BGGM: Bayesian Gaussian Graphical Models

The R package BGGM provides tools for making Bayesian inference in Gaussian graphical models (GGM). The methods are organized around two general approaches for Bayesian inference: (1) estimation (Williams 2018) and (2) hypothesis testing (Williams and Mulder 2019) . The key distinction is that the former focuses on either the posterior or posterior predictive distribution, whereas the latter focuses on model comparison with the Bayes factor.

The methods in BGGM build upon existing algorithms that are well-known in the literature. The central contribution of BGGM is to extend those approaches:

1. Bayesian estimation with the novel matrix-F prior distribution (Mulder and Pericchi 2018) .

   • Estimation estimate.

2. Bayesian hypothesis testing with the novel matrix-F prior distribution (Mulder and Pericchi 2018) .

   • Exploratory hypothesis testing explore.

   • Confirmatory hypothesis testing confirm.

3. Comparing GGMs (Williams et al. 2020)

   • Partial correlation differences ggm_compare_estimate.

   • Posterior predictive check ggm_compare_ppc.

   • Exploratory hypothesis testing ggm_compare_explore.

   • Confirmatory hypothesis testing ggm_compare_confirm.

4. Extending inference beyond the conditional (in)dependence structure

   • Predictability with Bayesian variance explained (Gelman et al. 2019) predictability.

   • Posterior uncertainty in the partial correlations estimate.

   • Custom Network Statistics roll_your_own.

Furthermore, the computationally intensive tasks are written in c++ via the R package Rcpp (Eddelbuettel et al. 2011) and the c++ library Armadillo (Sanderson and Curtin 2016) , there are plotting functions for each method, control variables can be included in the model, and there is support for missing values bggm_missing.

Supported Data Types:

• Continuous: The continuous method was described in Williams and Mulder (2019) .

• Binary: The binary method builds directly upon in Talhouk et al. (2012) , that, in turn, built upon the approaches of Lawrence et al. (2008) and Webb and Forster (2008) (to name a few).

• Ordinal: Ordinal data requires sampling thresholds. There are two approach included in BGGM: (1) the customary approach described in in Albert and Chib (1993) (the default) and the 'Cowles' algorithm described in in Cowles (1996) .

• Mixed: The mixed data (a combination of discrete and continuous) method was introduced in Hoff (2007) . This is a semi-parametric copula model (i.e., a copula GGM) based on the ranked likelihood. Note that this can be used for data consisting entirely of ordinal data.

Additional Features:

The primary focus of BGGM is Gaussian graphical modeling (the inverse covariance matrix). The residue is a suite of useful methods not explicitly for GGMs:

1. Bivariate correlations for binary (tetrachoric), ordinal (polychoric), mixed (rank based), and continous (Pearson's) data zero_order_cors.

2. Multivariate regression for binary (probit), ordinal (probit), mixed (rank likelihood), and continous data (estimate).

3. Multiple regression for binary (probit), ordinal (probit), mixed (rank likelihood), and continous data (e.g., coef.estimate).

Note on Conditional (In)dependence Models for Latent Data:

All of the data types (besides continuous) model latent data. That is, unoboserved data that is assumed to be Gaussian distributed. For example, a tetrachoric correlation (binary data) is a special case of a polychoric correlation (ordinal data). Both relations are between "theorized normally distributed continuous latent variables" (Wikipedia). In both instances, the correpsonding partial correlation between observed variables is conditioned on the remaining variables in the latent space. This implies that interpration is similar to continuous data, but with respect to latent variables. We refer interested users to page 2364, section 2.2, in Webb and Forster (2008) .

High Dimensional Data?

BGGM was builit specificially for social-behvarioal scientists. Of course, the methods can be used by all researchers. However, there is not support for high-dimensonal data (i.e., more variables than observations) that are common place in the genetics literature. These data are rare in the social-behavioral sciences. In the future, support for high-dimensional data may be added to BGGM.

References

Albert JH, Chib S (1993). “Bayesian analysis of binary and polychotomous response data.” Journal of the American statistical Association, 88(422), 669--679.

Cowles MK (1996). “Accelerating Monte Carlo Markov chain convergence for cumulative-link generalized linear models.” Statistics and Computing, 6(2), 101--111.

Eddelbuettel D, Fran攼㸷ois R, Allaire J, Ushey K, Kou Q, Russel N, Chambers J, Bates D (2011). “Rcpp: Seamless R and C++ integration.” Journal of Statistical Software, 40(8), 1--18.

Gelman A, Goodrich B, Gabry J, Vehtari A (2019). “R-squared for Bayesian Regression Models.” American Statistician, 73(3), 307--309. ISSN 15372731, doi: 10.1080/00031305.2018.1549100 .

Hoff PD (2007). “Extending the rank likelihood for semiparametric copula estimation.” The Annals of Applied Statistics, 1(1), 265--283.

Lawrence E, Bingham D, Liu C, Nair VN (2008). “Bayesian inference for multivariate ordinal data using parameter expansion.” Technometrics, 50(2), 182--191.

Mulder J, Pericchi L (2018). “The Matrix-F Prior for Estimating and Testing Covariance Matrices.” Bayesian Analysis, 1--22. ISSN 19316690, doi: 10.1214/17-BA1092 .

Sanderson C, Curtin R (2016). “Armadillo: a template-based C++ library for linear algebra.” Journal of Open Source Software, 1(2), 26.

Talhouk A, Doucet A, Murphy K (2012). “Efficient Bayesian inference for multivariate probit models with sparse inverse correlation matrices.” Journal of Computational and Graphical Statistics, 21(3), 739--757.

Webb EL, Forster JJ (2008). “Bayesian model determination for multivariate ordinal and binary data.” Computational statistics \& data analysis, 52(5), 2632--2649.

Williams DR (2018). “Bayesian Estimation for Gaussian Graphical Models: Structure Learning, Predictability, and Network Comparisons.” arXiv. doi: 10.31234/OSF.IO/X8DPR .

Williams DR, Mulder J (2019). “Bayesian Hypothesis Testing for Gaussian Graphical Models: Conditional Independence and Order Constraints.” PsyArXiv. doi: 10.31234/osf.io/ypxd8 .

Williams DR, Rast P, Pericchi LR, Mulder J (2020). “Comparing Gaussian graphical models with the posterior predictive distribution and Bayesian model selection.” Psychological Methods.