Inference In Graphical Models Research Articles

A Statistical Test for Differential Network Analysis Based on Inference of Gaussian Graphical Model

Differential network analysis investigates how the network of connected genes changes from one condition to another and has become a prevalent tool to provide a deeper and more comprehensive understanding of the molecular etiology of complex diseases. Based on the asymptotically normal estimation of large Gaussian graphical model (GGM) in the high-dimensional setting, we developed a computationally efficient test for differential network analysis through testing the equality of two precision matrices, which summarize the conditional dependence network structures of the genes. Additionally, we applied a multiple testing procedure to infer the differential network structure with false discovery rate (FDR) control. Through extensive simulation studies with different combinations of parameters including sample size, number of vertices, level of heterogeneity and graph structure, we demonstrated that our method performed much better than the current available methods in terms of accuracy and computational time. In real data analysis on lung adenocarcinoma, we revealed a differential network with 3503 nodes and 2550 edges, which consisted of 50 clusters with an FDR threshold at 0.05. Many of the top gene pairs in the differential network have been reported relevant to human cancers. Our method represents a powerful tool of network analysis for high-dimensional biological data.

Variational BEJG Solvers for Marginal-MAP Inference with Accurate Approximation of B-Conditional Entropy

Previously proposed variational techniques for approximate MMAP inference in complex graphical models of high-order factors relax a dual variational objective function to obtain its tractable approximation, and further perform MMAP inference in the resulting simplified graphical model, where the sub-graph with decision variables is assumed to be a disconnected forest. In contrast, we developed novel variational MMAP inference algorithms and proximal convergent solvers, where we can improve the approximation accuracy while better preserving the original MMAP query by designing such a dual variational objective function that an upper bound approximation is applied only to the entropy of decision variables. We evaluate the proposed algorithms on both simulated synthetic datasets and diagnostic Bayesian networks taken from the UAI inference challenge, and our solvers outperform other variational algorithms in a majority of reported cases. Additionally, we demonstrate the important real-life application of the proposed variational approaches to solve complex tasks of policy optimization by MMAP inference, and performance of the implemented approximation algorithms is compared. Here, we demonstrate that the original task of optimizing POMDP controllers can be approached by its reformulation as the equivalent problem of marginal-MAP inference in a novel single-DBN generative model, which guarantees that the control policies computed by probabilistic inference over this model are optimal in the traditional sense. Our motivation for approaching the planning problem through probabilistic inference in graphical models is explained by the fact that by transforming a Markovian planning problem into the task of probabilistic inference (a marginal MAP problem) and applying belief propagation techniques in generative models, we can achieve a computational complexity reduction from PSPACE-complete or NEXP-complete to NPPP-complete in comparison to solving the POMDP and Dec-POMDP models respectively search vs. dynamic programming).

Lifted Hinge-Loss Markov Random Fields

Statistical relational learning models are powerful tools that combine ideas from first-order logic with probabilistic graphical models to represent complex dependencies. Despite their success in encoding large problems with a compact set of weighted rules, performing inference over these models is often challenging. In this paper, we show how to effectively combine two powerful ideas for scaling inference for large graphical models. The first idea, lifted inference, is a wellstudied approach to speeding up inference in graphical models by exploiting symmetries in the underlying problem. The second idea is to frame Maximum a posteriori (MAP) inference as a convex optimization problem and use alternating direction method of multipliers (ADMM) to solve the problem in parallel. A well-studied relaxation to the combinatorial optimization problem defined for logical Markov random fields gives rise to a hinge-loss Markov random field (HLMRF) for which MAP inference is a convex optimization problem. We show how the formalism introduced for coloring weighted bipartite graphs using a color refinement algorithm can be integrated with the ADMM optimization technique to take advantage of the sparse dependency structures of HLMRFs. Our proposed approach, lifted hinge-loss Markov random fields (LHL-MRFs), preserves the structure of the original problem after lifting and solves lifted inference as distributed convex optimization with ADMM. In our empirical evaluation on real-world problems, we observe up to a three times speed up in inference over HL-MRFs.

Marginal Inference in Continuous Markov Random Fields Using Mixtures

Exact marginal inference in continuous graphical models is computationally challenging outside of a few special cases. Existing work on approximate inference has focused on approximately computing the messages as part of the loopy belief propagation algorithm either via sampling methods or moment matching relaxations. In this work, we present an alternative family of approximations that, instead of approximating the messages, approximates the beliefs in the continuous Bethe free energy using mixture distributions. We show that these types of approximations can be combined with numerical quadrature to yield algorithms with both theoretical guarantees on the quality of the approximation and significantly better practical performance in a variety of applications that are challenging for current state-of-the-art methods.

