R/nLinearReg.R
In erlisR/iLaplaceExamples: Extra Utilities and Examples for iLaplace
##' @name nLinearReg
##'
##' @title Posterior distribution (and related quantities) for two nonlinear regression models
##' @description Posterior distribution, and related quantities, for two non linear regression models each based on a different dataset. For each dataset, two error distributions are considered: normal and Student's \eqn{t}. The functions ending in "lub" refer to the \code{\link[iLaplaceExamples]{Lubricant}} dataset, those with ending "bod2" refer to the \code{\link[iLaplaceExamples]{BOD2}} dataset. The functions with an inner \code{T} refer to the model based on Student's \eqn{t}-error distribution. The functions starting with \code{nlpost} give the negative log-posterior density, those starting with \code{grad} give the gradient of \code{nlpost} and the functions starting with \code{hess} provide the Hessian matrix.
##'
##' @aliases nlpostT_lub, grad_lub, gradT_lub, hess_lub, hessT_lub, nlpostT_bod2, grad_bod2, gradT_bod2, hess_bod2, hessT_bod2
##'
##' @param beta The vector of regression parameters. For the Lubricant dataset beta is a 9-variate vector, for the BOD2 data beta is a bivariate vector
##' @param lsig The logarithm of the scale parameter
##' @param lnu For the models with Student's \eqn{t}-errors only: the logarithm of the degrees of freedom
##' @param y The response vector
##' @param x1 For Lubricant data only: the temperature of the lubricant
##' @param x2 For Lubricant data only: the pressure of the lubricant
##' @param Time For BOD2 data only: the time
##' @param n The sample size
##' @param muBeta The center of the prior for beta. See Details
##' @param SigBeta The scale matrix of the prior for beta
##' @param sigScale For the models with Student's \eqn{t}-errors only: the value of the scale of the prior for the scale parameter.
##'
##'
##' @return double
##' @export
nlpost_lub <- function(beta, lsig, y, x1, x2, n, muBeta, SigBeta, sigScale) {
.Call('iLaplaceExamples_nlpost_lub', PACKAGE = 'iLaplaceExamples', beta, lsig, y, x1, x2, n, muBeta, SigBeta, sigScale)
}
##' @return double
##' @rdname nLinearReg
##' @export
nlpostT_lub <- function(beta, lsig, lnu, y, x1, x2, n, muBeta, SigBeta, sigScale) {
.Call('iLaplaceExamples_nlpostT_lub', PACKAGE = 'iLaplaceExamples', beta, lsig, lnu, y, x1, x2, n, muBeta, SigBeta, sigScale)
}
##' @return 10-variate vector
##' @rdname nLinearReg
##' @export
grad_lub <- function(beta, lsig, y, x1, x2, n, muBeta, SigBeta, sigScale) {
.Call('iLaplaceExamples_grad_lub', PACKAGE = 'iLaplaceExamples', beta, lsig, y, x1, x2, n, muBeta, SigBeta, sigScale)
}
##' @return 11-variate vector
##' @rdname nLinearReg
##' @export
