mvrandn() R function from [TruncatedNormal]

Truncated multivariate normal generator

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.


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

Arguments

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

Returns

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

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.

Note

The algorithm may not work or be very inefficient if $\Sigma$ is close to being rank deficient.

Examples


# 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
## Not run:

# 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
## End(Not run)

mvrandn function