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#### Bayesian variable selection for the negative binomial model
#### using a Dirac spike and Student-t/ or normal slab
####
#### (invisible function)
#### last change: 2016/03/24
select_negbin <- function(y, X, offset, mcomp, compmix.pois = NULL, cm1, model,
prior, mcmc, param, imc){
# linear predicor in Poisson model
muP <- X%*%param$beta
linp <- muP
n <- length(y)
#### Step A --- data augmentation for the Poisson model
lambda <- exp(linp)*param$g
## A.1 sample the inter-arrival times of assumed Poisson process in [0,1]
# IAMS - mixture components
if (model$family == "pogit"){
compmix.pois <- get_mixcomp(y, mcomp)
#compmix.pois <- do.call(get_mixcomp, list(y, mcomp))
}
augPois <- iams1_poisson(y, offset*lambda, compmix.pois)
tau1 <- augPois$t1
tau2 <- augPois$t2
logMu <- linp + log(param$g) + log(offset)
logMugz <- logMu[compmix.pois$igz]
## A.2 --- sample the component indicators
R <- iams2_poisson(n, tau1, tau2, logMu, logMugz, cm1, compmix.pois)
# mixture component means and variances
m1 <- cm1$comp$m[R[1:n]]
m2 <- compmix.pois$my[cbind(seq_len(compmix.pois$ngz), R[-(1:n)])]
mR <- as.matrix(c(m1,m2), (n + compmix.pois$ngz))
v1 <- cm1$comp$v[R[1:n]]
v2 <- compmix.pois$vy[cbind(seq_len(compmix.pois$ngz), R[-(1:n)])]
invSig <- 1/sqrt(c(v1,v2))
# stacking and standardizing
tauS <- c(tau1, tau2)
offsetS <- c(offset, offset[compmix.pois$igz])
xS <- rbind(X, X[compmix.pois$igz, , drop = FALSE])
gS <- c(param$g, param$g[compmix.pois$igz])
yS <- (-log(tauS) - mR - log(gS) - log(offsetS))*invSig
Xall <- xS*kronecker(matrix(1, 1, model$d + 1), invSig)
# inverse prior variance of regression effects (updated)
invA0 <- diag(c(prior$invM0, 1/param$psi), nrow = model$d + 1)
#### Step B --- starts Bayesian variable selection
if (imc > mcmc$startsel && sum(!model$deltafix) > 0){
## (1) update mixture weights
incfix <- sum(param$delta == 1)
omega <- rbeta(1, prior$w['wa0'] + incfix, prior$w['wb0'] + model$d - incfix)
## (2) sample the indicators, the regression coefficients and the scale parameters
## --- (i) sample the indicators delta_{beta,j}, gamma_{beta} for the slab component
indic <- draw_indicators_nb(yS, Xall, param$delta, omega, model, prior, invA0)
delta <- indic$deltanew
pdelta <- indic$pdeltanew
} else {
delta <- param$delta
pdelta <- param$pdelta
omega <- param$omega
}
## --- (ii_A) sample the (selected) regression effects
index <- c(1, which(delta == 1) + 1)
Zsel <- Xall[, index, drop = FALSE] # Z*=[1, W^delta,atilde]*sqrt(Sigma^-1)
dsel <- length(index)
invA0_sel <- invA0[index, index, drop = FALSE]
a0_sel <- prior$a0[index, , drop = FALSE]
AP <- solve(invA0_sel + t(Zsel)%*%Zsel) # A = (A0^-1 + (Z*)'Sigma^-1 Z*)
aP <- AP%*%(invA0_sel%*%a0_sel + t(Zsel)%*%yS) # a = A(A0^-1*a0 + (Z*)'Sigma^-1*y
zetaP <- t(chol(AP))%*%matrix(rnorm(dsel), dsel, 1) + aP
v1 <- matrix(0, 1, model$d + 1)
v1[index] <- t(zetaP)
mu_beta <- v1[1]
if (model$d > 0){
beta <- v1[2:(model$d + 1)]
} else {
beta <- matrix(0, 1, model$d)
}
muP <- X%*%c(mu_beta, beta)
## --- (iii) sample the variance parameter of Student-t/ or normal slab
effBeta <- t(beta)
psi <- draw_psi(effBeta, delta, prior)
#### Step C ---
## (a) Sample number of degrees of freedom rho (MH step)
lambda <- exp(muP)*offset
rhoMH <- exp(rnorm(1, log(param$rho), prior$eps))
llik1 <- sum(lgamma(y + rhoMH)) - n*lgamma(rhoMH) - sum(lgamma(y + 1)) + n*rhoMH*log(rhoMH) + sum(y*log(lambda)) - sum((y + rhoMH)*log(lambda + rhoMH))
llik0 <- sum(lgamma(y + param$rho)) - n*lgamma(param$rho) - sum(lgamma(y + 1)) + n*param$rho*log(param$rho) + sum(y*log(lambda)) - sum((y + param$rho)*log(lambda + param$rho))
D1 <- llik1 + dgamma(rhoMH, shape = prior$c0, rate = prior$C0, log = TRUE)
D0 <- llik0 + dgamma(param$rho, shape = prior$c0, rate = prior$C0, log = TRUE)
q1 <- log(rhoMH)
q0 <- log(param$rho)
rho.alpha <- min(D1 - q0 - D0 + q1, 0)
u <- runif(1, 0, 1)
if(log(u) <= rho.alpha){
rho <- rhoMH
rho.acc <- 1
} else {
rho <- param$rho
rho.acc <- 0
}
## (b) Sample parameter g
g <- rgamma(n, shape = (y + rho), rate = (lambda + rho))
# returns updated par-list with parameters used for subsequent step
return(list(beta = c(mu_beta, beta), delta = delta, pdelta = pdelta,
omega = omega, psi = psi, g = g, rho = rho, rho.acc = rho.acc))
}
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