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Bayesian Structure Learning in Sparse Gaussian Graphical Models
A. Mohammadi, E. C. Wit
TL;DR
The paper tackles Bayesian graph structure learning when many variables must be analyzed from relatively few observations. It introduces a continuous-time birth-death MCMC framework for Gaussian graphical models, with rates chosen so the stationary distribution is the joint posterior. The method is reported to converge quickly, avoid rejection of graph moves, and remain scalable for large graphs, while non-decomposable models still require approximate G-Wishart normalizing constants.
Problem
Bayesian Gaussian graphical modeling must search an astronomical graph space to learn conditional-independence structure from high-dimensional, relatively small datasets.
Method
The paper uses continuous-time birth-death MCMC, adding or removing graph edges with rates designed so the stationary distribution equals the joint posterior.
Results
The method is reported to outperform alternative Bayesian approaches in convergence, graph-space mixing, and computing time, while graph moves are always accepted.
Takeaways & Limitations
The framework provides a scalable Bayesian approach that distinguishes important from irrelevant edges and detects the true model with high accuracy in large graphs.
Takeaways & Limitations
For non-decomposable graphs, the G-Wishart normalizing constant has no explicit form and must be numerically approximated.
Abstract
from arXiv · showhide
Decoding complex relationships among large numbers of variables with relatively few observations is one of the crucial issues in science. One approach to this problem is Gaussian graphical modeling, which describes conditional independence of variables through the presence or absence of edges in the underlying graph. In this paper, we introduce a novel and efficient Bayesian framework for Gaussian graphical model determination which is a trans-dimensional Markov Chain Monte Carlo (MCMC) approach based on a continuous-time birth-death process. We cover the theory and computational details of the method. It is easy to implement and computationally feasible for high-dimensional graphs. We show our method outperforms alternative Bayesian approaches in terms of convergence, mixing in the graph space and computing time. Unlike frequentist approaches, it gives a principled and, in practice, sensible approach for structure learning. We illustrate the efficiency of the method on a broad range of simulated data. We then apply the method on large-scale real applications from human and mammary gland gene expression studies to show its empirical usefulness. In addition, we implemented the method in the R package BDgraph which is freely available at http://CRAN.R-project.org/package=BDgraph
1 Introduction
The paper addresses Bayesian structure learning for Gaussian graphical models when many variables are observed only a few times. It introduces a continuous-time birth-death MCMC framework intended to search large graph spaces while learning graph structure and parameters.
- Gaussian graphical models represent conditional independence through edges determined by nonzero off-diagonal precision-matrix entries.
- For p variables, the graph space contains 2^p(p−1)/2 possible conditional-independence graphs, exceeding 10^100 graphs in high-dimensional genetic networks.
- Existing reversible-jump MCMC methods can be inefficient in high-dimensional settings because proposed model moves may have low acceptance probabilities.
- The proposed BDMCMC framework adds or removes edges through continuous-time birth and death events modeled as independent Poisson processes.The event waiting times are exponential, and relative birth and death rates determine the stationary distribution.
- The paper develops theory, implementation details, and an efficient precision-matrix sampler, then evaluates the method in simulations and applications.
2 Bayesian Gaussian graphical models
This section defines Gaussian graphical models, their likelihood and priors, and the G-Wishart precision-matrix prior. It also identifies computational difficulties for non-decomposable graphs and motivates the subsequent trans-dimensional sampler.
- An undirected Gaussian graphical model uses graph nodes as variables and constrains precision-matrix entries to zero for non-existing edges.
- The likelihood is based on an independent and identically distributed sample x=(x_1,...,x_n), with sufficient statistic S=x′x.
- The paper proposes graph priors including a discrete uniform distribution over graph space and a truncated Poisson distribution on graph size |E|.
- The G-Wishart prior is conjugate for normally distributed data and yields a posterior G-Wishart distribution conditional on the graph and observations.The posterior parameters are b*=b+n and D*=D+S.
- For non-decomposable graphs, the G-Wishart normalizing constant lacks an explicit form and requires numerical approximation such as Monte Carlo or Laplace methods.
- The following section introduces an efficient trans-dimensional MCMC sampler for the joint posterior distribution.
3 The birth-death MCMC method
The paper’s BDMCMC method explores graph and precision-matrix states through continuous-time edge births and deaths, with rates chosen so the joint posterior is stationary. Inference uses waiting-time weights, while computational procedures address rate evaluation and precision-matrix sampling.
- Birth-death process: Continuous-time birth and death events add or remove edges while updating the associated precision matrix.Each edge has an independent Poisson birth or death process, and the process jumps to the corresponding graph state.
- Event simulation: The time between successive events is exponentially distributed with mean 1/(β(K)+δ(K)), and event type probabilities use the total birth and death rates.The overall rates are sums of the edge-specific rates.
