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:
p
mu
Sigma
df
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 (2008, 2009). 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. (2008). The AdMit Package. Econometric Institute report 2008-17. http://publishing.eur.nl/ir/repub/asset/13053/EI2008-17.pdf (forthcoming in Rnews)
Ardia, D., Hoogerheide, L.F., van Dijk, H.K. (2009). Adaptive Mixture of Student-t Distributions as a Flexible Candidate Distribution for Efficient Simulation: The R Package AdMit. Journal of Statistical Software 29(3). 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)