The function returns the posterior mode of the power parameter \(\delta\) in normal linear model. It calculates the log of the posterior density (up to a normalizing constant), and conduct a grid search to find the approximate mode.
ModeDeltaLMNPP(y.Cur, y.Hist, x.Cur = NULL, x.Hist = NULL, npoints = 1000,
prior = list(a = 1.5, b = 0, mu0 = 0, Rinv = matrix(1, nrow = 1),
delta.alpha = 1, delta.beta = 1))a vector of individual level of the response y in current data.
a vector of individual level of the response y in historical data.
a vector or matrix or data frame of covariate observed in the current data. If more than 1 covariate available, the number of rows is equal to the number of observations.
a vector or matrix or data frame of covariate observed in the historical data. If more than 1 covariate available, the number of rows is equal to the number of observations.
is a non-negative integer scalar indicating number of points on a regular spaced grid between [0, 1], where we calculate the log of the posterior and search for the mode.
a list of the hyperparameters in the prior for model parameters \((\beta, \sigma^2)\) and \(\delta\). The form of the prior for model parameter \((\beta, \sigma^2)\) is in the section "Details".
a a positive hyperparameter for prior on model parameters. It is the power \(a\) in formula \((1/\sigma^2)^a\);
See details.
b equals 0 if a flat prior is used for \(\beta\). Equals 1 if a normal prior is used for \(\beta\); See details.
mu0 a vector of the mean for prior \(\beta|\sigma^2\). Only applicable if b = 1.
Rinv inverse of the matrix \(R\). The covariance matrix of the prior for \(\beta|\sigma^2\) is \(\sigma^2 R^{-1}\).
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 \(\delta\).
Zifei Han hanzifei1@gmail.com
If \(b = 1\), prior for \((\beta, \sigma)\) is \((1/\sigma^2)^a * N(mu0, \sigma^2 R^{-1})\), which includes the g-prior. If \(b = 0\), prior for \((\beta, \sigma)\) is \((1/\sigma^2)^a\). The outputs include posteriors of the model parameter(s) and power parameter, acceptance rate when sampling \(\delta\), and the deviance information criteria.
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.
ModeDeltaBerNPP;
ModeDeltaNormalNPP;
ModeDeltaMultinomialNPP;
ModeDeltaNormalNPP