Parallel sampling of decomposable graphs using Markov chain on junction trees

Bayesian inference for undirected graphical models is mostly restricted to the class of decomposable graphs, as they enjoy a rich set of properties making them amenable to high-dimensional problems. While parameter inference is straightforward in this setup, inferring the underlying graph is a challenge driven by the computational difficultly in exploring the space of decomposable graphs. This work makes two contributions to address this problem. First, we provide sufficient and necessary conditions for when multi-edge perturbations maintain decomposability of the graph. Using these, we characterize a simple class of partitions that efficiently classify all edge perturbations by whether they maintain decomposability. Second, we propose a new parallel non-reversible Markov chain Monte Carlo sampler for distributions over junction tree representations of the graph, where at every step, all edge perturbations within a partition are executed simultaneously. Through simulations, we demonstrate the efficiency of our new edge perturbation conditions and class of partitions. We find that our parallel sampler yields improved mixing properties in comparison to the single-move variate, and outperforms current methods. The implementation of our work is available in a Python package.Comment: 20 pages, 10 figures, with appendix and supplementary materia

A Skew-Normal Copula-Driven Generalized Linear Mixed Model for Longitudinal Data

Using the advancements of Arellano-Valle et al. [2005], which characterize the likelihood function of a linear mixed model (LMM) under a skew-normal distribution for the random effects, this thesis attempt to construct a copula-driven generalized linear mixed model (GLMM). Assuming a multivariate distribution from the exponential family for the response variable and a skew-normal copula, we drive a complete characterization of the general likelihood function. For estimation, we apply a Monte Carlo expectation maximization (MC-EM) algorithm. Some special cases are discussed, in particular, the exponential and gamma distributions. Simulations with multiple link functions are shown alongside a real data example from the Framingham Heart Study

A hierarchical Bayesian model for predicting ecological interactions using scaled evolutionary relationships

Identifying undocumented or potential future interactions among species is a challenge facing modern ecologists. Recent link prediction methods rely on trait data, however large species interaction databases are typically sparse and covariates are limited to only a fraction of species. On the other hand, evolutionary relationships, encoded as phylogenetic trees, can act as proxies for underlying traits and historical patterns of parasite sharing among hosts. We show that using a network-based conditional model, phylogenetic information provides strong predictive power in a recently published global database of host-parasite interactions. By scaling the phylogeny using an evolutionary model, our method allows for biological interpretation often missing from latent variable models. To further improve on the phylogeny-only model, we combine a hierarchical Bayesian latent score framework for bipartite graphs that accounts for the number of interactions per species with the host dependence informed by phylogeny. Combining the two information sources yields significant improvement in predictive accuracy over each of the submodels alone. As many interaction networks are constructed from presence-only data, we extend the model by integrating a correction mechanism for missing interactions, which proves valuable in reducing uncertainty in unobserved interactions.Comment: To appear in the Annals of Applied Statistic

Fast global convergence of gradient descent for low-rank matrix approximation

This paper investigates gradient descent for solving low-rank matrix approximation problems. We begin by establishing the local linear convergence of gradient descent for symmetric matrix approximation. Building on this result, we prove the rapid global convergence of gradient descent, particularly when initialized with small random values. Remarkably, we show that even with moderate random initialization, which includes small random initialization as a special case, gradient descent achieves fast global convergence in scenarios where the top eigenvalues are identical. Furthermore, we extend our analysis to address asymmetric matrix approximation problems and investigate the effectiveness of a retraction-free eigenspace computation method. Numerical experiments strongly support our theory. In particular, the retraction-free algorithm outperforms the corresponding Riemannian gradient descent method, resulting in a significant 29\% reduction in runtime

Prediction intervals for travel time on transportation networks

Estimating travel-time is essential for making travel decisions in transportation networks. Empirically, single road-segment travel-time is well studied, but how to aggregate such information over many edges to arrive at the distribution of travel time over a route is still theoretically challenging. Understanding travel-time distribution can help resolve many fundamental problems in transportation, quantifying travel uncertainty as an example. We develop a novel statistical perspective to specific types of dynamical processes that mimic the behavior of travel time on real-world networks. We show that, under general conditions, travel-time normalized by distance, follows a Gaussian distribution with route-invariant (universal) location and scale parameters. We develop efficient inference methods for such parameters, with which we propose asymptotic universal confidence and prediction intervals of travel time. We further develop our theory to include road-segment level information to construct route-specific location and scale parameter sequences that produce tighter route-specific Gaussian-based prediction intervals. We illustrate our methods with a real-world case study using precollected mobile GPS data, where we show that the route-specific and route-invariant intervals both achieve the 95\% theoretical coverage levels, where the former result in tighter bounds that also outperform competing models.Comment: 24 main pages, 4 figures and 4 tables. This version includes many changes to the previous on

Hyperoxemia and excess oxygen use in early acute respiratory distress syndrome : Insights from the LUNG SAFE study

