Generate Truncated Multivariate Normal Samples via Hamiltonian Monte Carlo

Generate \(p\)-dimensional truncated multivariate normal samples that satisfy an inequality constraint $$F \times x + g \ge 0,$$ where \(x\) is a column vector of the generated sample. The sampler is an exact Hamiltonian Monte Carlo (HMC) sampler as described in Pakman and Paninski (2014).

Usage

rtmvnorm_hmc(n, mean, cov, initial, Fmat, g, burn)

Arguments

n

a positive integer with the number of samples. C++: an int object.

mean

a \(p\)-dimensional mean vector. C++: an Eigen::VectorXd object.

cov

a \(p \times p\) covariance matrix of the normal distribution. C++: an Eigen::MatrixXd object.

initial

a \(p\)-dimensional initial value vector for the Markov chain. C++: an Eigen::VectorXd object.

Fmat

an \(m \times p\) constraint matrix \(F\) defining linear inequalities, where \(m\) is the number of constraints. C++: an Eigen::MatrixXd object.

g

an \(m\)-dimensional constraint vector defining linear inequalities. C++: an Eigen::VectorXd object.

burn

a non-negative integer with the number of burn-in iterations before collecting samples. C++: an int object.

Value

An \(n \times p\) matrix with each row corresponding to a sample. C++: an Eigen::MatrixXd object.

Details

The function generates samples from a truncated multivariate normal distribution with mean mean and covariance cov, subject to linear constraints defined by Fmat and g.

The user should supply an initial value that strictly satisfies the inequality constraints, although the generated samples only satisfy them weakly.

No check for symmetry is performed on the covariance matrix.

It is advisable to use a small burn-in period (e.g., 10) to allow the Markov chain to reach stationarity.

This function is a wrapper around a C++ implementation adapted from the 'tnorm' R package by Kenyon Ng.

References

Pakman, A. and Paninski, L. (2014). Exact Hamiltonian Monte Carlo for Truncated Multivariate Gaussians. Journal of Computational and Graphical Statistics, 23(2), 518–542. <doi:10.1080/10618600.2013.788448>

Ng, K. (2024). tnorm: Generate Multivariate Truncated Normal Samples. R package version 0.0.1. <https://github.com/weiyaw/tnorm>

Bates, D. and Eddelbuettel, D. (2013). Fast and Elegant Numerical Linear Algebra Using the RcppEigen Package. Journal of Statistical Software, 52(5), 1–24. <doi:10.18637/jss.v052.i05>

Examples

rtmvnorm_hmc(1, c(0, 0), diag(2), c(0, 2), diag(2), c(1, -1), 1)
#>           [,1]     [,2]
#> [1,] 0.9463479 1.316826