Multiple ordered historical data are incorporated together. Conduct posterior sampling for Poisson population with normalized power prior. For the power parameter \(\gamma\), a Metropolis-Hastings algorithm with independence proposal is used. For the model parameter \(\lambda\), Gibbs sampling is used.
PoiOMNPP_MCMC1(n0,n,prior_gamma,prior_lambda, gamma_ind_prop,
gamma_ini,nsample,burnin,thin)A list of class "NPP" with three elements:
the acceptance rate in MCMC sampling for \(\gamma\) using Metropolis-Hastings algorithm.
posterior of the model parameter \(\lambda\).
posterior of the power parameter \(\delta\). It is equal to the cumulative sum of \(\gamma\)
a natural number vector : number of successes in historical data.
a natural number : number of successes in the current data.
a vector of the hyperparameters in the prior distribution \(Dirichlet(\alpha_1, \alpha_2, ... ,\alpha_K)\) for \(\gamma\).
a vector of the hyperparameters in the prior distribution \(Gamma(\alpha, \beta)\) for \(\lambda\).
a vector of the hyperparameters in the proposal distribution \(Dirichlet(\alpha_1, \alpha_2, ... ,\alpha_K)\) for \(\gamma\).
the initial value of \(\gamma\) in MCMC sampling.
specifies the number of posterior samples in the output.
the number of burn-ins. The output will only show MCMC samples after bunrin.
the thinning parameter in MCMC sampling.
Qiang Zhang zqzjf0408@163.com
The outputs include posteriors of the model parameter(s) and power parameter, acceptance rate in sampling \(\gamma\). The normalized power prior distribution is $$\frac{\pi_0(\gamma)\pi_0(\lambda)\prod_{k=1}^{K}L(\lambda|D_{0k})^{(\sum_{i=1}^{k}\gamma_i)}}{\int \pi_0(\lambda)\prod_{k=1}^{K}L(\lambda|D_{0k})^{(\sum_{i=1}^{k}\gamma_i)}d\lambda }.$$
Here \(\pi_0(\gamma)\) and \(\pi_0(\lambda)\) are the initial prior distributions of \(\gamma\) and \(\lambda\), respectively. \(L(\lambda|D_{0k})\) is the likelihood function of historical data \(D_{0k}\), and \(\sum_{i=1}^{k}\gamma_i\) is the corresponding power parameter.
Ibrahim, J.G., Chen, M.-H., Gwon, Y. and Chen, F. (2015). The Power Prior: Theory and Applications. Statistics in Medicine 34:3724-3749.
Duan, Y., Ye, K. and Smith, E.P. (2006). Evaluating Water Quality: Using Power Priors to Incorporate Historical Information. Environmetrics 17:95-106.
PoiMNPP_MCMC1;
PoiMNPP_MCMC2;
PoiOMNPP_MCMC2
PoiOMNPP_MCMC1(n0=c(0,3,5),n=3,prior_gamma=c(1/2,1/2,1/2,1/2), prior_lambda=c(1,1/10),
gamma_ind_prop=rep(1,4),gamma_ini=NULL, nsample = 2000, burnin = 500, thin = 2)
Run the code above in your browser using DataLab