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###############################################
# Function to implement Gibbs sampler for the #
# Dirichlet-Laplace prior for normal means #
###############################################
######################
# FUNCTION ARGUMENTS #
######################
# x = n*1 noisy vector
#
# tau.est = method for selecting the hyperparmeter tau in Dir(a) scale parameter
# Option #1: 'fixed'. User-specified (not recommended)
# Option #1: 'est.sparsity'
# Option #2: 'reml'
# NOTE: Default is MMLE if user does not specify either
#
# tau = parameter in Dir(tau,..., tau) scale parameter
# Default is 1/n. Ignored if 'est.sparsity', 'reml', 'uniform',
# or 'halfCauchy' is used
#
# sigma2 = variance term
# User specifies this. Default is 1
#
# var.select = method for selecting variables
# Can select "threshold" for thresholding rule. Defaults to this
# Can also use "intervals" to select based on marginal credible intervals
# NOTE: Default is "threshold" if user does not specify either
#
# max.steps = # of iterations to run MCMC. Default is 10,000
#
# burnin = # of samples in burn-in. Default is 5,000
#
#
##################
# RETURNS A LIST #
##################
# theta.hat = posterior mean estimate for theta
# theta.med = posterior median estimate for theta
# theta.var = posterior varince estimate for theta
# theta.intervals = 95% posterior credible intervals for each component
# theta.classifications = binary vector of classifications, according to
# classification method chosen by user
# tau.est = empirical Bayes estimate of the hyperparameter 'tau'
dl.normalmeans = function(x, tau.est=c("fixed", "est.sparsity", "reml", "uniform", "truncatedCauchy"),
tau=1/length(x), sigma2=1, var.select = c("threshold", "intervals"),
max.steps=10000, burnin=5000){
#####################################
# Check that burnin < max.steps and #
# that vector is non-empty #
#####################################
if (burnin > max.steps)
stop("Burnin cannot be greater than # of iterations.")
if (length(x) == 0)
stop("Please enter a vector length greater than 0.")
#####################################
# Number of samples in noisy vector #
#####################################
n <- length(x)
################################################
# For the hyperparameter a, renamed as a.param #
################################################
a.param <- tau
if( tau.est=="fixed" ) # If 'fixed' method is used.
a.param <- tau
if( tau.est=="est.sparsity" ) # if 'est.sparsity' method is used.
a.param <- est.sparsity(x, sigma2)
if( tau.est=="reml" ) # If 'reml' method is used.
a.param <- dl.MMLE(x, sigma2)
# If user specified a different 'a,' it must be greater than 0.
if ( tau <= 0 )
stop("ERROR: 'tau' should be positive. \n")
# Check that sigma2 is greater than 0.
if (sigma2 <=0 )
stop("ERROR: sigma2 should be greater than 0. \n")
if ( tau.est=="uniform" ){ # if 'uniform' method is used.
a.param <- 1/n # starting value of the random walk
a.samples <- rep(NA, max.steps) # Create a vector for samples of 'a'
a.samples[1] <- a.param
}
if ( tau.est=="truncatedCauchy" ){ # if 'truncatedCauchy' method is used.
a.param <- 1/n # starting value of the random walk
a.samples <- rep(NA, max.steps) # Create a vector for samples of 'a'
a.samples[1] <- a.param
}
############################################
# Initial guesses for theta, psi, phi, tau #
############################################
theta <- rep(mean(x), n)
psi <- rep(1, n)
phi <- rep(1, n)
tau <- rep(1, n)
aux.t <- rep(NA, n) # For the auxiliary T_1, ..., T_n
###########################
# For storing the samples #
###########################
theta.samples <- rep(list(rep(NA,n)), max.steps)
kappa.samples <- rep(list(rep(NA,n)), max.steps)
###########################
# Start the Gibbs sampler #
###########################
j <- 0
while (j < max.steps) {
j <- j + 1
###########
# Counter #
###########
if (j %% 1000 == 0) {
cat("Iteration:", j, "\n")
}
####################
# Sample theta_i's #
####################
kappa <- 1/(1+psi*phi^2*tau^2)
theta.means <- (1-kappa)*x # conditional means for thetas
theta.sds <- sqrt(sigma2*(1-kappa)) # conditional variances for thetas
theta <- rnorm(n, mean=theta.means, sd=theta.sds) # sample thetas as a block
# Store theta samples
theta.samples[[j]] <- theta
# Store kappa samples
kappa.samples[[j]] <- kappa
# Function to block-sample from the GIG density
gig.sample <- function(x){
GIGrvg::rgig(1, lambda=u, chi=x, psi=w)
}
gig.sample.vec <- Vectorize(gig.sample) # handles a vector as input
##################
