MultivariateNormal: The Multivariate Normal Distribution

In nimble: MCMC, Particle Filtering, and Programmable Hierarchical Modeling

The Multivariate Normal Distribution

Description

Density and random generation for the multivariate normal distribution, using the Cholesky factor of either the precision matrix or the covariance matrix (when not using AD) or the precision or covariance matrix (when using AD).

Usage

dmnorm_chol(x, mean, cholesky, prec_param = TRUE, log = FALSE)

rmnorm_chol(n = 1, mean, cholesky, prec_param = TRUE)

dmnorm_inv_ld(x, mean, mat, inv_ld, prec_param = TRUE, log = FALSE)

rmnorm_inv_ld(n = 1, mean, mat, inv_ld, prec_param = TRUE)

Arguments

x	vector of values.
mean	vector of values giving the mean of the distribution.
cholesky	in standard cases (when not using AD), upper-triangular Cholesky factor of either the precision matrix (when prec_param is TRUE) or covariance matrix (otherwise).
prec_param	logical; in standard cases (when not using AD): if TRUE then cholesky is that of the precision matrix; otherwise, of the covariance matrix. When using AD: if TRUE then mat is the precision matrix; otherwise the covariance matrix.
log	logical; if TRUE, probability density is returned on the log scale.
n	number of observations (only n=1 is handled currently).
mat	when using AD, either the precision matrix (when prec_param is TRUE) or covariance matrix (otherwise).
inv_ld	when using AD, the inverse of mat, as a vector (ordered column-major), with lower diagonal elements ignored (and typically set to 0), appended with an additional element containing the log of the determinant of the inverse of mat. inv_ld can be obtained by inv_ld = PDinverse_logdet(mat). When prec_param is FALSE (or 0), only the last element (log determinant) is used for density calculations (but the rest is still used for derivative calculations). inv_ld is not used for random generation but is included in the argument list because for use in nimble models the available arguments for density and random number generation must match.

Details

See Gelman et al., Appendix A or the BUGS manual for mathematical details. The rate matrix as used here is defined as the inverse of the scale matrix, S^{-1}, given in Gelman et al. Note that if one wishes to take derivatives (using AD) involving the distribution, one should use the "inv_ld" versions, as use of the Cholesky is computationally inefficient.

Value

dmnorm_chol gives the density and rmnorm_chol generates random deviates.

Author(s)

Christopher Paciorek, Perry de Valpine

References

Gelman, A., Carlin, J.B., Stern, H.S., and Rubin, D.B. (2004) Bayesian Data Analysis, 2nd ed. Chapman and Hall/CRC.

See Also

Distributions for other standard distributions

Examples

mean <- c(-10, 0, 10)
covmat <- matrix(c(1, .9, .3, .9, 1, -0.1, .3, -0.1, 1), 3)
ch <- chol(covmat)
x <- rmnorm_chol(1, mean, ch, prec_param = FALSE)
dmnorm_chol(x, mean, ch, prec_param = FALSE)

nimble documentation built on Feb. 13, 2026, 9:07 a.m.