BerMNPP_MCMC2 {NPP} | R Documentation |
MCMC Sampling for Bernoulli Population of multiple historical data using Normalized Power Prior
Description
Multiple historical data are combined individually.
The NPP of multiple historical data is the product of the NPP of each historical data.
Conduct posterior sampling for Bernoulli population with normalized power prior.
For the power parameter \delta
, a Metropolis-Hastings algorithm with either independence proposal, or a random walk proposal on its logit scale is used.
For the model parameter p
, Gibbs sampling is used.
Usage
BerMNPP_MCMC2(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)
Arguments
n0 |
a non-negative integer vector: number of trials in historical data. |
y0 |
a non-negative integer vector: number 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_p |
a vector of the hyperparameters in the prior distribution |
prior_delta_alpha |
a vector of the hyperparameter |
prior_delta_beta |
a vector of the hyperparameter |
prop_delta_alpha |
a vector of the hyperparameter |
prop_delta_beta |
a vector of the hyperparameter |
delta_ini |
the initial value of |
prop_delta |
the class of proposal distribution for |
rw_delta |
the stepsize (variance of the normal distribution) for the random walk proposal of logit |
nsample |
specifies the number of posterior samples in the output. |
burnin |
the number of burn-ins. The output will only show MCMC samples after burnin. |
thin |
the thinning parameter in MCMC sampling. |
Details
The outputs include posteriors of the model parameter(s) and power parameter, acceptance rate in sampling \delta
.
The normalized power prior distribution is
\pi_0(\delta)\prod_{k=1}^{K}\frac{\pi_0(\theta)L(\theta|D_{0k})^{\delta_{k}}}{\int \pi_0(\theta)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.
Value
A list of class "NPP" with three elements:
acceptrate |
the acceptance rate in MCMC sampling for |
p |
posterior of the model parameter |
delta |
posterior of the power parameter |
Author(s)
Qiang Zhang zqzjf0408@163.com
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
;
BerOMNPP_MCMC1
;
BerOMNPP_MCMC2
Examples
BerMNPP_MCMC2(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)