Graphical models have found widespread applications in many areas of modern statistics and machine learning. Iterative Proportional Fitting (IPF) and its variants have become the default method for undirected graphical model estimation, and are thus ubiquitous in the field. As the IPF is an iterative approach, it is not always readily scalable to modern high-dimensional data regimes. In this paper we propose a novel and fast non-iterative method for positive definite graphical model estimation in high dimensions, one that directly addresses the shortcomings of IPF and its variants. In addition, the proposed method has a number of other attractive properties. First, we show formally that as the dimension p grows, the proportion of graphs for which the proposed method will outperform the state-of-the-art in terms of computational complexity and performance tends to 1, affirming its efficacy in modern settings. Second, the proposed approach can be readily combined with scalable non-iterative thresholding-based methods for high-dimensional sparsity selection. Third, the proposed method has high-dimensional statistical guarantees. Moreover, our numerical experiments also show that the proposed method achieves scalability without compromising on statistical precision. Fourth, unlike the IPF, which depends on the Gaussian likelihood, the proposed method is much more robust.

Over recent years Dirichlet processes and the associated Chinese restaurant process (CRP) have found many applications in clustering while the Indian buffet process (IBP) is increasingly used to describe latent feature models. In the clustering case, we associate to each data point a latent allocation variable. These latent variables can share the same value and this induces a partition of the data set. The CRP is a prior distribution on such partitions. In latent feature models, we associate to each data point a potentially infinite number of binary latent variables indicating the possession of some features and the IBP is a prior distribution on the associated infinite binary matrix.

Describing the relationship between the variables in a study domain and modelling the data generating mechanism is a fundamental problem in many empirical sciences. Probabilistic graphical models are one common approach to tackle the problem. Learning the graphical structure is computationally challenging and a fervent area of current research with a plethora of algorithms being developed. To facilitate the benchmarking of different methods, we present a novel automated workflow, called benchpress for producing scalable, reproducible, and platform-independent benchmarks of structure learning algorithms for probabilistic graphical models. Benchpress is interfaced via a simple JSON-file, which makes it accessible for all users, while the code is designed in a fully modular fashion to enable researchers to contribute additional methodologies. Benchpress currently provides an interface to a large number of state-of-the-art algorithms from libraries such as BDgraph, BiDAG, bnlearn, GOBNILP, pcalg, r.blip, scikit-learn, TETRAD, and trilearn as well as a variety of methods for data generating models and performance evaluation. Alongside user-defined models and randomly generated datasets, the software tool also includes a number of standard datasets and graphical models from the literature, which may be included in a benchmarking workflow. We demonstrate the applicability of this workflow for learning Bayesian networks in four typical data scenarios. The source code and documentation is publicly available from http://github.com/felixleopoldo/benchpress.

Over recent years Dirichlet processes and the associated Chinese restaurant process (CRP) have found many applications in clustering while the Indian buffet process (IBP) is increasingly used to describe latent feature models. In the clustering case, we associate to each data point a latent allocation variable. These latent variables can share the same value and this induces a partition of the data set. The CRP is a prior distribution on such partitions. In latent feature models, we associate to each data point a potentially infinite number of binary latent variables indicating the possession of some features and the IBP is a prior distribution on the associated infinite binary matrix.

In this study, we present a multi-class graphical Bayesian predictive classifier that incorporates the uncertainty in the model selection into the standard Bayesian formalism. For each class, the dependence structure underlying the observed features is represented by a set of decomposable Gaussian graphical models. Emphasis is then placed on the Bayesian model averaging which takes full account of the class-specific model uncertainty by averaging over the posterior graph model probabilities. Even though the decomposability assumption severely reduces the model space, the size of the class of decomposable models is still immense, rendering the explicit Bayesian averaging over all the models infeasible. To address this issue, we consider the particle Gibbs strategy of Olsson et al. (2016) for posterior sampling from decomposable graphical models which utilizes the Christmas tree algorithm of Rios et al. (2016) as proposal kernel. We also derive the a strong hyper Markov law which we call the hyper normal Wishart law that allow to perform the resultant Bayesian calculations locally. The proposed predictive graphical classifier reveals superior performance compared to the ordinary Bayesian predictive rule that does not account for the model uncertainty, as well as to a number of out-of-the-box classifiers.