- Stationarity: The birth-death process is constructed so its stationary distribution equals the joint posterior of the graph and precision matrix.Theorem 3.1 provides a sufficient condition for this stationary-distribution property.
- Algorithm: The algorithm alternates simulating a birth or death jump with sampling a new precision matrix from its posterior distribution.The main computational components are calculating birth and death rates and directly sampling from the G-Wishart distribution.
- Inference: Posterior graph probabilities are estimated from each graph’s proportion of total waiting time in the continuous-time trajectory.Waiting-time weights produce a Rao-Blackwellized sample-mean estimator.
- Normalizing constants: An exact G-Wishart sampler supplies an auxiliary precision matrix so intractable prior normalizing constants can be replaced by evaluating H.This procedure is used when calculating death rates.
4 Statistical performance
Across simulations, BDMCMC showed faster convergence and strong graph-selection performance than Bayesian and frequentist alternatives, while implementation costs varied by comparison.
- Bayesian comparisons: BDMCMC converged after approximately 10,000 iterations, compared with approximately 30,000 for Lenkoski and no convergence after 60,000 for WL.Convergence was assessed using cumulative occupancy fractions for all possible edges.
- Bayesian comparisons: The estimated posterior probability of the true graph was 0.66 for BDMCMC, compared with 0.40 for Lenkoski and 0.35 for WL.In the same simulation, BDMCMC visited around 450 graphs, Lenkoski around 400, and WL only 23.
- Bayesian comparisons: BDMCMC had the best F1-score and calibration error in most simulated cases, while BDMCMC and Lenkoski generally outperformed WL.The authors attribute the pattern mainly to faster convergence.
- Computational comparison: BDMCMC was slower than Lenkoski because it scans all possible edges for birth/death rates, but faster than WL in the reported computing-time comparison.Randomly selecting one edge makes Lenkoski computationally faster but less efficient according to the authors.
- Comparison with frequentist methods: BDMCMC’s F1-score typically outperformed frequentist methods, except in the high-observation scenario where it roughly matched the mb method with stars criterion.Frequentist methods performed well in some settings and poorly in others, depending on graph structure and criterion.
- Comparison with frequentist methods: For precision-matrix estimation, BDMCMC achieved the best KL performance in all scenarios except one.KL is positive-valued, with 0 as the minimum and smaller values preferred.
5 Discussion
The paper presents a scalable Bayesian BDMCMC framework for Gaussian graph structure learning that addresses growing-dimensional computational and methodological challenges. Simulations show benefits over reversible-jump approaches, while the framework remains extensible to broader graph models.
- 5 Discussion: The proposed BDMCMC method targets Gaussian graph structure learning and has a stationary distribution equal to the joint posterior of graph and precision matrix.The method is designed to distinguish important from irrelevant edges and detect the true model with high accuracy.
- 5 Discussion: Moves between graphs are always accepted, avoiding the high rejection rates that discrete-time trans-dimensional MCMC can suffer in high-dimensional graphs.This benefit comes with additional computational cost for calculating birth and death rates.
- 5 Discussion: The approach addresses convergence, prior-normalizing-constant computation, and posterior precision-matrix sampling as dimensionality grows.It uses BDMCMC for convergence, an exchange algorithm for the intractable normalizing constant, and an exact sampler for the precision matrix.
- 5 Discussion: The framework is flexible enough to accommodate different priors and extensions to multivariate time-series, log-linear, non-Gaussian, and copula regression graphical models.Extensions beyond normally distributed data are identified as future work.
Appendix 1: Proof of theorem 1
The appendix formulates the continuous-time birth-death process through state-dependent birth and death rates and transition kernels. It establishes detailed balance so the process has the desired posterior stationary distribution.
- Appendix 1: Proof of theorem 1: The process state consists of a graph G and precision matrix K, with behavior defined by birth rates, death rates, and transition kernels.Birth and death transitions move between graph states differing by one edge.
- Appendix 1: Proof of theorem 1: Birth and death kernels describe the probabilities of jumping from the current state to states with one added or removed edge.The kernels are expressed over the corresponding precision-matrix spaces.
- Appendix 1: Proof of theorem 1: Detailed balance requires transitions caused by deaths and births to match in both directions for measurable state sets.The appendix explains the two balance expressions as reciprocal flow conditions.
- Appendix 1: Proof of theorem 1: The new precision-matrix element is assigned a proposed density before deriving the theorem’s balance expression.The appendix introduces notation and then specifies the proposal used in the construction.
Appendix 2: Proposition
The proposition characterizes a 2 × 2 Wishart-distributed random matrix and cites a standard result for its properties. It also identifies when the G-Wishart normalizing constant is explicit or requires approximation.
- Appendix 2: Proposition: The proposition assumes a 2 × 2 random matrix A with Wishart distribution W(b, D).The cited proof relies on Muirhead’s theorem for the relevant Wishart result.