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. |
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).
Further information on importance sampling can be found in Geweke (1989) or Koop (2003).
David Ardia <david.ardia@unifr.ch>
Ardia, D., Hoogerheide, L.F., van Dijk, H.K. (2008) `Adaptive mixture of Student-t distributions as a flexible candidate distribution for efficient simulation: The R package AdMit', Working paper, Econometric Institute, Erasmus University Rotterdam (NL). http://www.tinbergen.nl/
Geweke, J.F. (1989) `Bayesian Inference in Econometric Models Using Monte Carlo Integration', Econometrica 57(6), pp.1317–1339. Reprinted in: Bayesian Inference, G. C. Box and N. Polson (Eds.), Edward Elgar Publishing, 1994.
Koop, G. (2003) Bayesian Econometrics, Wiley-Interscience (London, UK), first edition, 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 (2001) 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(GelmanMeng, mu0=c(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)