats=rbind(summary_stats,cbind(Parameter=title,Mean=mean(x),Median=median(x),Quantile_025=as.numeric(quantile(x,probs=0.025)),Quantile_975=as.numeric(quantile(x,probs=0.975)),Min=min(x),Max=max(x),R_hat=gelman_rubin_statistic(cbind(x1,x2))))

#SIGMA_TAU_3
title="SIGMA_TAU_3_2"
x1=sigma_tau_3_2_out_thin
x2=sigma_tau_3_2_out_2_thin
x=c(x1,x2)
summary_stats=rbind(summary_stats,cbind(Parameter=title,Mean=mean(x),Median=median(x),Quantile_025=as.numeric(quantile(x,probs=0.025)),Quantile_975=as.numeric(quantile(x,probs=0.975)),Min=min(x),Max=max(x),R_hat=gelman_rubin_statistic(cbind(x1,x2))))

## Output of summary_stats file:
#       Parameter         Mean                 Median               Quantile_025         Quantile_975        Min                  Max                 R_hat              
# [1,] "MU_THETA_1"      "2.42265079761195"   "2.42345971493161"   "2.33733619159105"   "2.5110128154507"   "2.28512225590524"   "2.55473742535874"  "0.997785952065068"
# [2,] "MU_THETA_2"      "3.08665065490421"   "3.08719887005176"   "2.98493674022227"   "3.18791069133147"  "2.91661303170407"   "3.2620384564064"   "1.00252546690478" 
# [3,] "MU_THETA_3"      "2.69165654093256"   "2.69353981894243"   "2.59649693931927"   "2.78830268713814"  "2.53175835604671"   "2.84782832961775"  "0.997062094426461"
# [4,] "SIGMA_THETA_1_2" "0.819779521313603"  "0.818459246303734"  "0.69388772000957"   "0.951902791907306" "0.64659717172341"   "1.01492019554871"  "0.999316043270197"
# [5,] "SIGMA_THETA_2_2" "0.802418035227214"  "0.797340470974681"  "0.662568168839702"  "0.983398102999743" "0.605136247628547"  "1.08934076036616"  "0.99703916484065" 
# [6,] "SIGMA_THETA_3_2" "0.742800074242186"  "0.742506076485214"  "0.621985475854145"  "0.875921838384537" "0.542371721806151"  "1.03876756327928"  "0.996965943284866"
# [7,] "SIGMA_1_2"       "0.231043765562215"  "0.230906423214962"  "0.216300709463928"  "0.245487368816233" "0.209599851744941"  "0.256635780332906" "0.997358272877599"
# [8,] "SIGMA_2_2"       "1.41201533839392"   "1.41141540212318"   "1.3266106775151"    "1.50375279426892"  "1.26220008985944"   "1.55344116301407"  "0.999474171776806"
# [9,] "SIGMA_3_2"       "0.821414769404232"  "0.821383606867718"  "0.769438690321111"  "0.875009094944501" "0.734458125722479"  "0.917813522765405" "0.999215803323573"
# [10,] "PI_1"            "0.484918853764735"  "0.48368990029278"   "0.388728297625293"  "0.582577378264063" "0.305981660631197"  "0.656038738852431" "0.997350540239281"
# [11,] "PI_2"            "0.487377475572537"  "0.486020444099123"  "0.379347061114915"  "0.596357097680914" "0.291035330154826"  "0.66166134893436"  "1.00033572265716" 
# [12,] "PI_3"            "0.479999167303335"  "0.476400457261915"  "0.370818425302909"  "0.592846946854756" "0.300372980307124"  "0.624281017037842" "0.997439435351002"
# [13,] "MU_GAMMA_1"      "2.39912892675307"   "2.40329798169505"   "2.04316181314241"   "2.72460349307235"  "1.87564331777991"   "2.88095619681287"  "1.08294162016205" 
# [14,] "MU_GAMMA_2"      "1.97929683603369"   "1.98064976379184"   "1.80571949234192"   "2.15478710272909"  "1.71433148895589"   "2.23857064513538"  "1.04640578536033" 
# [15,] "MU_GAMMA_3"      "1.86998208954534"   "1.86983267907192"   "1.55517096338649"   "2.25997192021376"  "1.2703630650998"    "2.36718382194828"  "1.03084258780834" 
# [16,] "SIGMA_GAMMA_1_2" "0.607651333477097"  "0.56204487303261"   "0.306080699421223"  "1.07175629627968"  "0.190802597547788"  "3.02095875882616"  "1.03111697980369" 
# [17,] "SIGMA_GAMMA_2_2" "0.0376222320042436" "0.030955316095767"  "0.01189513143154"   "0.10527827192307"  "0.0074231133241828" "0.279626350958302" "1.00032371773791" 
# [18,] "SIGMA_GAMMA_3_2" "0.0474296559860711" "0.0347172487947403" "0.0132380188323077" "0.151695890581098" "0.0106073810551432" "0.23831491745329"  "1.02121340967292" 
# [19,] "MU_TAU_1"        "0.993177416179372"  "0.991171668796741"  "0.516741765537291"  "1.47739411017481"  "0.0727012120609896" "1.95456195923995"  "1.03017671703281" 
# [20,] "MU_TAU_2"        "0.821454892262101"  "0.812754703689805"  "0.383133546203766"  "1.30463048773079"  "0.133695500577832"  "1.59247649067619"  "1.00352820903992" 
# [21,] "MU_TAU_3"        "0.58573031559857"   "0.58396103555976"   "0.119568643970449"  "1.1029275860521"   "-0.144430115083946" "1.54309613537044"  "0.998192047261044"
# [22,] "SIGMA_TAU_1_2"   "0.717029655657677"  "0.676473946291498"  "0.381394753724617"  "1.39772428393838"  "0.293280727634081"  "1.84399570460075"  "0.997464416766973"
# [23,] "SIGMA_TAU_2_2"   "0.668450765349369"  "0.627112799801704"  "0.355281989787936"  "1.19753192300242"  "0.295543714594954"  "1.52426822596089"  "1.00220518802802" 
# [24,] "SIGMA_TAU_3_2"   "0.552891833342835"  "0.518796394726842"  "0.308981432413263"  "0.958255284758113" "0.250982490731318"  "1.77243323252337"  "0.997720611748452"

