Li 2019 - Simulation Study and Gene Expression Application

Summary

GHS has the smallest Stein loss in 11 of 12 and the smallest Frobenius error in 10 of 12 simulation settings, with a very low false-positive rate (about 0.0001-0.0013) but lower true-positive rate when $n=50$. On 100 CEU gene expressions it returns a sparse graph (83 vertices, 109 edges) versus 1,135 edges (GL), 976 (BGL) and 4,950 (GSCAD, nothing zeroed).

Overview

Design: $p=100,n=50$ (Table 1); $p=100,n=120$ (Table 2); $p=200,n=120$ (Table 3); true ${\Omega }_0$ with unit diagonal and Random (prob 0.01 non-zero, magnitude $-\text{Unif}(0.2,1)$; 35 non-zero pairs at $p=100$), Hubs (10 groups of 10, ${\omega }_{ik}=0.25$; 90 pairs), Cliques (10 groups of 3, $\omega =-0.45$ positive partial correlation or $0.75$ negative; 30 pairs). 50 datasets each. Tuning (GL, GSCAD): 5-fold CV on log-likelihood. GHS/BGL: posterior mean estimate; selection by whether the 50% central credible interval excludes zero. Stein’s loss = 2 x KL of $\Omega$ from ${\Omega }_0$.

Main Content

Table 1 (excerpt): $p=100$, $n=50$, Random structure (mean, sd in parentheses)

Metric	GL1	GL2	GSCAD	BGL	GHS
Stein’s loss	10.20 (0.53)	13.42 (1.06)	10.05 (0.55)	80.92 (1.63)	6.44 (0.85)
F norm	4.33 (0.18)	5.30 (0.20)	4.31 (0.16)	5.58 (0.26)	3.31 (0.29)
TPR	.8246	.7097	.9977	.8709	.5903
FPR	.0947	.0374	.9955	.1055	.0004
CPU (min)	0.30	0.35	6.24	40.94	38.32

Hubs ($p=100,n=50$): Stein’s loss 10.12, 12.78, 10.01, 77.85, 12.56 (GHS not best here); F norm 3.95, 4.63, 3.94, 5.97, 3.96; TPR GHS .2687; FPR GHS .0013. Cliques positive: Stein GHS 5.87 vs GL1 9.16; cliques negative: GHS 6.28 vs 11.00 (F norm 3.64 vs 6.00).

Table 3 (excerpt): $p=200$, $n=120$, Random

Metric	GL1	GL2	GSCAD	BGL	GHS
Stein’s loss	10.06	15.80	9.96	116.61	2.94
F norm	4.49	5.97	4.45	6.76	2.44
TPR	.9476	.8393	1	.9855	.8421
FPR	.0514	.0159	.9951	.1035	.0001

Hubs: Stein 12.49, 15.12, 12.40, 122.87, 11.67; cliques (positive/negative): GHS 4.09/3.77 vs GL1 11.59/14.56.

Interpretation: BGL has huge Stein loss despite similar structure; GSCAD’s CV selects an almost dense matrix (FPR about 0.99); GHS is best when non-zero elements are large and $n-p$ is large (consistent with Thm 4.1) and, with $n=120$ and $p=100$, errors are comparable to an unpenalised refit of GL-selected variables. The supplement adds sensitivity/specificity/precision tables, a larger $p=400,n=120$ case and MCMC diagnostics. Caveat: 50% intervals are a pragmatic selection rule; theory for GHS discoveries is lacking (cf. van der Pas et al. 2017 for coverage in sparse normal means).

Section 6: human gene expression (CEU)

100 genes, 60 unrelated Utah individuals of Northern and Western European ancestry (data from Bhadra & Mallick 2013); centred expressions, zero-mean multivariate normal.

Method	Vertices	Edges
Graphical horseshoe	83	109
Graphical lasso (penalised diagonal)	100	1,135
Bayesian graphical lasso	100	976
Graphical SCAD	100	4,950 (no element zero)

GHS yields disconnected, much sparser clusters; GL and BGL share dense major clusters. Comparison with Bhadra & Mallick’s SNP-adjusted network shows which edges are robust to conditioning on SNPs.

Section 7 conclusions

All methods fail eventually when KL is the target in high dimensions; sample-size gap affects bias; screen variables first; open: theory for discoveries; compare with spike-and-slab lasso, horseshoe+, Dirichlet-Laplace.

Examples

Selection rule in code: given draws d of shape (nmc,p,p) from the sampler, lo, hi = np.percentile(d, [25, 75], axis=0); edge $(i,j)$ is declared if lo[i,j] > 0 or hi[i,j] < 0.

Connections

Bayesian Gaussian Graphical Models - G-Wishart and Neighbourhood Selection (alternative priors and selection rules); Stability Selection Simulation Study - Designs A-G and Results (frequentist comparison on graphical lasso).

See Also

Li 2019 - Graphical Horseshoe Estimator - Overview