BCDAG (version 1.1.2)

rDAGWishart: Random samples from a compatible DAG-Wishart distribution

Description

This function implements a direct sampling from a compatible DAG-Wishart distribution with parameters a and U.

Usage

rDAGWishart(n, DAG, a, U)

Value

A list of two elements: a \((q,q,n)\) array collecting \(n\) sampled matrices \(L\) and a \((q,q,n)\) array collecting \(n\) sampled matrices \(D\)

Arguments

n

number of samples

DAG

\((q, q)\) adjacency matrix of the DAG

a

common shape hyperparameter of the compatible DAG-Wishart, \(a > q - 1\)

U

position hyperparameter of the compatible DAG-Wishart, a \((q, q)\) s.p.d. matrix

Author

Federico Castelletti and Alessandro Mascaro

Details

Assume the joint distribution of random variables \(X_1, \dots, X_q\) is zero-mean Gaussian with covariance matrix Markov w.r.t. a Directed Acyclic Graph (DAG). The allied Structural Equation Model (SEM) representation of a Gaussian DAG-model allows to express the covariance matrix as a function of the (Cholesky) parameters \((D,L)\), collecting the regression coefficients and conditional variances of the SEM.

The DAG-Wishart distribution (Cao et. al, 2019) with shape hyperparameter \(a = (a_1, ..., a_q)\) and position hyperparameter \(U\) (a s.p.d. \((q,q)\) matrix) provides a conjugate prior for parameters \((D,L)\). In addition, to guarantee compatibility among Markov equivalent DAGs (same marginal likelihood), the default choice (here implemented) \(a_j = a + |pa(j)| - q + 1\) \((a > q - 1)\), with \(|pa(j)|\) the number of parents of node \(j\) in the DAG, was introduced by Peluso and Consonni (2020).

References

F. Castelletti and A. Mascaro (2021). Structural learning and estimation of joint causal effects among network-dependent variables. Statistical Methods and Applications, Advance publication.

X. Cao, K. Khare and M. Ghosh (2019). Posterior graph selection and estimation consistency for high-dimensional Bayesian DAG models. The Annals of Statistics 47 319-348.

S. Peluso and G. Consonni (2020). Compatible priors for model selection of high-dimensional Gaussian DAGs. Electronic Journal of Statistics 14(2) 4110 - 4132.

Examples

# Randomly generate a DAG on q = 8 nodes with probability of edge inclusion w = 0.2
q = 8
w = 0.2
set.seed(123)
DAG = rDAG(q = q, w = w)
# Draw from a compatible DAG-Wishart distribution with parameters a = q and U = diag(1,q)
outDL = rDAGWishart(n = 5, DAG = DAG, a = q, U = diag(1, q))
outDL