# Sample psi_i's #
##################
u <- 1/2
v <- (1/(sigma2*tau^2))*(theta^2/phi^2) # nx1 vector
v <- pmax(v, .Machine$double.eps) # For numerical stability
w <- 1
# Update psi's as a block
psi <- gig.sample.vec(v)
########################
# Sample tau and phi's #
########################
# Sample auxiliary T_i's
u <- a.param-1
v <- (2/sqrt(sigma2))*abs(theta) #nx1 vector
v <- pmax(v, .Machine$double.eps) # For numerical stability
w <- 1
# Update auxiliary T_i's as a block
aux.t <- gig.sample.vec(v)
# Set tau = sum of T_i's and phi_i = T_i/tau
tau <- sum(aux.t)
phi <- aux.t/tau
# If 'uniform' was the method of 'tau.est'
if ( (tau.est == "uniform") & (j >= 2) ){
# Draw from proposal density
prop.sd <- 1e-4
a.star <- rtruncnorm(1, a=1/n, b=1, mean=a.samples[j-1], sd=prop.sd)
# Ratio q(a.star | a)/q(a | a.star)
q.ratio <- (pnorm((1-a.samples[j-1])/prop.sd)-pnorm((1/n-a.samples[j-1])/prop.sd))/(pnorm((1-a.star)/prop.sd)-pnorm((1/n-a.star)/prop.sd))
# Ratio pi(a.star | rest)/pi(a | rest)
pi.ratio <- 2^(n*(a.samples[j-1]-a.star))*(gamma(a.samples[j-1])/gamma(a.star))^n * prod(psi^(a.star-a.samples[j-1]))
# Acceptance probability
alpha <- min(1, (q.ratio*pi.ratio) )
# Accept/reject algorithm
if ( runif(1) < alpha){
a.samples[j] <- a.star
} else {
a.samples[j] <- a.samples[j-1]
}
# Update a.param
a.param <- a.samples[j]
}
# If 'truncatedCauchy' was the method of 'tau.est'
if ( (tau.est == "truncatedCauchy") & (j >= 2) ){
# Draw from proposal density
prop.sd <- 1e-4
a.star <- rtruncnorm(1, a=1/n, b=1, mean=a.samples[j-1], sd=prop.sd)
# Ratio q(a.star | a)/q(a | a.star)
q.ratio <- (pnorm((1-a.samples[j-1])/prop.sd)-pnorm((1/n-a.samples[j-1])/prop.sd))/(pnorm((1-a.star)/prop.sd)-pnorm((1/n-a.star)/prop.sd))
# Ratio pi(a.star | rest)/pi(a | rest)
pi.ratio <- 2^(n*(a.samples[j-1]-a.star))*(gamma(a.samples[j-1])/gamma(a.star))^n * prod(psi^(a.star-a.samples[j-1]))
# Acceptance probability
alpha <- min(1, (q.ratio*pi.ratio) )
# Accept/reject algorithm
if ( runif(1) < alpha){
a.samples[j] <- a.star
} else {
a.samples[j] <- a.samples[j-1]
}
# Update a.param
a.param <- a.samples[j]
}
}
###################
# Discard burn-in #
###################
theta.samples <- tail(theta.samples,max.steps-burnin)
kappa.samples <- tail(kappa.samples,max.steps-burnin)
# Return estimate of tau if fully Bayes method of estimating it.
if( (tau.est == "uniform") | (tau.est == "truncatedCauchy") ){
a.param <- mean(tail(a.samples,max.steps-burnin))
}
#######################################
# Extract the posterior mean, median, #
# 2.5th, and 97.5th percentiles #
#######################################
theta.sample.mat <- simplify2array(theta.samples)
rm(theta.samples)
# Posterior mean
theta.hat <- rowMeans(theta.sample.mat)
# Posterior median
theta.med <- apply(theta.sample.mat, 1, median)
# Posterior var
theta.sd <- apply(theta.sample.mat, 1, sd)
theta.var <- theta.sd^2
# endpoints of 95% posterior credible intervals
theta.intervals <- apply(theta.sample.mat, 1, function(x) quantile(x, prob=c(.025,.975)))
##############################
# Perform variable selection #
##############################
# Initialize vector of binary entries: 0 for inactive variable b_i, 1 for active b_j
dl.classifications <- rep(0,n)
if(var.select=="threshold"){
# Estimate the shrinkage factor kappa_i's from the MCMC samples
kappa.sample.mat <- simplify2array(kappa.samples)
rm(kappa.samples)
kappa.estimates <- rowMeans(kappa.sample.mat)
# Return indices of the signals according to our classification rule
signal.indices <- which((1-kappa.estimates)>=0.5)
# Reset classified signals as 1
dl.classifications[signal.indices] <- 1
}
if(var.select=="intervals"){
# Find the active covariates
for(k in 1:n){
if(theta.intervals[1,k] < 0 && theta.intervals[2,k] < 0){
dl.classifications[k] <- 1
}
else if(theta.intervals[1,k] > 0 && theta.intervals[2,k] > 0){
dl.classifications[k] <- 1
}
}
}
###############################
# Return list of theta.hat, #
# theta.med, theta.intervals, #
# and dl.classifications #
###############################
# theta.hat = posterior mean point estimator
# theta.med = posterior median point estimator
# theta.var = posterior variance estimate
# theta.intervals = endpoints of 95% posterior credible intervals
# dl.classifications = selected variables
# tau.estimate = estimate of 'tau'
dl.output <- list(theta.hat = theta.hat,
theta.med = theta.med,
theta.var = theta.var,
theta.intervals = theta.intervals,
dl.classifications = dl.classifications,
tau.estimate = a.param)
# Return list
return(dl.output)
}
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