Model-free tracker for multiple objects using joint appearance and motion inference.

Model-free tracking is a widely-accepted approach to track an arbitrary object in a video using a single frame annotation with no further prior knowledge about the object of interest. Extending this problem to track multiple objects is really challenging because: a) the tracker is not aware of the objects' type while trying to distinguish them from background (detection task), and b) The tracker needs to distinguish one object from other potentially similar objects (data association task) to generate stable trajectories. In order to track multiple arbitrary objects, most existing model-free tracking approaches rely on tracking each target individually by updating their appearance model independently. Therefore, in this scenario they often fail to perform well due to confusion between the appearance of similar objects, their sudden appearance changes and occlusion. To tackle this problem, we propose to use both appearance and motion models, and to learn them jointly using graphical models and deep neural networks features. We introduce an indicator variable to predict sudden appearance change and/or occlusion. When these happen, our model does not update the appearance model thus avoiding using the background and/or incorrect object to update the appearance of the object of interest mistakenly, and relies on our motion model to track. Moreover, we consider the correlation among all targets, and seek the joint optimal locations for all targets simultaneously as a graphical model inference problem. We learn the joint parameters for both appearance model and motion model in an online fashion under the framework of LaRank. Experiment results show that our method achieved superior performance compared to the competitive methods.

Microbial Networks in SPRING - Semi-parametric Rank-Based Correlation and Partial Correlation Estimation for Quantitative Microbiome Data.

High-throughput microbial sequencing techniques, such as targeted amplicon-based and metagenomic profiling, provide low-cost genomic survey data of microbial communities in their natural environment, ranging from marine ecosystems to host-associated habitats. While standard microbiome profiling data can provide sparse relative abundances of operational taxonomic units or genes, recent advances in experimental protocols give a more quantitative picture of microbial communities by pairing sequencing-based techniques with orthogonal measurements of microbial cell counts from the same sample. These tandem measurements provide absolute microbial count data albeit with a large excess of zeros due to limited sequencing depth. In this contribution we consider the fundamental statistical problem of estimating correlations and partial correlations from such quantitative microbiome data. To this end, we propose a semi-parametric rank-based approach to correlation estimation that can naturally deal with the excess zeros in the data. Combining this estimator with sparse graphical modeling techniques leads to the Semi-Parametric Rank-based approach for INference in Graphical model (SPRING). SPRING enables inference of statistical microbial association networks from quantitative microbiome data which can serve as high-level statistical summary of the underlying microbial ecosystem and can provide testable hypotheses for functional species-species interactions. Due to the absence of verified microbial associations we also introduce a novel quantitative microbiome data generation mechanism which mimics empirical marginal distributions of measured count data while simultaneously allowing user-specified dependencies among the variables. SPRING shows superior network recovery performance on a wide range of realistic benchmark problems with varying network topologies and is robust to misspecifications of the total cell count estimate. To highlight SPRING's broad applicability we infer taxon-taxon associations from the American Gut Project data and genus-genus associations from a recent quantitative gut microbiome dataset. We believe that, as quantitative microbiome profiling data will become increasingly available, the semi-parametric estimators for correlation and partial correlation estimation introduced here provide an important tool for reliable statistical analysis of quantitative microbiome data.

Exact or approximate inference in graphical models: why the choice is dictated by the treewidth, and how variable elimination can be exploited

SummaryProbabilistic graphical models offer a powerful framework to account for the dependence structure between variables, which is represented as a graph. However, the dependence between variables may render inference tasks intractable. In this paper, we review techniques exploiting the graph structure for exact inference, borrowed from optimisation and computer science. They are built on the principle of variable elimination whose complexity is dictated in an intricate way by the order in which variables are eliminated. The so‐called treewidth of the graph characterises this algorithmic complexity: low‐treewidth graphs can be processed efficiently. The first point that we illustrate is therefore the idea that for inference in graphical models, the number of variables is not the limiting factor, and it is worth checking the width of several tree decompositions of the graph before resorting to the approximate method. We show how algorithms providing an upper bound of the treewidth can be exploited to derive a ‘good' elimination order enabling to realise exact inference. The second point is that when the treewidth is too large, algorithms for approximate inference linked to the principle of variable elimination, such as loopy belief propagation and variational approaches, can lead to accurate results while being much less time consuming than Monte‐Carlo approaches. We illustrate the techniques reviewed in this article on benchmarks of inference problems in genetic linkage analysis and computer vision, as well as on hidden variables restoration in coupled Hidden Markov Models.

