| AdMitIS {AdMit} | R Documentation |
Performs importance sampling using an adaptive mixture of Student-t distributions as the importance density
AdMitIS(N = 1e5, KERNEL, G = function(theta){theta}, mit = list(), ...)
N |
number of draws used in importance sampling (positive
integer number). Default: N = 1e5. |
KERNEL |
kernel function of the target density on which the
adaptive mixture of Student-t distributions is fitted. This
function should be vectorized for speed purposes (i.e., its first
argument should be a matrix and its output a vector). Moreover, the function must contain
the logical argument log. If log = TRUE, the function
returns (natural) logarithm values of the kernel function. NA
and NaN values are not allowed. (See the function
AdMit for examples of KERNEL implementation.) |
G |
function of interest used in importance sampling (see *Details*). |
mit |
list containing information on the mixture approximation (see *Details*). |
... |
further arguments to be passed to KERNEL and/or G. |
The AdMitIS function estimates
E_p[g(theta)], where p is the target
density, g is an (integrable w.r.t. p) function and E denotes
the expectation operator, by importance sampling using an adaptive
mixture of Student-t distributions as the importance density.
By default, the function G is given by:
G <- function(theta)
{
theta
}
and therefore, AdMitIS estimates the mean of
theta by importance sampling. For other definitions of
G, see *Examples*.
The argument mit is a list containing information on the
mixture approximation. The following components must be provided:
pmuSigmadf
where H (>=1) is the number of components of the
adaptive mixture of Student-t distributions and
d (>=1) is the dimension of the first argument in KERNEL. Typically,
mit is estimated by the function AdMit.
A list with the following components:
ghat: a vector containing the importance sampling estimates.
NSE: a vector containing the numerical standard error of the components of ghat.
RNE: a vector containing the relative numerical efficiency of the
components of ghat.
Further details and examples of the R package AdMit
can be found in Ardia, Hoogerheide, van Dijk (2009a,b). See also
the package vignette by typing vignette("AdMit") and the
files ‘AdMitJSS.txt’ and ‘AdMitRnews.txt’ in the ‘/doc’ package's folder.
Further information on importance sampling can be found in Geweke (1989) or Koop (2003).
Please cite the package in publications. Use citation("AdMit").
David Ardia <david.ardia@unifr.ch>
Ardia, D., Hoogerheide, L.F., van Dijk, H.K. (2009a). AdMit: Adaptive Mixture of Student-t Distributions. The R Journal 1(1), pp.25–30. http://journal.r-project.org/2009-1/
Ardia, D., Hoogerheide, L.F., van Dijk, H.K. (2009b). Adaptive Mixture of Student-t Distributions as a Flexible Candidate Distribution for Efficient Simulation: The R Package AdMit. Journal of Statistical Software 29(3), pp.1–32. http://www.jstatsoft.org/v29/i03/
Geweke, J.F. (1989). Bayesian Inference in Econometric Models Using Monte Carlo Integration. Econometrica 57(6), pp.1317–1339.
Koop, G. (2003). Bayesian Econometrics. Wiley-Interscience (London, UK). ISBN: 0470845678.
AdMit for fitting an adaptive mixture of Student-t
distributions to a target density through its KERNEL function,
AdMitMH for the independence chain Metropolis-Hastings
algorithm using an adaptive mixture of Student-t distributions
as the candidate density.
## Gelman and Meng (1991) kernel function
GelmanMeng <- function(x, A = 1, B = 0, C1 = 3, C2 = 3, log = TRUE)
{
if (is.vector(x))
x <- matrix(x, nrow = 1)
r <- -.5 * (A * x[,1]^2 * x[,2]^2 + x[,1]^2 + x[,2]^2
- 2 * B * x[,1] * x[,2] - 2 * C1 * x[,1] - 2 * C2 * x[,2])
if (!log)
r <- exp(r)
as.vector(r)
}
## Run the AdMit function to fit the mixture approximation
set.seed(1234)
outAdMit <- AdMit(KERNEL = GelmanMeng,
mu0 = c(0.0, 0.1))
## Use importance sampling with the mixture approximation as the
## importance density
outAdMitIS <- AdMitIS(KERNEL = GelmanMeng, mit = outAdMit$mit)
print(outAdMitIS)
## Covariance matrix estimated by importance sampling
G.cov <- function(theta, mu)
{
G.cov_sub <- function(x)
(x - mu)
theta <- as.matrix(theta)
tmp <- apply(theta, 1, G.cov_sub)
if (length(mu) > 1)
t(tmp)
else
as.matrix(tmp)
}
outAdMitIS <- AdMitIS(KERNEL = GelmanMeng,
G = G.cov,
mit = outAdMit$mit,
mu = c(1.459, 1.459))
print(outAdMitIS)
## Covariance matrix
V <- matrix(outAdMitIS$ghat, 2, 2)
print(V)
## Correlation matrix
cov2cor(V)