Truncated multivariate normal generator
Simulate n independent and identically distributed random vectors from the d-dimensional N(0,Σ) distribution (zero-mean normal with covariance Σ) conditional on l<X<u. Infinite values for l and u are accepted.
mvrandn(l, u, Sig, n, mu = NULL)
l |
lower truncation limit |
u |
upper truncation limit |
Sig |
covariance matrix |
n |
number of simulated vectors |
mu |
location parameter |
Bivariate normal: Suppose we wish to simulate a bivariate X from N(μ,Σ), conditional on X_1-X_2<-6. We can recast this as the problem of simulation of Y from N(0,AΣ A^\top) (for an appropriate matrix A) conditional on l-Aμ < Y < u-Aμ and then setting X=μ+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 β be N(0,ν^2 I). Then, to simulate from the Bayesian
posterior exactly, we first simulate
Z from N(0,Σ), where Σ=I+ν^2 X X^\top,
conditional on Z≥ 0. Then, we simulate the posterior regression coefficients, β, of the Probit regression
by drawing (β|Z) from N(C X^\top Z,C), where C^{-1}=I/ν^2+X^\top X.
See the example code below.
a d by n matrix storing the random vectors, X, drawn from N(0,Σ), conditional on l<X<u;
The algorithm may not work or be very inefficient if Σ is close to being rank deficient.
# 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)Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.