# rgwish: Sampling from G-Wishart distribution In BDgraph: Bayesian Structure Learning in Graphical Models using Birth-Death MCMC

## Description

Generates random matrices, distributed according to the G-Wishart distribution with parameters b and D, W_G(b, D) with respect to the graph structure G. Note this fuction works for both non-decomposable and decomposable graphs.

## Usage

 1  rgwish( n = 1, adj = NULL, b = 3, D = NULL, threshold = 1e-8 ) 

## Arguments

 n The number of samples required. adj The adjacency matrix corresponding to the graph structure which can be non-decomposable or decomposable. It should be an upper triangular matrix in which a_{ij}=1 if there is a link between notes i and j, otherwise a_{ij}=0. adj could be an object of class "graph", from function graph.sim. It also could be an object of class "sim", from function bdgraph.sim. It also could be an object of class "bdgraph", from functions bdgraph.mpl or bdgraph. b The degree of freedom for G-Wishart distribution, W_G(b, D). D The positive definite (p \times p) "scale" matrix for G-Wishart distribution, W_G(b, D). The default is an identity matrix. threshold The threshold value for the convergence of sampling algorithm from G-Wishart.

## Details

Sampling from G-Wishart distribution, K \sim W_G(b, D), with density:

Pr(K) \propto |K| ^ {(b - 2) / 2} \exp ≤ft\{- \frac{1}{2} \mbox{trace}(K \times D)\right\},

which b > 2 is the degree of freedom and D is a symmetric positive definite matrix.

## Value

A numeric array, say A, of dimension (p \times p \times n), where each A[,,i] is a positive definite matrix, a realization of the G-Wishart distribution, W_G(b, D). Note, for the case n=1, the output is a matrix.

## References

Lenkoski, A. (2013). A direct sampler for G-Wishart variates, Stat, 2:119-128

Mohammadi, R. and Wit, E. C. (2019). BDgraph: An R Package for Bayesian Structure Learning in Graphical Models, Journal of Statistical Software, 89(3):1-30

Mohammadi, A. and Wit, E. C. (2015). Bayesian Structure Learning in Sparse Gaussian Graphical Models, Bayesian Analysis, 10(1):109-138

Letac, G., Massam, H. and Mohammadi, R. (2018). The Ratio of Normalizing Constants for Bayesian Graphical Gaussian Model Selection, arXiv preprint arXiv:1706.04416v2

Mohammadi, A. et al (2017). Bayesian modelling of Dupuytren disease by using Gaussian copula graphical models, Journal of the Royal Statistical Society: Series C, 66(3):629-645

gnorm, rwish

## Examples

 1 2 3 4 5 6 7 8 9 # Generating a 'circle' graph as a non-decomposable graph adj <- graph.sim( p = 5, graph = "circle" ) adj # adjacency of graph with 5 nodes sample <- rgwish( n = 1, adj = adj, b = 3, D = diag( 5 ) ) round( sample, 2 ) sample <- rgwish( n = 5, adj = adj ) round( sample, 2 ) 

### Example output

     [,1] [,2] [,3] [,4] [,5]
[1,]    0    1    0    0    1
[2,]    1    0    1    0    0
[3,]    0    1    0    1    0
[4,]    0    0    1    0    1
[5,]    1    0    0    1    0
attr(,"class")
[1] "graph"
[,1]  [,2]  [,3]  [,4]  [,5]
[1,]  0.59  0.06  0.00  0.00 -0.04
[2,]  0.06  1.68 -0.21  0.00  0.00
[3,]  0.00 -0.21  2.64 -2.70  0.00
[4,]  0.00  0.00 -2.70  3.49  0.48
[5,] -0.04  0.00  0.00  0.48  7.42
, , 1

[,1] [,2] [,3] [,4] [,5]
[1,] 8.26 2.63 0.00 0.00 0.58
[2,] 2.63 8.92 1.27 0.00 0.00
[3,] 0.00 1.27 3.17 0.62 0.00
[4,] 0.00 0.00 0.62 1.80 1.86
[5,] 0.58 0.00 0.00 1.86 2.65

, , 2

[,1]  [,2]  [,3] [,4] [,5]
[1,] 3.78  1.85  0.00 0.00 3.11
[2,] 1.85  3.14 -0.46 0.00 0.00
[3,] 0.00 -0.46  5.63 1.63 0.00
[4,] 0.00  0.00  1.63 5.35 2.41
[5,] 3.11  0.00  0.00 2.41 8.05

, , 3

[,1]  [,2] [,3]  [,4]  [,5]
[1,]  1.64 -0.82 0.00  0.00 -0.62
[2,] -0.82  3.02 1.04  0.00  0.00
[3,]  0.00  1.04 4.55  0.07  0.00
[4,]  0.00  0.00 0.07  3.97 -1.31
[5,] -0.62  0.00 0.00 -1.31  1.08

, , 4

[,1] [,2]  [,3]  [,4] [,5]
[1,] 19.21 2.16  0.00  0.00 2.21
[2,]  2.16 1.75  2.27  0.00 0.00
[3,]  0.00 2.27  6.17 -3.45 0.00
[4,]  0.00 0.00 -3.45  8.40 1.92
[5,]  2.21 0.00  0.00  1.92 2.40

, , 5

[,1]  [,2]  [,3]  [,4] [,5]
[1,] 8.26  0.81  0.00  0.00 3.15
[2,] 0.81  2.86 -2.05  0.00 0.00
[3,] 0.00 -2.05  9.55  3.22 0.00
[4,] 0.00  0.00  3.22 15.38 3.22
[5,] 3.15  0.00  0.00  3.22 5.75


BDgraph documentation built on May 3, 2021, 9:08 a.m.