AdMitIS {AdMit}R Documentation

Importance Sampling using an Adaptive Mixture of Student-t Distributions as the Importance Density

Description

Performs importance sampling using an adaptive mixture of Student-t distributions as the importance density

Usage

AdMitIS(N=1e5, KERNEL, G=function(theta){theta}, mit=list(), ...)

Arguments

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.

Details

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
vector (of length H) of mixing probabilities.
mu
matrix (of size Hxd) containing the vectors of modes (in row) of the mixture components.
Sigma
matrix (of size Hxd*d) containing the scale matrices (in row) of the mixture components.
df
degrees of freedom parameter of the Student-t components (real number not smaller than one).

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.

Value

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.

Note

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).

Author(s)

David Ardia <david.ardia@unifr.ch>

References

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.

See Also

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.

Examples

  ## 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)

[Package AdMit version 1-00.02 Index]