A parametrization of hypergraphs based on the geometry of points in $\mathbf{R}^d$ is developed. Informative prior distributions on hypergraphs are induced through this parametrization by priors on point configurations via spatial processes. This prior specification is used to infer conditional independence models or Markov structure of multivariate distributions. Specifically, we can recover both the junction tree factorization as well as the hyper Markov law. This approach offers greater control on the distribution of graph features than Erd\"os-R\'enyi random graphs, supports inference of factorizations that cannot be retrieved by a graph alone, and leads to new Metropolis\slash Hastings Markov chain Monte Carlo algorithms with both local and global moves in graph space. We illustrate the utility of this parametrization and prior specification using simulations.

We consider the problem of learning the structure of undirected graphical models with bounded treewidth, within the maximum likelihood framework. This is an NP-hard problem and most approaches consider local search techniques. In this paper, we pose it as a combinatorial optimization problem, which is then relaxed to a convex optimization problem that involves searching over the forest and hyperforest polytopes with special structures, independently. A supergradient method is used to solve the dual problem, with a run-time complexity of $O(k^3 n^{k+2} \log n)$ for each iteration, where $n$ is the number of variables and $k$ is a bound on the treewidth. We compare our approach to state-of-the-art methods on synthetic datasets and classical benchmarks, showing the gains of the novel convex approach.

This paper is concerned with the inference of the conditional independence graph G of a multivariate random vector Y of dimension n, a problem sometimes referred to as structure learning. We focus here on undirected decomposable graphs, whose popularity is mainly due to the tractable factorization they allow for the likelihood ([9, 20]); related work for directed graphical models can be found in [18]. Learning the conditional 1 independence graph G is an onerous task due to the large number of graphs on a set of n nodes, or variables. It is possible using optimization methods to find the graph which best fits the data according to some metric [23, 30, 13]; alternatively Bayesian model averaging may be used to accommodate for uncertainty in the estimated graph, or maximum a posteriori estimation may be used to select a given model from the posterior over graphs. Such an approach relies on a prior distribution π(G) over the set of decomposable graphs of a given size; through Bayes theorem, this prior is updated based on the data to give an a posteriori estimate of the distribution over graphs.

In this paper we make several contributions towards accelerating approximate Bayesian structural inference for non-decomposable GGMs. Our first contribution is to show how to efficiently compute a BIC or Laplace approximation to the marginal likelihood of non-decomposable graphs using convex methods for precision matrix estimation. This optimization technique can be used as a fast scoring function inside standard Stochastic Local Search (SLS) for generating posterior samples. Our second contribution is a novel framework for efficiently generating large sets of high-quality graph topologies without performing local search. This graph proposal method, which we call Neighborhood Fusion" (NF), samples candidate Markov blankets at each node using sparse regression techniques. Our final contribution is a hybrid method combining the complementary strengths of NF and SLS. Experimental results in structural recovery and prediction tasks demonstrate that NF and hybrid NF/SLS out-perform state-of-the-art local search methods, on both synthetic and real-world datasets, when realistic computational limits are imposed."

Over recent years Dirichlet processes and the associated Chinese restaurant process (CRP) have found many applications in clustering while the Indian buffet process (IBP) is increasingly used to describe latent feature models. In the clustering case, we associate to each data point a latent allocation variable. These latent variables can share the same value and this induces a partition of the data set. The CRP is a prior distribution on such partitions. In latent feature models, we associate to each data point a potentially infinite number of binary latent variables indicating the possession of some features and the IBP is a prior distribution on the associated infinite binary matrix. These prior distributions are attractive because they ensure exchangeability (over samples). We propose here extensions of these models to decomposable graphs. These models have appealing properties and can be easily learned using Monte Carlo techniques.