NPP (version 0.6.0)

BerOMNPP_MCMC2: 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)

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 \(\gamma\)

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.

Author

Qiang Zhang

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.

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)