Multiple ordered historical data are incorporated together. 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.
BerOMNPP_MCMC1(n0, y0, n, y, prior_gamma, prior_p, gamma_ind_prop,
gamma_ini, nsample, burnin, thin, adjust = FALSE)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 \(p\).
posterior of the power parameter \(\delta\). It is equal to the cumulative sum of \(\gamma\).
a non-negative integer vector: number of trials in historical data.
a non-negative integer vector: number of successes in historical data.
a non-negative integer: number of trials in the current data.
a non-negative integer: 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 \(Beta(\alpha, \beta)\) for \(p\).
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 burnin.
the thinning parameter in MCMC sampling.
Logical, indicating whether or not to adjust the parameters of the proposal distribution.
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 given by: $$\frac{\pi_0(\gamma)\pi_0(\theta)\prod_{k=1}^{K}L(\theta|D_{0k})^{(\sum_{i=1}^{k}\gamma_i)}}{\int \pi_0(\theta)\prod_{k=1}^{K}L(\theta|D_{0k})^{(\sum_{i=1}^{k}\gamma_i)}d\theta}.$$ 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.
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.
BerMNPP_MCMC1,
BerMNPP_MCMC2,
BerOMNPP_MCMC2
BerOMNPP_MCMC1(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)
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