#Summary of Gelman-Rubin statistics
sum(summary_stats[,8]>1.01)
summary_stats[summary_stats[,8]>1.01,c(1,8)]
#All less than 1.09

#Extract samples from posterior distributions of hyper parameters to use in implementation of screening in the validation cohort
sigma_1_posterior=c(sigma_1_2_out_thin,sigma_1_2_out_2_thin)
sigma_2_posterior=c(sigma_2_2_out_thin,sigma_2_2_out_2_thin)
sigma_3_posterior=c(sigma_3_2_out_thin,sigma_3_2_out_2_thin)
mu_theta_1_posterior=c(mu_theta_1_out_thin,mu_theta_1_out_2_thin)
mu_theta_2_posterior=c(mu_theta_2_out_thin,mu_theta_2_out_2_thin)
mu_theta_3_posterior=c(mu_theta_3_out_thin,mu_theta_3_out_2_thin)
sigma_theta_1_2_posterior=c(sigma_theta_1_2_out_thin,sigma_theta_1_2_out_2_thin)
sigma_theta_2_2_posterior=c(sigma_theta_2_2_out_thin,sigma_theta_2_2_out_2_thin)
sigma_theta_3_2_posterior=c(sigma_theta_3_2_out_thin,sigma_theta_3_2_out_2_thin)
Pi_1_posterior=c(Pi_1_out_thin,Pi_1_out_2_thin)
Pi_2_posterior=c(Pi_2_out_thin,Pi_2_out_2_thin)
Pi_3_posterior=c(Pi_3_out_thin,Pi_3_out_2_thin)
mu_gamma_1_posterior=c(mu_gamma_1_out_thin,mu_gamma_1_out_2_thin)
mu_gamma_2_posterior=c(mu_gamma_2_out_thin,mu_gamma_2_out_2_thin)
mu_gamma_3_posterior=c(mu_gamma_3_out_thin,mu_gamma_3_out_2_thin)
sigma_gamma_1_2_posterior=c(sigma_gamma_1_2_out_thin,sigma_gamma_1_2_out_2_thin)
sigma_gamma_2_2_posterior=c(sigma_gamma_2_2_out_thin,sigma_gamma_2_2_out_2_thin)
sigma_gamma_3_2_posterior=c(sigma_gamma_3_2_out_thin,sigma_gamma_3_2_out_2_thin)
mu_tau_1_posterior=c(mu_tau_1_out_thin,mu_tau_1_out_2_thin)
mu_tau_2_posterior=c(mu_tau_2_out_thin,mu_tau_2_out_2_thin)
mu_tau_3_posterior=c(mu_tau_3_out_thin,mu_tau_3_out_2_thin)
sigma_tau_1_2_posterior=c(sigma_tau_1_2_out_thin,sigma_tau_1_2_out_2_thin)
sigma_tau_2_2_posterior=c(sigma_tau_2_2_out_thin,sigma_tau_2_2_out_2_thin)
sigma_tau_3_2_posterior=c(sigma_tau_3_2_out_thin,sigma_tau_3_2_out_2_thin)
S=length(Pi_1_posterior)

########################################################################################################################################################################################################################
#Implement screening in the validation data (pg 11 Supplementary Materials)
########################################################################################################################################################################################################################
#Get sample size 
N=length(unique(data_validation$ID)) #total number of patients
n_0=length(unique(data_validation$ID[(data_validation$D==0)])) #number of control patients

#Extract vectors
d=data_validation$d[(data_validation$obs_number==1)]
D=data_validation$D[(data_validation$obs_number==1)]
J=rle(data_validation$ID)$lengths
subject_ID=unique(data_validation$ID)

#observed HCC diagnosis times in training cohort
d_HCC=sort(unique(data$d[data$D==1]))

p_Y1_noHCC=rep(NA,length(data_validation$ID))
p_Y2_noHCC=rep(NA,length(data_validation$ID))
p_Y3_noHCC=rep(NA,length(data_validation$ID))
p_Y1_HCC=rep(NA,length(data_validation$ID))
p_Y2_HCC=rep(NA,length(data_validation$ID))
p_Y3_HCC=rep(NA,length(data_validation$ID))

l=1
for(i in 1:N)
{
  subject_data=subset(data_validation,data_validation$ID==subject_ID[i]) 
  
  for(j in 1:J[i])
  {    
    temp_Y1=rep(subject_data$Y1[1:j],rep(S,j))
    temp_Y2=rep(subject_data$Y2[1:j],rep(S,j))
    temp_Y3=rep(subject_data$Y3[1:j],rep(S,j))
    temp_t=rep(subject_data$t[1:j],rep(S,j))
    
    #Calculate Pr(Y1|No HCC) for each patient at each screening time
    theta_1_posterior=rnorm(n=S,mean=mu_theta_1_posterior,sd=sqrt(sigma_theta_1_2_posterior))
    
    prob_Y1_noHCC=dnorm(temp_Y1,mean=theta_1_posterior,sd=sqrt(sigma_1_posterior))
    prob_Y1_noHCC_1=array(prob_Y1_noHCC,c(S,j))
    prob_Y1_noHCC_2=apply(prob_Y1_noHCC_1,1,prod,na.rm=TRUE)
    p_Y1_noHCC[l]=mean(prob_Y1_noHCC_2)
    