gradT_lub <- function(beta, lsig, lnu, y, x1, x2, n, muBeta, SigBeta, sigScale) {
.Call('iLaplaceExamples_gradT_lub', PACKAGE = 'iLaplaceExamples', beta, lsig, lnu, y, x1, x2, n, muBeta, SigBeta, sigScale)
}
##' @return \eqn{10\times 10}{10 by 10} matrix
##' @rdname nLinearReg
##' @export
hess_lub <- function(beta, lsig, y, x1, x2, n, muBeta, SigBeta, sigScale) {
.Call('iLaplaceExamples_hess_lub', PACKAGE = 'iLaplaceExamples', beta, lsig, y, x1, x2, n, muBeta, SigBeta, sigScale)
}
##' @return \eqn{11\times 11}{11 by 11} matrix
##' @rdname nLinearReg
##' @export
hessT_lub <- function(beta, lsig, lnu, y, x1, x2, n, muBeta, SigBeta, sigScale) {
.Call('iLaplaceExamples_hessT_lub', PACKAGE = 'iLaplaceExamples', beta, lsig, lnu, y, x1, x2, n, muBeta, SigBeta, sigScale)
}
##' @return double
##' @rdname nLinearReg
##' @export
nlpost_bod2 <- function(beta, lsig, y, Time, n, muBeta, SigBeta, sigScale) {
.Call('iLaplaceExamples_nlpost_bod2', PACKAGE = 'iLaplaceExamples', beta, lsig, y, Time, n, muBeta, SigBeta, sigScale)
}
##' @return double
##' @rdname nLinearReg
##' @export
nlpostT_bod2 <- function(beta, lsig, lnu, y, Time, n, muBeta, SigBeta, sigScale) {
.Call('iLaplaceExamples_nlpostT_bod2', PACKAGE = 'iLaplaceExamples', beta, lsig, lnu, y, Time, n, muBeta, SigBeta, sigScale)
}
##' @return 3-variate vector
##' @rdname nLinearReg
##' @export
grad_bod2 <- function(beta, lsig, y, Time, n, muBeta, SigBeta, sigScale) {
.Call('iLaplaceExamples_grad_bod2', PACKAGE = 'iLaplaceExamples', beta, lsig, y, Time, n, muBeta, SigBeta, sigScale)
}
##' @return 4-variate vector
##' @rdname nLinearReg
##' @export
gradT_bod2 <- function(beta, lsig, lnu, y, Time, n, muBeta, SigBeta, sigScale) {
.Call('iLaplaceExamples_gradT_bod2', PACKAGE = 'iLaplaceExamples', beta, lsig, lnu, y, Time, n, muBeta, SigBeta, sigScale)
}
##' @return \eqn{3\times 3}{3 by 3} matrix
##' @rdname nLinearReg
##' @export
hess_bod2 <- function(beta, lsig, y, Time, n, muBeta, SigBeta, sigScale) {
.Call('iLaplaceExamples_hess_bod2', PACKAGE = 'iLaplaceExamples', beta, lsig, y, Time, n, muBeta, SigBeta, sigScale)
}
##' @examples
##' \dontrun{
##'
##' library(iLaplace)
##'
##' ## regression models with BOD2 data
##' # set the data
##' data(BOD2)
##' y = BOD2$demand
##' Time = BOD2$Time
##' n = length(y)
##' muBeta = rep(0, 2)
##' SigBeta = matrix(0, 2,2)
##' diag(SigBeta) = 10
##' sigScale = 10
##' dfprop = 3
##'
##' ## Model with normal errors
##'
##' # the objective function and derivatives
##' ff = function(pp, ...) nlpost_bod2(beta = pp[1:2], lsig = pp[3], ...)
##' ff.gr = function(pp, ...) grad_bod2(beta = pp[1:2], lsig = pp[3], ...)
##' ff.hess = function(pp, ...) hess_bod2(beta = pp[1:2], lsig = pp[3], ...)
