BerMNPP_MCMC1 {NPP} | R Documentation |
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)
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 |
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 bunrin. |
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
\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.
Value
A list of class "NPP" comprising:
acceptrate |
Acceptance rate in MCMC sampling for |
p |
Posterior distribution of the model parameter |
delta |
Posterior distribution 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_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)