    #Calculate Pr(Y2|No HCC) for each patient at each screening time
    theta_2_posterior=rnorm(n=S,mean=mu_theta_2_posterior,sd=sqrt(sigma_theta_2_2_posterior))
    
    prob_Y2_noHCC=dnorm(temp_Y2,mean=theta_2_posterior,sd=sqrt(sigma_2_posterior))
    prob_Y2_noHCC_1=array(prob_Y2_noHCC,c(S,j))
    prob_Y2_noHCC_2=apply(prob_Y2_noHCC_1,1,prod,na.rm=TRUE)
    p_Y2_noHCC[l]=mean(prob_Y2_noHCC_2)
    
    #Calculate Pr(Y3|No HCC) for each patient at each screening time
    theta_3_posterior=rnorm(n=S,mean=mu_theta_3_posterior,sd=sqrt(sigma_theta_3_2_posterior))
    
    prob_Y3_noHCC=dnorm(temp_Y3,mean=theta_3_posterior,sd=sqrt(sigma_3_posterior))
    prob_Y3_noHCC_1=array(prob_Y3_noHCC,c(S,j))
    prob_Y3_noHCC_2=apply(prob_Y3_noHCC_1,1,prod,na.rm=TRUE)
    p_Y3_noHCC[l]=mean(prob_Y3_noHCC_2)
    
    #Get draws of vector c(I_1,I_2)
    I_1_posterior=rbinom(n=S,size=1,prob=Pi_1_posterior)
    I_2_posterior=rbinom(n=S,size=1,prob=Pi_2_posterior)
    I_3_posterior=rbinom(n=S,size=1,prob=Pi_3_posterior)
    
    ############ Draw d from empirical distribution ############
    u_temp=runif(S)
    d_draws=quantile(d_HCC,u_temp,type=1)
    
    #Calculate Pr(Y1|HCC) for each patient at each screening time
    theta_1_posterior=rnorm(n=S,mean=mu_theta_1_posterior,sd=sqrt(sigma_theta_1_2_posterior))
    gamma_1_posterior=exp(rnorm(n=S,mean=mu_gamma_1_posterior,sd=sqrt(sigma_gamma_1_2_posterior)))
    Tau_1_posterior=rtruncnorm(n=S,a=(d_draws-Tau_star_1),b=d_draws,mean=d_draws-mu_tau_1_posterior,sd=sqrt(sigma_tau_1_2_posterior))    
    mean_Y1=rep(theta_1_posterior,j) + rep(I_1_posterior,j)*rep(gamma_1_posterior,j)*(temp_t-Tau_1_posterior)*as.numeric(temp_t>Tau_1_posterior)
    
    prob_Y1_HCC=dnorm(temp_Y1,mean=mean_Y1,sd=sqrt(sigma_1_posterior))
    prob_Y1_HCC_1=array(prob_Y1_HCC,c(S,j))
    prob_Y1_HCC_2=apply(prob_Y1_HCC_1,1,prod,na.rm=TRUE)
    p_Y1_HCC[l]=mean(prob_Y1_HCC_2)
    
    #Calculate Pr(Y2|HCC) for each patient at each screening time
    theta_2_posterior=rnorm(n=S,mean=mu_theta_2_posterior,sd=sqrt(sigma_theta_2_2_posterior))
    gamma_2_posterior=exp(rnorm(n=S,mean=mu_gamma_2_posterior,sd=sqrt(sigma_gamma_2_2_posterior)))
    Tau_2_posterior=rtruncnorm(n=S,a=(d_draws-Tau_star_2),b=d_draws,mean=d_draws-mu_tau_2_posterior,sd=sqrt(sigma_tau_2_2_posterior))
    mean_Y2=rep(theta_2_posterior,j) + rep(I_2_posterior,j)*rep(gamma_2_posterior,j)*(temp_t-Tau_2_posterior)*as.numeric(temp_t>Tau_2_posterior)
    
    prob_Y2_HCC=dnorm(temp_Y2,mean=mean_Y2,sd=sqrt(sigma_2_posterior))
    prob_Y2_HCC_1=array(prob_Y2_HCC,c(S,j))
    prob_Y2_HCC_2=apply(prob_Y2_HCC_1,1,prod,na.rm=TRUE)
    p_Y2_HCC[l]=mean(prob_Y2_HCC_2)
    