##'
##' # find the mode
##' init <- c(5, 2, 0.5)
##' opt.post = nlminb(init, obj = ff, gradient = ff.gr, hessian = ff.hess,
##' y = y, Time = Time, n = n, muBeta = muBeta,
##' SigBeta = SigBeta, sigScale = sigScale, control = list(trace = 1))
##' opt.post$hessian = ff.hess(opt.post$par, y = y, Time = Time, n = n, muBeta = muBeta,
##' SigBeta = SigBeta, sigScale = sigScale)
##'
##' # simulate from the posterior
##' mcmc.post = MHmcmc(logfun = function(x) -ff(x, y = y, Time = Time, n = n, muBeta = muBeta,
##' SigBeta = SigBeta, sigScale = sigScale),
##' burnin = 20000, mcmc = 1e+6,
##' thin = 5, tune = 0.98, V = solve(opt.post$hessian),
##' df = dfprop, theta.init = init, verbose = 50000)
##'
##'
##' # marginal likelihood by Chib's method
##' logM.Chib <- ChibML(logfun = function(x) -ff(x, y = y, Time = Time, n = n,
##' muBeta = muBeta, SigBeta = SigBeta,
##' sigScale = sigScale),
##' theta.star = opt.post$par,
##' tune = 1.5, V = solve(opt.post$hessian), t(mcmc.post),
##' df = dfprop, verbose = 100000)
##'
##' # marginal likelihood by the improved Laplace
##' logM.iLap <- iLaplace(fullOpt = opt.post, ff = ff, ff.gr = ff.gr, ff.hess = ff.hess,
##' control = list(sp.points = 100, delta = 13, n.cores = 3),
##' clEvalQ = c("iLaplaceExamples"), y = y, Time = Time, n = n,
##' muBeta = muBeta, SigBeta = SigBeta,
##' sigScale = sigScale)
##'
##' logM.Lap <- 3*log(2*pi)/2 - opt.post$objective - 0.5*determinant(opt.post$hessian)$mod
##' # marginal likelihood by importance samling
##' logM.IS <- isML(logfun = function(x) -ff(x, y = y, Time = Time, n = n,
##' muBeta = muBeta, SigBeta = SigBeta,
##' sigScale = sigScale), nsim = 1e+6,
##' theta.hat = opt.post$par, tune = 1.5,
##' V = solve(opt.post$hessian),df = dfprop, verbose = 1000)
##' logM.iLap
##' logM.Lap
##' logM.IS
##' logM.Chib
##'
##' ## Model with Student's t errors
##'
##' # the objective function and derivatives
##' tff = function(pp, ...) nlpostT_bod2(beta = pp[1:2], lsig = pp[3], lnu = pp[4], ...)
##' tff.gr = function(pp, ...) gradT_bod2(beta = pp[1:2], lsig = pp[3], lnu = pp[4], ...)
##' tff.hess = function(pp, ...) hessT_bod2(beta = pp[1:2], lsig = pp[3], lnu = pp[4], ...)
##'
##' # find the mode
##' init <- c(2, 5, 0.5, 0)
##' opt.postt = nlminb(init, obj = tff, gradient = tff.gr, hessian = tff.hess,
##' y = y, Time = Time, n = n, muBeta = muBeta,
##' SigBeta = SigBeta, sigScale = sigScale, control = list(trace = 1))
##' opt.postt$hessian = tff.hess(opt.postt$par, y = y, Time = Time, n = n, muBeta = muBeta,
##' SigBeta = SigBeta, sigScale = sigScale)
##'
##' # simulate from the posterior
##' mcmc.postt = MHmcmc(logfun = function(x) -tff(x, y = y, Time = Time, n = n, muBeta = muBeta,
##' SigBeta = SigBeta, sigScale = sigScale),
##' burnin = 20000, mcmc = 1e+6,
##' thin = 5, tune = 1.2, V = solve(opt.postt$hessian),
##' df = dfprop, theta.init = opt.postt$par, verbose = 50000)
##'
##' logM.Lapt <- 4*log(2*pi)/2 - opt.postt$objective - 0.5*determinant(opt.postt$hessian)$mod
##'
##' # marginal likelihood by Chib's method
##' logM.Chibt <- ChibML(logfun = function(x) -tff(x, y = y, Time = Time, n = n,
##' muBeta = muBeta, SigBeta = SigBeta,
##' sigScale = sigScale),
##' theta.star = opt.postt$par,
##' tune = 1.5, V = solve(opt.postt$hessian), t(mcmc.postt),
##' df = dfprop, verbose = 100000)
##'
##' logM.iLapt <- iLaplace(fullOpt = opt.postt, ff = tff, ff.gr = tff.gr, ff.hess = tff.hess,
##' control = list(sp.points = 100, delta = 9, n.cores = 4),
##' clEvalQ = c("iLaplaceExamples"), y = y, Time = Time, n = n,
##' muBeta = muBeta, SigBeta = SigBeta,
##' sigScale = sigScale)
##'
##' logM.ISt <- isML(logfun = function(x) -tff(x, y = y, Time = Time, n = n,
##' muBeta = muBeta, SigBeta = SigBeta,
##' sigScale = sigScale), nsim = 1e+6,
##' theta.hat = opt.postt$par, tune = 1.5,
##' V = solve(opt.postt$hessian),df = dfprop, verbose = 10000)
##'
##' logM.iLapt
##' logM.Lapt
##' logM.ISt
##' logM.Chibt
##' }
##'
##'
##' @seealso \code{\link[iLaplaceExamples]{MHmcmc}}
##'
##' @details The prior for \code{beta} is the multivariate Studnet's \eqn{t}{t} with the appropriate dimension, with location \code{muBeta}, scale matrix \code{SigBeta} and degrees of freedom equal to 5. The prior for the \code{scale} is the HalfCauchy distribution with scale \code{sigScale} and the prior for the degress of freedom is the Jeffreys' rule prior of Fonseca et al. (2008).
