create_TMVN_sampler {mcmcsae}

Set up a sampler object for sampling from a possibly truncated and degenerate multivariate normal distribution

Description

This function sets up an object for multivariate normal sampling based on a specified precision matrix. Linear equality and inequality restrictions are supported. For sampling under inequality restrictions four algorithms are available. The default in that case is an exact Hamiltonian Monte Carlo algorithm (Pakman and Paninski, 2014). A related algorithm is the zig-zag Hamiltonian Monte Carlo method (Nishimura et al., 2021) in which momentum is sampled from a Laplace instead of normal distribution. Alternatively, a Gibbs sampling algorithm can be used (Rodriguez-Yam et al., 2004). The fourth option is a data augmentation method that samples from a smooth approximation to the truncated multivariate normal distribution (Souris et al., 2018).

Usage

create_TMVN_sampler(
  Q,
  mu = NULL,
  Xy = NULL,
  update.Q = FALSE,
  update.mu = update.Q,
  name = "x",
  coef.names = NULL,
  R = NULL,
  r = NULL,
  S = NULL,
  s = NULL,
  lower = NULL,
  upper = NULL,
  check.constraints = FALSE,
  method = NULL,
  reduce = NULL,
  chol.control = chol_control()
)

Arguments

Q

precision matrix of the (unconstrained) multivariate normal distribution.

mu

mean of the (unconstrained) multivariate normal distribution.

Xy

alternative to specifying mu; in this case mu is computed as Q^{-1}\code{Xy}.

update.Q

whether Q is updated for each draw.

update.mu

whether mu is updated for each draw. By default equal to update.Q.

name

name of the TMVN vector parameter.

coef.names

optional labels for the components of the vector parameter.

R

equality restriction matrix.

r

rhs vector for equality constraints R'x = r, where R' denotes the transpose of R.

S

inequality restriction matrix.

s

rhs vector for inequality constraints S'x >= s, where S' denotes the transpose of S.

lower

alternative to s for two-sided inequality restrictions \code{lower} <= S'x <= \code{upper}.

upper

alternative to s for two-sided inequality restrictions \code{lower} <= S'x <= \code{upper}.

check.constraints

if TRUE check whether the starting values satisfy all constraints.

method

sampling method. The options are "direct" for direct sampling from the unconstrained or equality constrained multivariate normal (MVN). For inequality constrained MVN sampling three methods are supported: "HMC" for (exact) Hamiltonian Monte Carlo, "HMCZigZag" for (exact) Hamiltonian Monte Carlo with Laplace momentum, "Gibbs" for a component-wise Gibbs sampling approach, and "softTMVN" for a data augmentation method that samples from a smooth approximation to the truncated MVN. Alternatively, the method setting functions m_direct, m_HMC, m_HMC_ZigZag, m_Gibbs or m_softTMVN can be used to select the method and possibly set some of its options to non-default values, see TMVN-methods.

reduce

whether to a priori restrict the simulation to the subspace defined by the equality constraints.

chol.control

options for Cholesky decomposition, see chol_control.

Details

The componentwise Gibbs sampler uses univariate truncated normal samplers as described in Botev and L'Ecuyer (2016). These samplers are implemented in R package TruncatedNormal, but here translated to C++ for an additional speed-up.

Value

An environment for sampling from a possibly degenerate and truncated multivariate normal distribution.

Author(s)

Harm Jan Boonstra, with help from Grzegorz Baltissen

References

Z.I. Botev and P. L'Ecuyer (2016). Simulation from the Normal Distribution Truncated to an Interval in the Tail. in VALUETOOLS.

Y. Cong, B. Chen and M. Zhou (2017). Fast simulation of hyperplane-truncated multivariate normal distributions. Bayesian Analysis 12(4), 1017-1037.

Y. Li and S.K. Ghosh (2015). Efficient sampling methods for truncated multivariate normal and student-t distributions subject to linear inequality constraints. Journal of Statistical Theory and Practice 9(4), 712-732.

A. Nishimura, Z. Zhang and M.A. Suchard (2021). Hamiltonian zigzag sampler got more momentum than its Markovian counterpart: Equivalence of two zigzags under a momentum refreshment limit. arXiv:2104.07694.

A. Pakman and L. Paninski (2014). Exact Hamiltonian Monte Carlo for truncated multivariate gaussians. Journal of Computational and Graphical Statistics 23(2), 518-542.

G. Rodriguez-Yam, R.A. Davis and L.L. Scharf (2004). Efficient Gibbs sampling of truncated multivariate normal with application to constrained linear regression. Unpublished manuscript.

H. Rue and L. Held (2005). Gaussian Markov Random Fields. Chapman & Hall/CRC.

A. Souris, A. Bhattacharya and P. Debdeep (2018). The Soft Multivariate Truncated Normal Distribution. arXiv:1807.09155.

K.A. Valeriano, C.E. Galarza and L.A. Matos (2023). Moments and random number generation for the truncated elliptical family of distributions. Statistics and Computing 33(1), 1-20.

Examples

S <- cbind(diag(2), c(-1, 1), c(1.1, -1))  # inequality matrix
# S'x >= 0 represents the wedge x1 <= x2 <= 1.1 x1
# example taken from Pakman and Paninski (2014)
# 1. exact Hamiltonian Monte Carlo (Pakman and Paninski, 2014)
sampler <- create_TMVN_sampler(Q=diag(2), mu=c(4, 4), S=S, method="HMC")
sim <- MCMCsim(sampler, n.iter=600, verbose=FALSE)
summary(sim)
plot(as.matrix(sim$x), pch=".")
# 2. exact Hamiltonian Monte Carlo with Laplace momentum (Nishimura et al., 2021)
sampler <- create_TMVN_sampler(Q=diag(2), mu=c(4, 4), S=S, method="HMCZigZag")
sim <- MCMCsim(sampler, n.iter=600, verbose=FALSE)
summary(sim)
plot(as.matrix(sim$x), pch=".")
# 3. Gibbs sampling approach (Rodriguez-Yam et al., 2004)
sampler <- create_TMVN_sampler(Q=diag(2), mu=c(4, 4), S=S, method="Gibbs")
sim <- MCMCsim(sampler, burnin=500, n.iter=2000, verbose=FALSE)
summary(sim)
plot(as.matrix(sim$x), pch=".")
# 4. soft TMVN approximation (Souris et al., 2018)
sampler <- create_TMVN_sampler(Q=diag(2), mu=c(4, 4), S=S, method="softTMVN")
sim <- MCMCsim(sampler, n.iter=600, verbose=FALSE)
summary(sim)
plot(as.matrix(sim$x), pch=".")