Introduction to Bayesian Statistics, Third Edition. By William M.Bolstad and James M.CurranHoboken, New Jersey: John Wiley and Sons. 2017. 601+xvi pages. AUD $219.95 (hardback). ISBN 9781118091562

We review techniques from computer science, exploiting the graph structure for exact and approximate inference in graphical models.

Bayesian Inference in Nonparanormal Graphical Models

Gaussian graphical models have been used to study intrinsic dependence among several variables, but the Gaussianity assumption may be restrictive in many applications. A nonparanormal graphical model is a semiparametric generalization for continuous variables where it is assumed that the variables follow a Gaussian graphical model only after some unknown smooth monotone transformations on each of them. We consider a Bayesian approach in the nonparanormal graphical model by putting priors on the unknown transformations through a random series based on B-splines where the coefficients are ordered to induce monotonicity. A truncated normal prior leads to partial conjugacy in the model and is useful for posterior simulation using Gibbs sampling. On the underlying precision matrix of the transformed variables, we consider a spike-and-slab prior and use an efficient posterior Gibbs sampling scheme. We use the Bayesian Information Criterion to choose the hyperparameters for the spike-and-slab prior. We present a posterior consistency result on the underlying transformation and the precision matrix. We study the numerical performance of the proposed method through an extensive simulation study and finally apply the proposed method on a real data set.

Fast Bayesian inference in large Gaussian graphical models.

Despite major methodological developments, Bayesian inference in Gaussian graphical models remains challenging in high dimension due to the tremendous size of the model space. This article proposes a method to infer the marginal and conditional independence structures between variables by multiple testing, which bypasses the exploration of the model space. Specifically, we introduce closed‐form Bayes factors under the Gaussian conjugate model to evaluate the null hypotheses of marginal and conditional independence between variables. Their computation for all pairs of variables is shown to be extremely efficient, thereby allowing us to address large problems with thousands of nodes as required by modern applications. Moreover, we derive exact tail probabilities from the null distributions of the Bayes factors. These allow the use of any multiplicity correction procedure to control error rates for incorrect edge inclusion. We demonstrate the proposed approach on various simulated examples as well as on a large gene expression data set from The Cancer Genome Atlas.

Approximate Counting, the Lovász Local Lemma, and Inference in Graphical Models

In this article, we introduce a new approach to approximate counting in bounded degree systems with higher-order constraints. Our main result is an algorithm to approximately count the number of solutions to a CNF formula Φ when the width is logarithmic in the maximum degree. This closes an exponential gap between the known upper and lower bounds. Moreover, our algorithm extends straightforwardly to approximate sampling, which shows that under Lovász Local Lemma-like conditions it is not only possible to find a satisfying assignment, it is also possible to generate one approximately uniformly at random from the set of all satisfying assignments. Our approach is a significant departure from earlier techniques in approximate counting, and is based on a framework to bootstrap an oracle for computing marginal probabilities on individual variables. Finally, we give an application of our results to show that it is algorithmically possible to sample from the posterior distribution in an interesting class of graphical models.

Combinatorial inference for graphical models

We propose a new family of combinatorial inference problems for graphical models. Unlike classical statistical inference where the main interest is point estimation or parameter testing, combinatorial inference aims at testing the global structure of the underlying graph. Examples include testing the graph connectivity, the presence of a cycle of certain size, or the maximum degree of the graph. To begin with, we study the information-theoretic limits of a large family of combinatorial inference problems. We propose new concepts including structural packing and buffer entropies to characterize how the complexity of combinatorial graph structures impacts the corresponding minimax lower bounds. On the other hand, we propose a family of novel and practical structural testing algorithms to match the lower bounds. We provide numerical results on both synthetic graphical models and brain networks to illustrate the usefulness of these proposed methods.

Hierarchical Normalized Completely Random Measures for Robust Graphical Modeling.