##' @references
##' Fonseca T. C., Ferreira M. A. R. & Migon H. S. (2008)
##' Objective Bayesian analysis for the Student-\eqn{t}{t} regression model.
##' \emph{Biometrika} \bold{95}, 325--333.
##'
##' @references
##' Ruli E., Sartori N. & Ventura L. (2015)
##' Improved Laplace approximation for marignal likelihoods.
##' \url{http://arxiv.org/abs/1502.06440}
##' @return \eqn{4\times 4}{4 by 4} matrix
##' @rdname nLinearReg
##' @export
hessT_bod2 <- function(beta, lsig, lnu, y, Time, n, muBeta, SigBeta, sigScale) {
.Call('iLaplaceExamples_hessT_bod2', PACKAGE = 'iLaplaceExamples', beta, lsig, lnu, y, Time, n, muBeta, SigBeta, sigScale)
}
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##' @name nLinearReg
##'
##' @title Posterior distribution (and related quantities) for two nonlinear regression models
##' @description Posterior distribution, and related quantities, for two non linear regression models each based on a different dataset. For each dataset, two error distributions are considered: normal and Student's \eqn{t}. The functions ending in "lub" refer to the \code{\link[iLaplaceExamples]{Lubricant}} dataset, those with ending "bod2" refer to the \code{\link[iLaplaceExamples]{BOD2}} dataset. The functions with an inner \code{T} refer to the model based on Student's \eqn{t}-error distribution. The functions starting with \code{nlpost} give the negative log-posterior density, those starting with \code{grad} give the gradient of \code{nlpost} and the functions starting with \code{hess} provide the Hessian matrix.
##'
##' @aliases nlpostT_lub, grad_lub, gradT_lub, hess_lub, hessT_lub, nlpostT_bod2, grad_bod2, gradT_bod2, hess_bod2, hessT_bod2
##'
##' @param beta The vector of regression parameters. For the Lubricant dataset beta is a 9-variate vector, for the BOD2 data beta is a bivariate vector
##' @param lsig The logarithm of the scale parameter
##' @param lnu For the models with Student's \eqn{t}-errors only: the logarithm of the degrees of freedom
##' @param y The response vector
##' @param x1 For Lubricant data only: the temperature of the lubricant
##' @param x2 For Lubricant data only: the pressure of the lubricant
##' @param Time For BOD2 data only: the time
##' @param n The sample size
##' @param muBeta The center of the prior for beta. See Details
##' @param SigBeta The scale matrix of the prior for beta
##' @param sigScale For the models with Student's \eqn{t}-errors only: the value of the scale of the prior for the scale parameter.