    #Calculate Pr(Y3|HCC) for each patient at each screening time
    theta_3_posterior=rnorm(n=S,mean=mu_theta_3_posterior,sd=sqrt(sigma_theta_3_2_posterior))
    gamma_3_posterior=exp(rnorm(n=S,mean=mu_gamma_3_posterior,sd=sqrt(sigma_gamma_3_2_posterior)))
    Tau_3_posterior=rtruncnorm(n=S,a=(d_draws-Tau_star_3),b=d_draws,mean=d_draws-mu_tau_3_posterior,sd=sqrt(sigma_tau_3_2_posterior))
    mean_Y3=rep(theta_3_posterior,j) + rep(I_3_posterior,j)*rep(gamma_3_posterior,j)*(temp_t-Tau_3_posterior)*as.numeric(temp_t>Tau_3_posterior)
    
    prob_Y3_HCC=dnorm(temp_Y3,mean=mean_Y3,sd=sqrt(sigma_3_posterior))
    prob_Y3_HCC_1=array(prob_Y3_HCC,c(S,j))
    prob_Y3_HCC_2=apply(prob_Y3_HCC_1,1,prod,na.rm=TRUE)
    p_Y3_HCC[l]=mean(prob_Y3_HCC_2)
    
    l=l+1
  }
  #print(c(i,j,l))
}

#Get results
p_HCC=mean(data$D[(data$obs_number==1)]) #prior probability of HCC in training data
p_noHCC=1-p_HCC
data_validation["posterior_risk_M1"]=(p_Y1_HCC*p_HCC)/(p_Y1_noHCC*p_noHCC)
data_validation["posterior_risk_M2"]=(p_Y2_HCC*p_HCC)/(p_Y2_noHCC*p_noHCC)
data_validation["posterior_risk_M3"]=(p_Y3_HCC*p_HCC)/(p_Y3_noHCC*p_noHCC)

########################################################################################################################################################################################################################
#Estimate ROC(0.1)
########################################################################################################################################################################################################################
specificity_fixed=0.9

cut_off_M1=quantile(data_validation$posterior_risk_M1[(data_validation$D==0)],probs=specificity_fixed,type=1)
cut_off_M2=quantile(data_validation$posterior_risk_M2[(data_validation$D==0)],probs=specificity_fixed,type=1)
cut_off_M3=quantile(data_validation$posterior_risk_M3[(data_validation$D==0)],probs=specificity_fixed,type=1)

at_least_one_positive_M1=rep(NA,N-n_0)
at_least_one_positive_M2=rep(NA,N-n_0)
at_least_one_positive_M3=rep(NA,N-n_0)

for(i in 1:(N-n_0))
{
  subject_data=subset(data_validation,data_validation$ID==subject_ID[i+n_0]) 
  
  at_least_one_positive_M1[i]=as.numeric(sum(subject_data$posterior_risk_M1>cut_off_M1,na.rm=T)>0)
  at_least_one_positive_M2[i]=as.numeric(sum(subject_data$posterior_risk_M2>cut_off_M2,na.rm=T)>0)
  at_least_one_positive_M3[i]=as.numeric(sum(subject_data$posterior_risk_M3>cut_off_M3,na.rm=T)>0)
}

round(mean(at_least_one_positive_M1)*100,2)
#71.43
round(mean(at_least_one_positive_M2)*100,2)
#67.35
round(mean(at_least_one_positive_M3)*100,2)
#63.27