Gaussian graphical models are useful tools for exploring network structures in multivariate normal data. In this paper we are interested in situations where data show departures from Gaussianity, therefore requiring alternative modeling distributions. The multivariate t-distribution, obtained by dividing each component of the data vector by a gamma random variable, is a straightforward generalization to accommodate deviations from normality such as heavy tails. Since different groups of variables may be contaminated to a different extent, Finegold and Drton (2014) introduced the Dirichlet t-distribution, where the divisors are clustered using a Dirichlet process. In this work, we consider a more general class of nonparametric distributions as the prior on the divisor terms, namely the class of normalized completely random measures (NormCRMs). To improve the effectiveness of the clustering, we propose modeling the dependence among the divisors through a nonparametric hierarchical structure, which allows for the sharing of parameters across the samples in the data set. This desirable feature enables us to cluster together different components of multivariate data in a parsimonious way. We demonstrate through simulations that this approach provides accurate graphical model inference, and apply it to a case study examining the dependence structure in radiomics data derived from The Cancer Imaging Atlas.

Tensor Graphical Model: Non-Convex Optimization and Statistical Inference.

We consider the estimation and inference of graphical models that characterize the dependency structure of high-dimensional tensor-valued data. To facilitate the estimation of the precision matrix corresponding to each way of the tensor, we assume the data follow a tensor normal distribution whose covariance has a Kronecker product structure. A critical challenge in the estimation and inference of this model is the fact that its penalized maximum likelihood estimation involves minimizing a non-convex objective function. To address it, this paper makes two contributions: (i) In spite of the non-convexity of this estimation problem, we prove that an alternating minimization algorithm, which iteratively estimates each sparse precision matrix while fixing the others, attains an estimator with an optimal statistical rate of convergence. (ii) We propose a de-biased statistical inference procedure for testing hypotheses on the true support of the sparse precision matrices, and employ it for testing a growing number of hypothesis with false discovery rate (FDR) control. The asymptotic normality of our test statistic and the consistency of FDR control procedure are established. Our theoretical results are backed up by thorough numerical studies and our real applications on neuroimaging studies of Autism spectrum disorder and users' advertising click analysis bring new scientific findings and business insights. The proposed methods are encoded into a publicly available R package Tlasso.

Discrete Graphical Models — An Optimization Perspective

This monograph is about discrete energy minimization for discrete graphical models. It considers graphical models, or, more precisely, maximum a posteriori inference for graphical models, purely as a combinatorial optimization problem. Modeling, applications, probabilistic interpretations and many other aspects are either ignored here or find their place in examples and remarks only. It covers the integer linear programming formulation of the problem as well as its linear programming, Lagrange and Lagrange decomposition-based relaxations. In particular, it provides a detailed analysis of the polynomially solvable acyclic and submodular problems, along with the corresponding exact optimization methods. Major approximate methods, such as message passing and graph cut techniques are also described and analyzed comprehensively. The monograph can be useful for undergraduate and graduate students studying optimization or graphical models, as well as for experts in optimization who want to have a look into graphical models. To make the monograph suitable for both categories of readers we explicitly separate the mathematical optimization background chapters from those specific to graphical models.

Incomplete graphical model inference via latent tree aggregation

Graphical network inference is used in many fields such as genomics or ecology to infer the conditional independence structure between variables, from measurements of gene expression or species abundances for instance. In many practical cases, not all variables involved in the network have been observed, and the samples are actually drawn from a distribution where some variables have been marginalized out. This challenges the sparsity assumption commonly made in graphical model inference, since marginalization yields locally dense structures, even when the original network is sparse. We present a procedure for inferring Gaussian graphical models when some variables are unobserved, that accounts both for the influence of missing variables and the low density of the original network. Our model is based on the aggregation of spanning trees, and the estimation procedure on the expectation-maximization algorithm. We treat the graph structure and the unobserved nodes as missing variables and compute posterior probabilities of edge appearance. To provide a complete methodology, we also propose several model selection criteria to estimate the number of missing nodes. A simulation study and an illustration on flow cytometry data reveal that our method has favourable edge detection properties compared to existing graph inference techniques. The methods are implemented in an R package.

SILGGM: An extensive R package for efficient statistical inference in large-scale gene networks.