##'
##'
##' @return double
##' @export
nlpost_lub <- function(beta, lsig, y, x1, x2, n, muBeta, SigBeta, sigScale) {
.Call('iLaplaceExamples_nlpost_lub', PACKAGE = 'iLaplaceExamples', beta, lsig, y, x1, x2, n, muBeta, SigBeta, sigScale)
}
##' @return double
##' @rdname nLinearReg
##' @export
nlpostT_lub <- function(beta, lsig, lnu, y, x1, x2, n, muBeta, SigBeta, sigScale) {
.Call('iLaplaceExamples_nlpostT_lub', PACKAGE = 'iLaplaceExamples', beta, lsig, lnu, y, x1, x2, n, muBeta, SigBeta, sigScale)
}
##' @return 10-variate vector
##' @rdname nLinearReg
##' @export
grad_lub <- function(beta, lsig, y, x1, x2, n, muBeta, SigBeta, sigScale) {
.Call('iLaplaceExamples_grad_lub', PACKAGE = 'iLaplaceExamples', beta, lsig, y, x1, x2, n, muBeta, SigBeta, sigScale)
}
##' @return 11-variate vector
##' @rdname nLinearReg
##' @export
gradT_lub <- function(beta, lsig, lnu, y, x1, x2, n, muBeta, SigBeta, sigScale) {
.Call('iLaplaceExamples_gradT_lub', PACKAGE = 'iLaplaceExamples', beta, lsig, lnu, y, x1, x2, n, muBeta, SigBeta, sigScale)
}
##' @return \eqn{10\times 10}{10 by 10} matrix
##' @rdname nLinearReg
##' @export
hess_lub <- function(beta, lsig, y, x1, x2, n, muBeta, SigBeta, sigScale) {
.Call('iLaplaceExamples_hess_lub', PACKAGE = 'iLaplaceExamples', beta, lsig, y, x1, x2, n, muBeta, SigBeta, sigScale)
}
##' @return \eqn{11\times 11}{11 by 11} matrix
##' @rdname nLinearReg
##' @export
hessT_lub <- function(beta, lsig, lnu, y, x1, x2, n, muBeta, SigBeta, sigScale) {
.Call('iLaplaceExamples_hessT_lub', PACKAGE = 'iLaplaceExamples', beta, lsig, lnu, y, x1, x2, n, muBeta, SigBeta, sigScale)
}
##' @return double
##' @rdname nLinearReg
##' @export
nlpost_bod2 <- function(beta, lsig, y, Time, n, muBeta, SigBeta, sigScale) {
.Call('iLaplaceExamples_nlpost_bod2', PACKAGE = 'iLaplaceExamples', beta, lsig, y, Time, n, muBeta, SigBeta, sigScale)
}
##' @return double
##' @rdname nLinearReg
##' @export
nlpostT_bod2 <- function(beta, lsig, lnu, y, Time, n, muBeta, SigBeta, sigScale) {
.Call('iLaplaceExamples_nlpostT_bod2', PACKAGE = 'iLaplaceExamples', beta, lsig, lnu, y, Time, n, muBeta, SigBeta, sigScale)
}
##' @return 3-variate vector
##' @rdname nLinearReg
##' @export
grad_bod2 <- function(beta, lsig, y, Time, n, muBeta, SigBeta, sigScale) {
.Call('iLaplaceExamples_grad_bod2', PACKAGE = 'iLaplaceExamples', beta, lsig, y, Time, n, muBeta, SigBeta, sigScale)
}
##' @return 4-variate vector
##' @rdname nLinearReg
##' @export
gradT_bod2 <- function(beta, lsig, lnu, y, Time, n, muBeta, SigBeta, sigScale) {
.Call('iLaplaceExamples_gradT_bod2', PACKAGE = 'iLaplaceExamples', beta, lsig, lnu, y, Time, n, muBeta, SigBeta, sigScale)
}
##' @return \eqn{3\times 3}{3 by 3} matrix
##' @rdname nLinearReg
##' @export
hess_bod2 <- function(beta, lsig, y, Time, n, muBeta, SigBeta, sigScale) {
.Call('iLaplaceExamples_hess_bod2', PACKAGE = 'iLaplaceExamples', beta, lsig, y, Time, n, muBeta, SigBeta, sigScale)
}
##' @examples
##' \dontrun{
##'
##' library(iLaplace)
##'
##' ## regression models with BOD2 data
##' # set the data
##' data(BOD2)
##' y = BOD2$demand
##' Time = BOD2$Time
##' n = length(y)
##' muBeta = rep(0, 2)
##' SigBeta = matrix(0, 2,2)
##' diag(SigBeta) = 10
##' sigScale = 10
##' dfprop = 3
##'
##' ## Model with normal errors
##'
##' # the objective function and derivatives
##' ff = function(pp, ...) nlpost_bod2(beta = pp[1:2], lsig = pp[3], ...)
