do.mixture.sample: Internal function. sample from the mixture distribution $q_{\alpha}$

Description

Internal function. sample from the mixture distribution $q_{\alpha}$

Usage

do.mixture.sample(seed, b, n, J, mixture.param, alpha, fname, rpname, rqname, dpname, dqname)

Arguments

seed

seed for sampling

batch index for the samples

total number of samples

number of mixture components, including the defensive one

mixture.param

mixture.param = list(p, J, ...), where $p$ is the dimension of the sample, and $J$ is the number of mixture components, including the defensive one. mixture.param should be compatible with user defined functions f(n, j, mixture.param), rp(n, mixture.param), rq(n, j, mixture.param), dp(xmat, mixture.param), dq(xmat, j, mixture.param)

alpha

vector of mixture proportions

fname

name of user defined function fname(xmat, j, mixture.param). xmat is an $n \times p$ matrix of $n$ samples with $p$ dimensions. fname returns a vector of function values for each row in xmat. fname is defined for $j = 1, \cdots, J$. $j = 1, \cdots, J - 1$ corresponds to different proposal mixture components, and $j = J$ corresponds to the defensive mixture component.

rpname

name of user definded function rpname(n, mixture.param). It generates $n$ random samples from target distribution pname. Parameters can be specified in mixture.param. rpname returns an $n \times p$ matrix.

rqname

name of user defined function rqname(n, j, mixture.param). It generate $n$ random samples from the $j^{th}$ mixture component of proposal mixture distribution. rqname returns an $n \times p$ matrix. rqname is defined for $j = 1, \cdots, J - 1$.

dpname

name of user defined function dpname(xmat, mixture.param). It returns the density of xmat from the target distribution $pname$ as a vector of length nrow(xmat). Note that only the ratio between dpname and dqname matters. So dpname and dqname can return values of $C \times$dpname and $C \times$dqname respectively.

dqname

name of user defined function dqname(xmat, j, mixture.param). It returns the density of xmat from the proposal distribution $q_j$ as a vector of length nrow(xmat). dqname is defined for $j = 1, \cdots, J - 1$. Note that only the ratio between dpname and dqname matters. So dpname and dqname can return values of $C \times$dpname and $C \times$dqname respectively.

Value

a list of

x: the matrix(size $n \times p$) of samples from the mixture distribution $q_{\alpha}$
fx: the vector of fname evaluated at sample matrix x
qx: the matrix of densities under each mixture component, i.e. qx[,j] is the density under $q_j$. The defensive component is the $J^{th}$ column
px: the vector of densities under proposal distribution pname
alpha: the rounded alpha used for sampling, i.e. $alpha2N(n, alpha)/n$