model{
  for (i in 1:p){
    for (j in 1:A){
      for (k in 1:T){
        dxt[i,j,k]~dpois(q[i,j,k]*ext[i,j,k])
        q[i,j,k]<-exp(alpha[j]+beta[i,j] * kappa[i,k])
      }
    }
  }

  for (i in 1:p){
  for (j in 2:T){
    kappa[i,j]<-kappa_rest_mat[i,j-1]
  }

    kappa[i,1]<--sum(kappa_rest_mat[i,])

  }

  for (i in 1:p){
    for (j in 2:(T-1)){
    prior_mean_kappa[i,j]<-eta_mean[j+1]+rho*(kappa_rest_mat[i,j-1]-eta_mean[j])	
    kappa_rest_mat[i,j]~dmnorm(prior_mean_kappa[i,j],i_sigma2_kappa)
    }
    prior_mean_kappa[i,1]<-eta_mean[2]+0 # the rho*(kappa[i,1]-eta_mean[1]) should have been -sum(kappa_rest_mat[i,2:T-1]) but it does not work because the model could not be uniquely defined	
    kappa_rest_mat[i,1]~dmnorm(prior_mean_kappa[i,1],i_sigma2_kappa)
  }

  eta_mean<-c(matrix_kappa_X%*%eta)

  for (i in 1:p){
  for (j in 2:A){
  beta[i,j]<-beta_rest_mat[i,j-1]
  }

  beta[i,1]<-1-sum(beta_rest_mat[i,])

  }

  for (i in 1:p){

    beta_rest_mat[i,1:(A-1)]~dmnorm(prior_mean_beta,prior_prec_beta)

  }
  
  for (j in 1:A){
  alpha[j]~dnorm(0,0.001)
  }

  eta~dmnorm(prior_mean_eta,prior_prec_eta) #supply prior_prec_eta<-solve(matrix(c(400,0,0,2),nrow=2));prior_mean_eta<-c(0,0)
  rho~dnorm(0,1) T(-1,1)
  sigma2_kappa<-1/i_sigma2_kappa
  i_sigma2_kappa~dgamma(1,0.0001)

}