##' ff.gr = function(pp, ...) grad_bod2(beta = pp[1:2], lsig = pp[3], ...)
##' ff.hess = function(pp, ...) hess_bod2(beta = pp[1:2], lsig = pp[3], ...)
##'
##' # find the mode
##' init <- c(5, 2, 0.5)
##' opt.post = nlminb(init, obj = ff, gradient = ff.gr, hessian = ff.hess,
##' y = y, Time = Time, n = n, muBeta = muBeta,
##' SigBeta = SigBeta, sigScale = sigScale, control = list(trace = 1))
##' opt.post$hessian = ff.hess(opt.post$par, y = y, Time = Time, n = n, muBeta = muBeta,
##' SigBeta = SigBeta, sigScale = sigScale)
##'
##' # simulate from the posterior
##' mcmc.post = MHmcmc(logfun = function(x) -ff(x, y = y, Time = Time, n = n, muBeta = muBeta,
##' SigBeta = SigBeta, sigScale = sigScale),
##' burnin = 20000, mcmc = 1e+6,
##' thin = 5, tune = 0.98, V = solve(opt.post$hessian),
##' df = dfprop, theta.init = init, verbose = 50000)
##'
##'
##' # marginal likelihood by Chib's method
##' logM.Chib <- ChibML(logfun = function(x) -ff(x, y = y, Time = Time, n = n,
##' muBeta = muBeta, SigBeta = SigBeta,
##' sigScale = sigScale),
##' theta.star = opt.post$par,
##' tune = 1.5, V = solve(opt.post$hessian), t(mcmc.post),
##' df = dfprop, verbose = 100000)
##'
##' # marginal likelihood by the improved Laplace
##' logM.iLap <- iLaplace(fullOpt = opt.post, ff = ff, ff.gr = ff.gr, ff.hess = ff.hess,
##' control = list(sp.points = 100, delta = 13, n.cores = 3),
##' clEvalQ = c("iLaplaceExamples"), y = y, Time = Time, n = n,
##' muBeta = muBeta, SigBeta = SigBeta,
##' sigScale = sigScale)
##'
##' logM.Lap <- 3*log(2*pi)/2 - opt.post$objective - 0.5*determinant(opt.post$hessian)$mod
##' # marginal likelihood by importance samling
##' logM.IS <- isML(logfun = function(x) -ff(x, y = y, Time = Time, n = n,
##' muBeta = muBeta, SigBeta = SigBeta,
##' sigScale = sigScale), nsim = 1e+6,
##' theta.hat = opt.post$par, tune = 1.5,
##' V = solve(opt.post$hessian),df = dfprop, verbose = 1000)
##' logM.iLap
##' logM.Lap
##' logM.IS
##' logM.Chib
##'
##' ## Model with Student's t errors
##'
##' # the objective function and derivatives
##' tff = function(pp, ...) nlpostT_bod2(beta = pp[1:2], lsig = pp[3], lnu = pp[4], ...)
##' tff.gr = function(pp, ...) gradT_bod2(beta = pp[1:2], lsig = pp[3], lnu = pp[4], ...)
##' tff.hess = function(pp, ...) hessT_bod2(beta = pp[1:2], lsig = pp[3], lnu = pp[4], ...)