Gene co-expression network analysis is extremely useful in interpreting a complex biological process. The recent droplet-based single-cell technology is able to generate much larger gene expression data routinely with thousands of samples and tens of thousands of genes. To analyze such a large-scale gene-gene network, remarkable progress has been made in rigorous statistical inference of high-dimensional Gaussian graphical model (GGM). These approaches provide a formal confidence interval or a p-value rather than only a single point estimator for conditional dependence of a gene pair and are more desirable for identifying reliable gene networks. To promote their widespread use, we herein introduce an extensive and efficient R package named SILGGM (Statistical Inference of Large-scale Gaussian Graphical Model) that includes four main approaches in statistical inference of high-dimensional GGM. Unlike the existing tools, SILGGM provides statistically efficient inference on both individual gene pair and whole-scale gene pairs. It has a novel and consistent false discovery rate (FDR) procedure in all four methodologies. Based on the user-friendly design, it provides outputs compatible with multiple platforms for interactive network visualization. Furthermore, comparisons in simulation illustrate that SILGGM can accelerate the existing MATLAB implementation to several orders of magnitudes and further improve the speed of the already very efficient R package FastGGM. Testing results from the simulated data confirm the validity of all the approaches in SILGGM even in a very large-scale setting with the number of variables or genes to a ten thousand level. We have also applied our package to a novel single-cell RNA-seq data set with pan T cells. The results show that the approaches in SILGGM significantly outperform the conventional ones in a biological sense. The package is freely available via CRAN at https://cran.r-project.org/package=SILGGM.

Selecting the tuning parameter in penalized Gaussian graphical models

Penalized inference of Gaussian graphical models is a way to assess the conditional independence structure in multivariate problems. In this setting, the conditional independence structure, corresponding to a graph, is related to the choice of the tuning parameter, which determines the model complexity or degrees of freedom. There has been little research on the degrees of freedom for penalized Gaussian graphical models. In this paper, we propose an estimator of the degrees of freedom in $$\ell _1$$ -penalized Gaussian graphical models. Specifically, we derive an estimator inspired by the generalized information criterion and propose to use this estimator as the bias term for two information criteria. We called these tuning parameter selectors GAIC and GBIC. These selectors can be used to choose the tuning parameter, i.e., the optimal tuning parameter is the minimizer of GAIC or GBIC. A simulation study shows that GAIC tends to improve the performance of both AIC-type and CV-type model selectors, in terms of estimation quality (entropy loss function) in high-dimensional setting. Moreover, GBIC model selector improves the performance of both BIC-type and CV-type model selectors, in terms of support recovery (F-score). A data analysis shows that GBIC selects a tuning parameter that produces a sparser graph with respect to BIC and a CV-type model selector (KLCV).

Bayesian Inference for Gaussian Graphical Models Beyond Decomposable Graphs

Summary Bayesian inference for graphical models has received much attention in the literature in recent years. It is well known that, when the graph G is decomposable, Bayesian inference is significantly more tractable than in the general non-decomposable setting. Penalized likelihood inference in contrast has made tremendous gains in the past few years in terms of scalability and tractability. Bayesian inference, however, has not had the same level of success, though a scalable Bayesian approach has its strengths, especially in terms of quantifying uncertainty. To address this gap, we propose a scalable and flexible novel Bayesian approach for estimation and model selection in Gaussian undirected graphical models. We first develop a class of generalized G-Wishart distributions with multiple shape parameters for an arbitrary underlying graph. This class contains the G-Wishart distribution as a special case. We then introduce the class of generalized Bartlett graphs and derive an efficient Gibbs sampling algorithm to obtain posterior draws from generalized G-Wishart distributions corresponding to a generalized Bartlett graph. The class of generalized Bartlett graphs contains the class of decomposable graphs as a special case but is substantially larger than the class of decomposable graphs. We proceed to derive theoretical properties of the proposed Gibbs sampler. We then demonstrate that the proposed Gibbs sampler is scalable to significantly higher dimensional problems compared with using an accept–reject or a Metropolis–Hasting algorithm. Finally, we show the efficacy of the proposed approach on simulated and real data. In particular, we demonstrate that our generalized Bartlett methodology can be used for efficient model selection by reducing the graph search space by using penalized likelihood and pseudolikelihood methods.

Probabilistic Planning With Influence Diagrams

Graphical models provide a powerful framework for reasoning under uncertainty, and an influence diagram (ID) is a graphical model of a sequential decision problem that maximizes the total expected utility of a non-forgetting agent. Relaxing the regular modeling assumptions, an ID can be flexibly extended to general decision scenarios involving a limited memory agent or multi-agents. The approach of probabilistic planning with IDs is expected to gain computational leverage by exploiting the local structure as well as representation flexibility of influence diagram frameworks. My research focuses on graphical model inference for IDs and its application to probabilistic planning, targeting online MDP/POMDP planning as testbeds in the evaluation.