mvtprqmc <- function(n, L, l, u, nu, mu){
# computes P(l<X<u), where X is student with
# Sig=L*L', zero mean vector, and degrees of freedom 'nu';
# exponential tilting uses parameter 'mu';
# Quasi Monte Carlo uses 'n' samples;
d <- length(l); # Initialization
eta <- mu[d];
Z <- matrix(0, nrow = d, ncol = n); # create array for variables
# QMC pointset
if(n*(d-1) > 2e7){
warning("High memory requirements for storage of QMC sequence\nConsider reducing n")
}
# x <- as.matrix(randtoolbox::sobol(n, dim = d - 1, init = TRUE, scrambling = 1, seed = ceiling(1e6 * runif(1))))
x <- as.matrix(qrng::sobol(n = n,
d = d - 1,
randomize = "digital.shift",
seed = ceiling(1e6 * runif(1))))
#Fixed 21.03.2018 to ensure that if d=2, no error returned
# Monte Carlo uses 'n' samples;
# precompute constants
const <- log(2*pi) / 2 - lgamma(nu / 2) - (nu / 2 - 1) * log(2) +
lnNpr(-eta, Inf) + 0.5 * eta^2;
R <- eta + trandn(rep(-eta, n), rep(Inf, n));
# simulate R~N(eta,1) with R>0
p <- (nu - 1) * log(R) - eta * R; # compute Likelihood Ratio for R
R <- R / sqrt(nu); # scale parameter divided by nu
for(k in 1:(d-1)){
# compute matrix multiplication L*Z
col <- c(L[k,1:k] %*% Z[1:k,])
#bottleneck, but hard to reduce
# compute limits of truncation
tl <- R * l[k] - mu[k] - col;
tu <- R * u[k] - mu[k] - col;
#simulate N(mu,1) conditional on [tl,tu]
Z[k,] <- mu[k] + norminvp(x[1:n,k], tl, tu);
# update likelihood ratio
p <- p + lnNpr(tl,tu) + .5*mu[k]^2-mu[k]*Z[k,];
}
# deal with final Z(d) which need not be simulated
col <- c(L[d,] %*% Z);
tl <- R * l[d] - col
tu <- R * u[d] - col;
p <- p + lnNpr(tl, tu); # update LR corresponding to Z(d)
return(exp(const)*mean(exp(p)))
}
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