# Metropolis-Hastings lasso sampler under a fixed active set. Metropolis-Hastings sampler to simulate from the sampling distribution of lasso given a fixed active set. ```r MHLS(X, PE, sig2, lbd, weights = rep(1, ncol(X)), B0, S0, A = which(B0 != 0), tau = rep(1, ncol(X)), niter = 2000, burnin = 0, PEtype = "coeff", updateS.itv = 1, verbose = FALSE, ...) ``` ## Arguments - `X`: predictor matrix. - `PE, sig2, lbd`: parameters of target distribution. (point estimate of beta or `E(y)` depends on `PEtype`, variance estimate of error and lambda). - `weights`: weight vector with length `p`(the number of covariates). Default is `weights = rep(1, p)`. - `B0`: numeric vector with length `p`. Initial value of lasso estimator. - `S0`: numeric vector with length `p`. Initial value of subgradients. If not given, this will be generated in a default way. - `A`: numeric vector. Active coefficient index. Every active coefficient index in `B0` must be included. Default is `A = which(B0 != 0)`. - `tau`: numeric vector with length `p`. Standard deviation of proposal distribution for each coefficient. - `niter`: integer. The number of iterations. Default is `niter = 2000` - `burnin`: integer. The length of burin-in periods. Default is `burnin = 0` - `PEtype`: Type of `PE` which is needed to characterize the target distribution. Users can choose either `"coeff"` or `"mu"`. - `updateS.itv`: integer. Update subgradients every `updateS.itv` iterations. Set this value larger than `niter` if one wants to skip updating subgradients. - `verbose`: logical. If true, print out the progress step. - `...`: complementary arguments. * `FlipSA :`optional parameter. This has to be a subset of active set, A. If the index is not listed in FlipSA, the sign of coefficients which correspond to the listed index will remain fixed. The default is `FlipSA=A` * `SFindex :`optional parameter. subgradient index for the free coordinate. * `randomSFindex :`logical. If `true`, resample `SFindex` every `updateSF.itv` iterations. * `updateSF.itv :`integer. In every `updateSF.itv` iterations, randomize `SFindex`. ## Returns `MHLS` returns an object of class `"MHLS"`. The functions `summary.MHLS` and `plot.MHLS` provide a brief summary and generate plots. - **beta**: lasso samples. - **subgrad**: subgradient samples. - **acceptHistory**: numbers of acceptance and proposal. - **niter, burnin, PE, type**: same as function arguments. ## Details Given appropriate initial value, provides Metropolis-Hastings samples under the fixed active set. From the initial values, `B0` and S0, `MHLS` draws `beta` and `subgrad` samples. In every iteration, given `t`-th iteration values, `t`-th `beta` and `t`-th `subgrad`, a new set of proposed beta and subgradient is sampled. We either accept the proposed sample and use that as `(t+1)`-th iteration values or reuse `t`-th iteration values. See Zhou(2014) for more details. ## Examples ```r #------------------------- # Low dim #------------------------- set.seed(123) n <- 10 p <- 5 X <- matrix(rnorm(n * p), n) Y <- X %*% rep(1, p) + rnorm(n) sigma2 <- 1 lbd <- .37 weights <- rep(1, p) LassoResult <- lassoFit(X = X, Y = Y, lbd = lbd, type = "lasso", weights = weights) B0 <- LassoResult$B0 S0 <- LassoResult$S0 MHLS(X = X, PE = rep(0, p), sig2 = 1, lbd = 1, weights = weights, B0 = B0, S0 = S0, niter = 50, burnin = 0, PEtype = "coeff") MHLS(X = X, PE = rep(0, n), sig2 = 1, lbd = 1, weights = weights, B0 = B0, S0 = S0, niter = 50, burnin = 0, PEtype = "mu") #------------------------- # High dim #------------------------- set.seed(123) n <- 5 p <- 10 X <- matrix(rnorm(n*p),n) Y <- X %*% rep(1,p) + rnorm(n) weights <- rep(1,p) LassoResult <- lassoFit(X = X,Y = Y,lbd = lbd, type = "lasso", weights = weights) B0 <- LassoResult$B0 S0 <- LassoResult$S0 MHLS(X = X, PE = rep(0, p), sig2 = 1, lbd = 1, weights = weights, B0 = B0, S0 = S0, niter = 50, burnin = 0, PEtype = "coeff") MHLS(X = X, PE = rep(0, n), sig2 = 1, lbd = 1, weights = weights, B0 = B0, S0 = S0, niter = 50, burnin = 0, PEtype = "mu") ``` ## References Zhou, Q. (2014), "Monte Carlo simulation for Lasso-type problems by estimator augmentation," Journal of the American Statistical Association, 109, 1495-1516.

MHLS function