BerOMNPP_MCMC2: MCMC Sampling for Bernoulli Population of multiple ordered...

In NPP: Normalized Power Prior Bayesian Analysis

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

Description

Multiple ordered historical data are combined individually. Conduct posterior sampling for Bernoulli population with normalized power prior. For the power parameter \gamma, a Metropolis-Hastings algorithm with independence proposal is used. For the model parameter p, Gibbs sampling is used.

Usage

BerOMNPP_MCMC2(n0, y0, n, y, prior_gamma, prior_p, gamma_ind_prop, gamma_ini,
               nsample, burnin, thin, adjust = FALSE)

Arguments

n0	a vector of non-negative integers: numbers of trials in historical data.
y0	a vector of non-negative integers: numbers of successes in historical data.
n	a non-negative integer: number of trials in the current data.
y	a non-negative integer: 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_p	a vector of the hyperparameters in the prior distribution Beta(\alpha, \beta) for p.
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 burn-in.
thin	the thinning parameter in MCMC sampling.
adjust	Whether or not to adjust the parameters of the proposal distribution.

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(\theta)L(\theta|D_{0k})^{(\sum_{i=1}^{k}\gamma_i)}}{\int \pi_0(\theta)L(\theta|D_{0k})^{(\sum_{i=1}^{k}\gamma_i)} d\theta}.

Here \pi_0(\gamma) and \pi_0(\theta) are the initial prior distributions of \gamma and \theta, respectively. L(\theta|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.
p	posterior of the model parameter p.
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

BerMNPP_MCMC1; BerMNPP_MCMC2; BerOMNPP_MCMC1

Examples

BerOMNPP_MCMC2(n0 = c(275, 287), y0 = c(92, 125), n = 39, y = 17, prior_gamma=c(1,1,1)/3,
               prior_p=c(1/2,1/2),  gamma_ind_prop=rep(1,3)/2, gamma_ini=NULL,
               nsample = 2000, burnin = 500, thin = 2, adjust = FALSE)

NPP documentation built on Sept. 18, 2023, 5:18 p.m.