hdIS {EAinference}

Compute importance weights for lasso, group lasso, scaled lasso or scaled group lasso estimator under high-dimensional setting

Description

hdIS computes importance weights using samples drawn by PBsampler. See the examples below for details.

Usage

hdIS(PBsample, PETarget, sig2Target, lbdTarget, TsA.method = "default",
  log = TRUE, parallel = FALSE, ncores = 2L)

Arguments

PBsample

bootstrap samples of class PB from PBsampler.

PETarget, sig2Target, lbdTarget

parameters of target distribution. (point estimate of beta or E(y), estimated variance of error and lambda)

TsA.method

method to construct T(eta(s),A) matrix. See Zhou and Min(2017) for details.

log

logical. If log = TRUE, importance weight is computed in log scale.

parallel

logical. If parallel = TRUE, uses parallelization. Default is parallel = FALSE.

ncores

integer. The number of cores to use for parallelization.

Details

computes importance weights which is defined as (target density)/(proposal density), when the samples are drawn from the proposal distribution with the function PBsampler while the parameters of the target distribution are (PETarget, sig2Target, lbdTarget).
Say that we are interested in computing the expectation of a function of a random variable, h(X). Let f(x) be the true or target distribution and g(x) be the proposal distribution. We can approximate the expectation, E[h(X)], by a weighted average of samples, x_i, drawn from the proposal distribution as follows, E[h(X)] = mean( h(x_i) * f(x_i)/h(x_i) ).

Value

importance weights of the proposed samples.

References

Zhou, Q. (2014), "Monte Carlo simulation for Lasso-type problems by estimator augmentation," Journal of the American Statistical Association, 109, 1495-1516.

Zhou, Q. and Min, S. (2017), "Estimator augmentation with applications in high-dimensional group inference," Electronic Journal of Statistics, 11(2), 3039-3080.

Examples

set.seed(1234)
n <- 10
p <- 30
Niter <-  10
Group <- rep(1:(p/10), each = 10)
Weights <- rep(1, p/10)
x <- matrix(rnorm(n*p), n)

# Target distribution parameter
PETarget <- rep(0, p)
sig2Target <- .5
lbdTarget <- .37

#
# Using non-mixture distribution
# ------------------------------
## Proposal distribution parameter
PEProp1 <- rep(1, p)
sig2Prop1 <- .5
lbdProp1 <- 1
PB <- PBsampler(X = x, PE_1 = PEProp1, sig2_1 = sig2Prop1,
 lbd_1 = lbdProp1, weights = Weights, group = Group, niter = Niter,
 type="grlasso", PEtype = "coeff")

hdIS(PB, PETarget = PETarget, sig2Target = sig2Target, lbdTarget = lbdTarget,
 log = TRUE)

#
# Using mixture distribution
# ------------------------------
# Target distribution parameters (coeff, sig2, lbd) = (rep(0,p), .5, .37)
# Proposal distribution parameters
#  (coeff, sig2, lbd) = (rep(0,p), .5, .37) & (rep(1,p), 1, .5)
#
#
PEProp1 <- rep(0,p); PEProp2 <- rep(1,p)
sig2Prop1 <- .5; sig2Prop2 <- 1
lbdProp1 <- .37; lbdProp2 <- .5

PBMixture <- PBsampler(X = x, PE_1 = PEProp1,
 sig2_1 = sig2Prop1, lbd_1 = lbdProp1, PE_2 = PEProp2,
 sig2_2 = sig2Prop2, lbd_2 = lbdProp2, weights = Weights, group = Group,
 niter = Niter, type = "grlasso", PEtype = "coeff")
hdIS(PBMixture, PETarget = PETarget, sig2Target = sig2Target, lbdTarget = lbdTarget,
 log = TRUE)