NPP (version 0.6.0)

BerMNPP_MCMC1: MCMC Sampling for Bernoulli Population with Multiple Historical Data using Normalized Power Prior

Description

Incorporate multiple historical data sets for posterior sampling of a Bernoulli population using the normalized power prior. The Metropolis-Hastings algorithm, with either an independence proposal or a random walk proposal on the logit scale, is applied for the power parameter \(\delta\). Gibbs sampling is utilized for the model parameter \(p\).

Usage

BerMNPP_MCMC1(n0, y0, n, y, prior_p, prior_delta_alpha,
                prior_delta_beta, prop_delta_alpha, prop_delta_beta,
                delta_ini, prop_delta, rw_delta, nsample, burnin, thin)

Value

A list of class "NPP" comprising:

acceptrate

Acceptance rate in MCMC sampling for \(\delta\) via the Metropolis-Hastings algorithm.

p

Posterior distribution of the model parameter \(p\).

delta

Posterior distribution of the power parameter \(\delta\).

Arguments

n0

A non-negative integer vector representing the number of trials in historical data.

y0

A non-negative integer vector denoting the number of successes in historical data.

n

A non-negative integer indicating the number of trials in the current data.

y

A non-negative integer for the number of successes in the current data.

prior_p

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

prior_delta_alpha

a vector of the hyperparameter \(\alpha\) in the prior distribution \(Beta(\alpha, \beta)\) for each \(\delta\).

prior_delta_beta

a vector of the hyperparameter \(\beta\) in the prior distribution \(Beta(\alpha, \beta)\) for each \(\delta\).

prop_delta_alpha

a vector of the hyperparameter \(\alpha\) in the proposal distribution \(Beta(\alpha, \beta)\) for each \(\delta\).

prop_delta_beta

a vector of the hyperparameter \(\beta\) in the proposal distribution \(Beta(\alpha, \beta)\) for each \(\delta\).

delta_ini

the initial value of \(\delta\) in MCMC sampling.

prop_delta

the class of proposal distribution for \(\delta\).

rw_delta

the stepsize(variance of the normal distribution) for the random walk proposal of logit \(\delta\). Only applicable if prop_delta = 'RW'.

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.

Author

Qiang Zhang

Details

The outputs include posteriors of the model parameter(s) and power parameter, acceptance rate in sampling \(\delta\). The normalized power prior distribution is $$\frac{\pi_0(\delta)\pi_0(\theta)\prod_{k=1}^{K}L(\theta|D_{0k})^{\delta_{k}}}{\int \pi_0(\theta)\prod_{k=1}^{K}L(\theta|D_{0k})^{\delta_{k}} d\theta}.$$

Here \(\pi_0(\delta)\) and \(\pi_0(\theta)\) are the initial prior distributions of \(\delta\) and \(\theta\), respectively. \(L(\theta|D_{0k})\) is the likelihood function of historical data \(D_{0k}\), and \(\delta_k\) 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_MCMC2; BerOMNPP_MCMC1; BerOMNPP_MCMC2

Examples

BerMNPP_MCMC1(n0 = c(275, 287), y0 = c(92, 125), n = 39, y = 17,
              prior_p = c(1/2,1/2), prior_delta_alpha = c(1/2,1/2),
              prior_delta_beta = c(1/2,1/2),
              prop_delta_alpha = c(1,1)/2, prop_delta_beta = c(1,1)/2,
              delta_ini = NULL, prop_delta = "IND",
              nsample = 2000, burnin = 500, thin = 2)