rNormal_Gamma_reg: The Bayesian Normal-Gamma Regression Distribution
In knygren/glmbayes: Bayesian Generalized Linear Models (iid Samples)

Description Usage Arguments Details Value References See Also Examples

rNormal_Gamma_reg is used to generate iid samples from Bayesian linear models with a normal-gamma prior. The model is specified by providing a data vector, a design matrix, and 4 prior constants.

rNormal_Gamma_reg(
  n,
  y,
  x,
  prior_list,
  offset = NULL,
  weights = 1,
  family = gaussian()
)

`n`	number of draws to generate. If `length(n) > 1`, the length is taken to be the number required.
`y`	a vector of observations of length `m`.
`x`	a design matrix of dimension `m * p`.
`prior_list`	a list with the prior parameters (mu, Sigma, shape and rate) for the prior distribution.
`offset`	this can be used to specify an a priori known component to be included in the linear predictor during fitting. This should be `NULL` or a numeric vector of length equal to the number of cases. One or more offset terms can be included in the formula instead or as well, and if more than one is specified their sum is used. See `model.offset`.
`weights`	an optional vector of ‘prior weights’ to be used in the fitting process. Should be NULL or a numeric vector.
`family`	a description of the error distribution and link function to be used in the model. For `glm` this can be a character string naming a family function, a family function or the result of a call to a family function. For `glm.fit` only the third option is supported. (See `family` for details of family functions.)

The rNormal_Gamma_reg function produces iid samples for Bayesian generalized linear models from the gaussian family (identity link) with a conjugate multivariate normal-gamma prior for the regression coefficients and the dispersion (variance). See Raiffa and Schlaifer (1961) for details on conjugate priors.

Core required inputs for the function include the data vector, the design matrix and a prior specification. The function returns the simulated Bayesian coefficients and some associated outputs. The iid samples from the posterior density is genererated using standard simulation procedures for multivariate normal and gamma distributions.

rNormal_Gamma_reg returns a object of class "rglmb". The function summary (i.e., summary.rglmb) can be used to obtain or print a summary of the results. The generic accessor functions coefficients, fitted.values, residuals, and extractAIC can be used to extract various useful features of the value returned by rNormal_Gamma_reg. An object of class "rglmb" is a list containing at least the following components:

`coefficients`	a `n` by `length(mu)` matrix with one sample in each row
`PostMode`	a vector of `length(mu)` with the estimated posterior mode coefficients
`Prior`	A list with two components. The first being the prior mean vector and the second the prior precision matrix
`iters`	an `n` by `1` matrix giving the number of candidates generated before acceptance for each sample.
`famfunc`	an object of class `"famfunc"`
`Envelope`	an object of class `"envelope"`
`dispersion`	an `n` by `1` matrix with simulated values for the dispersion
`loglike`	a `n` by `1` matrix containing the negative loglikelihood for each sample.

Dobson, A. J. (1990) An Introduction to Generalized Linear Models. London: Chapman and Hall.

Hastie, T. J. and Pregibon, D. (1992) Generalized linear models. Chapter 6 of Statistical Models in S eds J. M. Chambers and T. J. Hastie, Wadsworth & Brooks/Cole. McCullagh P. and Nelder, J. A. (1989) Generalized Linear Models. London: Chapman and Hall.

Nygren, K.N. and Nygren, L.M (2006) Likelihood Subgradient Densities. Journal of the American Statistical Association. vol.101, no.475, pp 1144-1156.

Raiffa, Howard and Schlaifer, R (1961) Applied Statistical Decision Theory. Boston: Clinton Press, Inc.

Venables, W. N. and Ripley, B. D. (2002) Modern Applied Statistics with S. New York: Springer.

pfamily for the list of available pfamilies and how they are specified. Each pfamily has a specific assigned simulation function. This particular simulation function is the simulation function for the dNormal_Gamma pfamily.

rglmb and glmb for functions that depend on simulation functions.

Other family simulation functions: rGamma_reg(), rNormal_reg(), rindependent_norm_gamma_reg()

## Annette Dobson (1990) "An Introduction to Generalized Linear Models".
## Page 9: Plant Weight Data.
ctl <- c(4.17,5.58,5.18,6.11,4.50,4.61,5.17,4.53,5.33,5.14)
trt <- c(4.81,4.17,4.41,3.59,5.87,3.83,6.03,4.89,4.32,4.69)
group <- gl(2, 10, 20, labels = c("Ctl","Trt"))
weight <- c(ctl, trt)

ps=Prior_Setup(weight ~ group)
mu=ps$mu
V=ps$Sigma
mu[1,1]=mean(weight)

Prior_Check(weight ~ group,family =gaussian(),
           pfamily=dNormal(mu=mu,Sigma=V))

## Will move this step inside the Prior_Check
lm.D9 <- lm(weight ~ group,x=TRUE,y=TRUE)
disp_ML=sigma(lm.D9)^2
n_prior=2
shape=n_prior/2
rate= disp_ML*shape

lmb.D9=lmb(weight ~ group,dNormal_Gamma(mu,V/disp_ML,shape=shape,rate=rate))
summary(lmb.D9)

n<-10000
y<-lmb.D9$y
x<-as.matrix(lmb.D9$x)
prior_list=list(mu=mu,Sigma=V/disp_ML,shape=shape,rate=rate)
ngamma.D9=rNormal_Gamma_reg(n=1000,y=y,x=x,
  prior_list=prior_list)
summary(ngamma.D9$coefficients)