PoiOMNPP_MCMC2 {NPP}

MCMC Sampling for Poisson Population of multiple ordered historical data using Normalized Power Prior

Description

Multiple ordered historical data are combined individually. 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.

Usage

PoiOMNPP_MCMC2(n0,n,prior_gamma,prior_lambda, gamma_ind_prop,gamma_ini,
               nsample, burnin, thin)

Arguments

n0

a natural number vector : number of successes in historical data.

n

a natural number : number of successes in the current data.

prior_gamma

a vector of the hyperparameters in the prior distribution Dirichlet(\alpha_1, \alpha_2, ... ,\alpha_K) for \gamma.

prior_lambda

a vector of the hyperparameters in the prior distribution Gamma(\alpha, \beta) for \lambda.

gamma_ind_prop

a vector of the hyperparameters in the proposal distribution Dirichlet(\alpha_1, \alpha_2, ... ,\alpha_K) for \gamma.

gamma_ini

the initial value of \gamma in MCMC sampling.

nsample

specifies the number of posterior samples in the output.

burnin

the number of burn-ins. The output will only show MCMC samples after bunrin.

thin

the thinning parameter in MCMC sampling.

Details

The outputs include posteriors of the model parameter(s) and power parameter, acceptance rate in sampling \gamma. The normalized power prior distribution is

\pi_0(\gamma)\prod_{k=1}^{K}\frac{\pi_0(\lambda)L(\lambda|D_{0k})^{(\sum_{i=1}^{k}\gamma_i)}}{\int \pi_0(\lambda)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.

Value

A list of class "NPP" with three elements:

acceptrate

the acceptance rate in MCMC sampling for \gamma using Metropolis-Hastings algorithm.

lambda

posterior of the model parameter \lambda.

delta

posterior of the power parameter \delta. It is equal to the cumulative sum of \gamma

Author(s)

Qiang Zhang [e-mail]

References

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.

See Also

PoiMNPP_MCMC1; PoiMNPP_MCMC2; PoiOMNPP_MCMC1

Examples

PoiOMNPP_MCMC2(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)