# Compute importance weights for lasso, group lasso, scaled lasso or scaled group lasso estimator under high-dimensional setting `hdIS` computes importance weights using samples drawn by `PBsampler`. See the examples below for details. ```r 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. ## Returns importance weights of the proposed samples. ## 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) )`. ## Examples ```r 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) ``` ## 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.

hdIS function