model{
  for (i in 1:p){
    for (j in 1:A){
      for (k in 1:(T+h)){
        dxt[i,j,k]~dnegbin(prob[i,j,k],phi)
        prob[i,j,k]<-phi/(phi+q[i,j,k]*ext[i,j,k])
        log(q[i,j,k])<-alpha[j]+beta[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 (k in 1:h){
    kappa[i,k+T]<-kappa_forecast_rest_mat[i,k]
  }

  }

  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 0 should have been rho*(kappa[i,1]-eta_mean[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)
  }

  for (i in 1:p){
    for (j in 2:h){
    prior_mean_kappa_forecast[i,j]<-eta_mean[T+j]+rho*(kappa_forecast_rest_mat[i,j-1]-eta_mean[T+j-1])	
    kappa_forecast_rest_mat[i,j]~dmnorm(prior_mean_kappa_forecast[i,j],i_sigma2_kappa)
    }
    prior_mean_kappa_forecast[i,1]<-eta_mean[T+1]+rho*(kappa_rest_mat[i,T-1]-eta_mean[T])	
    kappa_forecast_rest_mat[i,1]~dmnorm(prior_mean_kappa_forecast[i,1],i_sigma2_kappa)
  }

  eta_mean<-c(matrix_kappa_X%*%eta)


  for (j in 2:A){
  beta[j]<-beta_rest[j-1]
  }
  beta[1]<-1-sum(beta_rest)

  beta_rest[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)

  phi~dgamma(25,0.05)

}