NormalNPP_MCMC {NPP}

MCMC Sampling for Normal Population using Normalized Power Prior

Description

Conduct posterior sampling for normal population with normalized power prior. The initial prior \pi(\mu|\sigma^2) is a flat 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 \mu and \sigma^2, Gibbs sampling is used.

Usage

NormalNPP_MCMC(Data.Cur, Data.Hist,
               CompStat = list(n0 = NULL, mean0 = NULL, var0 = NULL,
                               n1 = NULL, mean1 = NULL, var1 = NULL),
               prior = list(a = 1.5, delta.alpha = 1, delta.beta = 1),
               MCMCmethod = 'IND', rw.logit.delta = 0.1,
               ind.delta.alpha= 1, ind.delta.beta= 1, nsample = 5000,
               control.mcmc = list(delta.ini = NULL, burnin = 0, thin = 1))

Arguments

Data.Cur

a vector of individual level current data.

Data.Hist

a vector of individual level historical data.

CompStat

a list of six elements(scalar) that represents the "compatibility(sufficient) statistics" for model parameters. Default is NULL so the fitting will be based on the data. If the CompStat is provided then the inputs in Data.Cur and Data.Hist will be ignored.

n0 is the sample size of historical data.

mean0 is the sample mean of the historical data.

var0 is the sample variance of the historical data.

n1 is the sample size of current data.

mean1 is the sample mean of the current data.

var1 is the sample variance of the current data.

prior

a list of the hyperparameters in the prior for both (\mu, \sigma^2) and \delta. The form of the prior for model parameter (\mu, \sigma^2) is (1/\sigma^2)^a. When a = 1 it corresponds to the reference prior, and when a = 1.5 it corresponds to the Jeffrey's prior.

a is the power a in formula (1/\sigma^2)^a, the prior for (\mu, \sigma^2) jointly.

delta.alpha is the hyperparameter \alpha in the prior distribution Beta(\alpha, \beta) for \delta.

delta.beta is the hyperparameter \beta in the prior distribution Beta(\alpha, \beta) for \delta.

MCMCmethod

sampling method for \delta in MCMC. It can be either 'IND' for independence proposal; or 'RW' for random walk proposal on logit scale.

rw.logit.delta

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

ind.delta.alpha

specifies the first parameter \alpha when independent proposal Beta(\alpha, \beta) for \delta is used. Only applicable if MCMCmethod = 'IND'

ind.delta.beta

specifies the first parameter \beta when independent proposal Beta(\alpha, \beta) for \delta is used. Only applicable if MCMCmethod = 'IND'

nsample

specifies the number of posterior samples in the output.

control.mcmc

a list of three elements used in posterior sampling.

delta.ini is the initial value of \delta in MCMC sampling.

burnin is the number of burn-ins. The output will only show MCMC samples after bunrin.

thin is the thinning parameter in MCMC sampling.

Details

The outputs include posteriors of the model parameter(s) and power parameter, acceptance rate in sampling \delta, and the deviance information criteria.

Value

A list of class "NPP" with five elements:

mu

posterior of the model parameter \mu.

sigmasq

posterior of the model parameter \sigma^2.

delta

posterior of the power parameter \delta.

acceptance

the acceptance rate in MCMC sampling for \delta using Metropolis-Hastings algorithm.

DIC

the deviance information criteria for model diagnostics.

Author(s)

Zifei Han [e-mail]

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.

Berger, J.O. and Bernardo, J.M. (1992). On the development of reference priors. Bayesian Statistics 4: Proceedings of the Fourth Valencia International Meeting, Bernardo, J.M, Berger, J.O., Dawid, A.P. and Smith, A.F.M. eds., 35-60, Clarendon Press:Oxford.

Jeffreys, H. (1946). An Invariant Form for the Prior Probability in Estimation Problems. Proceedings of the Royal Statistical Society of London, Series A 186:453-461.

See Also

BerNPP_MCMC; MultinomialNPP_MCMC; PoissonNPP_MCMC;

Examples

set.seed(1234)
NormalData0 <- rnorm(n = 100, mean= 20, sd = 1)

set.seed(12345)
NormalData1 <- rnorm(n = 50, mean= 30, sd = 1)

NormalNPP_MCMC(Data.Cur = NormalData1, Data.Hist = NormalData0,
               CompStat = list(n0 = 100, mean0 = 10, var0 = 1,
               n1 = 100, mean1 = 10, var1 = 1),
               prior = list(a = 1.5, delta.alpha = 1, delta.beta = 1),
               MCMCmethod = 'RW', rw.logit.delta = 1,
               ind.delta.alpha= 1, ind.delta.beta= 1, nsample = 10000,
               control.mcmc = list(delta.ini = NULL, burnin = 0, thin = 1))