AdMit (version 2.1.12)

AdMitIS: 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(), ...)

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. It is NA for a component of G

whose estimated variance is zero, that is, when NSE is zero and the relative efficiency is undefined.

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. (See the function AdMit for examples of KERNEL implementation.)

G

function of interest used in importance sampling (see *Details*). It is applied to the \(N \times d\) matrix of draws and may return either a vector (scalar-valued function of interest) or a matrix with \(N\) rows.

mit

list containing information on the mixture approximation (see *Details*).

...

further arguments to be passed to KERNEL and/or G.

Author

David Ardia

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*. Additional arguments required by G are passed through ....

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 \(H \times d\)) containing the vectors of modes (in row) of the mixture components.

Sigma

matrix (of size \(H \times d^2\)) containing the scale matrices (in row) of the mixture components.

df

degrees of freedom parameter of the Student-t components (positive real number; AdMit itself fits mixtures with df not smaller than one).

where \(H (\geq 1)\) is the number of components of the adaptive mixture of Student-t distributions and \(d (\geq 1)\) is the dimension of the first argument in KERNEL. Typically, mit is estimated by the function AdMit.

References

Ardia, D., Hoogerheide, L.F., van Dijk, H.K. (2009a). AdMit: Adaptive Mixture of Student-t Distributions. R Journal 1(1), pp.25-30. tools:::Rd_expr_doi("10.32614/RJ-2009-003")

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. tools:::Rd_expr_doi("10.18637/jss.v029.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.

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

  ## NB : Low number of draws for speedup. Consider using more draws!
  ## 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),
                    control = list(Ns = 2e3, Np = 5e2, Hmax = 4))

  ## Use importance sampling with the mixture approximation as the
  ## importance density
  outAdMitIS <- AdMitIS(N = 1e4, KERNEL = GelmanMeng, mit = outAdMit$mit)
  print(outAdMitIS)

  ## A scalar-valued function of interest: E[theta_1^2]
  G.square <- function(theta) theta[,1]^2
  print(AdMitIS(N = 1e4, KERNEL = GelmanMeng, G = G.square, mit = outAdMit$mit))

  ## A matrix-valued function of interest, with an extra argument passed
  ## through '...': the posterior covariance matrix around 'mu'
  G.cov <- function(theta, mu)
  {
    G.cov_sub <- function(x)
      (x - mu) %*% t(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(N = 1e4, KERNEL = GelmanMeng, G = G.cov,
                        mit = outAdMit$mit, mu = c(1.459, 1.459))
  print(matrix(outAdMitIS$ghat, 2, 2))