mvrandn: Truncated multivariate normal generator

Description

Simulate \(n\) independent and identically distributed random vectors from the \(d\)-dimensional \(N(0,\Sigma)\) distribution (zero-mean normal with covariance \(\Sigma\)) conditional on \(l<X<u\). Infinite values for \(l\) and \(u\) are accepted.

Usage

mvrandn(l, u, Sig, n, mu = NULL)

Value

a \(d\) by \(n\) matrix storing the random vectors, \(X\), drawn from \(N(0,\Sigma)\), conditional on \(l<X<u\);

Arguments

l: lower truncation limit
u: upper truncation limit
Sig: covariance matrix
n: number of simulated vectors
mu: location parameter

Details

Bivariate normal: Suppose we wish to simulate a bivariate \(X\) from \(N(\mu,\Sigma)\), conditional on \(X_1-X_2<-6\). We can recast this as the problem of simulation of \(Y\) from \(N(0,A\Sigma A^\top)\) (for an appropriate matrix \(A\)) conditional on \(l-A\mu < Y < u-A\mu\) and then setting \(X=\mu+A^{-1}Y\). See the example code below.
Exact posterior simulation for Probit regression: Consider the Bayesian Probit Regression model applied to the lupus dataset. Let the prior for the regression coefficients \(\beta\) be \(N(0,\nu^2 I)\). Then, to simulate from the Bayesian posterior exactly, we first simulate \(Z\) from \(N(0,\Sigma)\), where \(\Sigma=I+\nu^2 X X^\top,\) conditional on \(Z\ge 0\). Then, we simulate the posterior regression coefficients, \(\beta\), of the Probit regression by drawing \((\beta|Z)\) from \(N(C X^\top Z,C)\), where \(C^{-1}=I/\nu^2+X^\top X\). See the example code below.

Examples

Run this code

 # Bivariate example.

 Sig <- matrix(c(1,0.9,0.9,1), 2, 2);
 mu <- c(-3,0); l <- c(-Inf,-Inf); u <- c(-6,Inf);
 A <- matrix(c(1,0,-1,1),2,2);
 n <- 1e3; # number of sampled vectors
 Y <- mvrandn(l - A %*% mu, u - A %*% mu, A %*% Sig %*% t(A), n);
 X <- rep(mu, n) + solve(A, diag(2)) %*% Y;
 # now apply the inverse map as explained above
 plot(X[1,], X[2,]) # provide a scatterplot of exactly simulated points
if (FALSE) {
# Exact Bayesian Posterior Simulation Example.

data("lupus"); # load lupus data
Y = lupus[,1]; # response data
X = lupus[,-1]  # construct design matrix
m=dim(X)[1]; d=dim(X)[2]; # dimensions of problem
 X=diag(2*Y-1) %*%X; # incorporate response into design matrix
 nu=sqrt(10000); # prior scale parameter
 C=solve(diag(d)/nu^2+t(X)%*%X);
 L=t(chol(t(C))); # lower Cholesky decomposition
 Sig=diag(m)+nu^2*X %*% t(X); # this is covariance of Z given beta
 l=rep(0,m);u=rep(Inf,m);
 est=mvNcdf(l,u,Sig,1e3);
 # estimate acceptance probability of Crude Monte Carlo
 print(est$upbnd/est$prob)
 # estimate the reciprocal of acceptance probability
 n=1e4 # number of iid variables
 z=mvrandn(l,u,Sig,n);
 # sample exactly from auxiliary distribution
 beta=L %*% matrix(rnorm(d*n),d,n)+C %*% t(X) %*% z;
 # simulate beta given Z and plot boxplots of marginals
 boxplot(t(beta))
 # plot the boxplots of the marginal
 # distribution of the coefficients in beta
 print(rowMeans(beta)) # output the posterior means
 }