##'
##' # find the mode
##' init <- c(2, 5, 0.5, 0)
##' opt.postt = nlminb(init, obj = tff, gradient = tff.gr, hessian = tff.hess,
##' y = y, Time = Time, n = n, muBeta = muBeta,
##' SigBeta = SigBeta, sigScale = sigScale, control = list(trace = 1))
##' opt.postt$hessian = tff.hess(opt.postt$par, y = y, Time = Time, n = n, muBeta = muBeta,
##' SigBeta = SigBeta, sigScale = sigScale)
##'
##' # simulate from the posterior
##' mcmc.postt = MHmcmc(logfun = function(x) -tff(x, y = y, Time = Time, n = n, muBeta = muBeta,
##' SigBeta = SigBeta, sigScale = sigScale),
##' burnin = 20000, mcmc = 1e+6,
##' thin = 5, tune = 1.2, V = solve(opt.postt$hessian),
##' df = dfprop, theta.init = opt.postt$par, verbose = 50000)
##'
##' logM.Lapt <- 4*log(2*pi)/2 - opt.postt$objective - 0.5*determinant(opt.postt$hessian)$mod
##'
##' # marginal likelihood by Chib's method
##' logM.Chibt <- ChibML(logfun = function(x) -tff(x, y = y, Time = Time, n = n,
##' muBeta = muBeta, SigBeta = SigBeta,
##' sigScale = sigScale),
##' theta.star = opt.postt$par,
##' tune = 1.5, V = solve(opt.postt$hessian), t(mcmc.postt),
##' df = dfprop, verbose = 100000)
##'
##' logM.iLapt <- iLaplace(fullOpt = opt.postt, ff = tff, ff.gr = tff.gr, ff.hess = tff.hess,
##' control = list(sp.points = 100, delta = 9, n.cores = 4),
##' clEvalQ = c("iLaplaceExamples"), y = y, Time = Time, n = n,
##' muBeta = muBeta, SigBeta = SigBeta,
##' sigScale = sigScale)
##'
##' logM.ISt <- isML(logfun = function(x) -tff(x, y = y, Time = Time, n = n,
##' muBeta = muBeta, SigBeta = SigBeta,
##' sigScale = sigScale), nsim = 1e+6,
##' theta.hat = opt.postt$par, tune = 1.5,
##' V = solve(opt.postt$hessian),df = dfprop, verbose = 10000)
##'
##' logM.iLapt
##' logM.Lapt
##' logM.ISt
##' logM.Chibt
##' }
##'
##'
##' @seealso \code{\link[iLaplaceExamples]{MHmcmc}}
##'
##' @details The prior for \code{beta} is the multivariate Studnet's \eqn{t}{t} with the appropriate dimension, with location \code{muBeta}, scale matrix \code{SigBeta} and degrees of freedom equal to 5. The prior for the \code{scale} is the HalfCauchy distribution with scale \code{sigScale} and the prior for the degress of freedom is the Jeffreys' rule prior of Fonseca et al. (2008).
##' @references
##' Fonseca T. C., Ferreira M. A. R. & Migon H. S. (2008)
##' Objective Bayesian analysis for the Student-\eqn{t}{t} regression model.
##' \emph{Biometrika} \bold{95}, 325--333.
##'
##' @references
##' Ruli E., Sartori N. & Ventura L. (2015)
##' Improved Laplace approximation for marignal likelihoods.
##' \url{http://arxiv.org/abs/1502.06440}
##' @return \eqn{4\times 4}{4 by 4} matrix
##' @rdname nLinearReg
##' @export
hessT_bod2 <- function(beta, lsig, lnu, y, Time, n, muBeta, SigBeta, sigScale) {
.Call('iLaplaceExamples_hessT_bod2', PACKAGE = 'iLaplaceExamples', beta, lsig, lnu, y, Time, n, muBeta, SigBeta, sigScale)
}
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