Publisher Copyright: © 2020 The Author(s). Copyright: Copyright 2020 Elsevier B.V., All rights reserved.Background: Concerns exist regarding the prevalence and impact of unnecessary oxygen use in patients with acute respiratory distress syndrome (ARDS). We examined this issue in patients with ARDS enrolled in the Large observational study to UNderstand the Global impact of Severe Acute respiratory FailurE (LUNG SAFE) study. Methods: In this secondary analysis of the LUNG SAFE study, we wished to determine the prevalence and the outcomes associated with hyperoxemia on day 1, sustained hyperoxemia, and excessive oxygen use in patients with early ARDS. Patients who fulfilled criteria of ARDS on day 1 and day 2 of acute hypoxemic respiratory failure were categorized based on the presence of hyperoxemia (PaO2 > 100 mmHg) on day 1, sustained (i.e., present on day 1 and day 2) hyperoxemia, or excessive oxygen use (FIO2 ≥ 0.60 during hyperoxemia). Results: Of 2005 patients that met the inclusion criteria, 131 (6.5%) were hypoxemic (PaO2 < 55 mmHg), 607 (30%) had hyperoxemia on day 1, and 250 (12%) had sustained hyperoxemia. Excess FIO2 use occurred in 400 (66%) out of 607 patients with hyperoxemia. Excess FIO2 use decreased from day 1 to day 2 of ARDS, with most hyperoxemic patients on day 2 receiving relatively low FIO2. Multivariate analyses found no independent relationship between day 1 hyperoxemia, sustained hyperoxemia, or excess FIO2 use and adverse clinical outcomes. Mortality was 42% in patients with excess FIO2 use, compared to 39% in a propensity-matched sample of normoxemic (PaO2 55-100 mmHg) patients (P = 0.47). Conclusions: Hyperoxemia and excess oxygen use are both prevalent in early ARDS but are most often non-sustained. No relationship was found between hyperoxemia or excessive oxygen use and patient outcome in this cohort. Trial registration: LUNG-SAFE is registered with ClinicalTrials.gov, NCT02010073publishersversionPeer reviewe

On decomposable random graphs and link prediction models

In combinatorial graph theory, decomposable graphs are such type of graphs that are guaranteed to be decomposable into conditionally independent components, known as maximal cliques. In statistics, decomposable graphs are widely used in the field of graphical models or Bayesian model determination, where the dependency structure among high dimensional data or model parameters is unknown. Decomposable graphs are hence used as functional priors over large covariance matrices or as priors over hierarchies of model parameters. One such example is the Gaussian graphical model (lauritzen 1996, whittaker 2009), which has seen success in a variety of applications. Beyond this framework, decomposable graphs are seldom used in statistical applications.Random graphs, on the other hand, have recently seen much research interest, where the focus is on developing methodologies for models on relational data in the form of random binary matrices. A principle component of such models is to assume a network framework by mapping the relations to edges of the network, and data sources to nodes. The likelihood of an edge is assumed to be driven by affinity parameters of the associated nodes. The first part of this work attempts to propose a framework for modelling random decomposable graphs, using similar tools as in random graphs. Rather than modelling edges between nodes, the framework models the bipartite links between the graph nodes and latent community nodes, through node affinity parameters. The latent communities are assumed to represent the maximal cliques in decomposable graphs. Under the proposed framework, simple Markov update rules are given with explicit lower bounds for its mixing time (time until convergence). Under a set of conditions, an exact expression for the expected number of maximal cliques per node is given. The second part of this work illustrates a new application of decomposable graphs that is motivated by the proposedframework. Combinatorially, there is a unique set of subgraphs of any maximal clique. Treating maximal cliques as latent communities allows the treatment of subgraphs of maximal cliques as sub-clusters within each community. The proposed framework is extended to incorporate a sub-clustering component, which enables the modelling of decomposable graphs and simultaneous modelling of the sub-clustering dynamics forming within each larger community.The final part of this work deals with the topic of link prediction in networks with presence-only data, where absence is only an indication of missing information and not a prohibited link. The work is motivated by a particular example of identifying undocumented or potential interactions among species from the set of available documented interactions, in an aim to help guide the sampling of ecological networks by identifying the most likely undocumented interactions. The problem is framed in bipartite graph structure, where edges represent interactions between pairs of species. The work first constructs a Bayesian latent score model, which ranks observed edges from the most probable down to the least certain. To improve scoring efficiency, and thus link prediction, the work incorporates a Markov random field component informed by phylogenetic relationships among species. The model is validated using two host-parasite networks constructed from published databases, the Global Mammal Parasite Database and the Enhanced Infectious Diseases database, each with thousands of pairwise interactions. Finally, the model is extended by integrating a correction mechanism for missing interactions in the observed data, which proves valuable in reducing uncertainty in unobserved interactions.En théorie des graphes combinatoire, les graphes décomposables sont un type de graphe dont il est garanti qu'ils sont décomposables en composantes conditionnellement indépendantes, appelées cliques maximum. En statistiques, les graphes décomposables sont communément utilisés dans le champ des modèles graphiques ou dans la détermination de modèles Bayésiens, pour lesquels la structure de dépendence entre variables à haute dimensionalité ou des paramètres du modèle est inconnue. Les graphes décomposables sont ainsi utilisés comme précédents fonctionnels par rapport aux matrices à large covariance ou en tant que précédents par rapport aux hierarchies des paramètres du modèle. Un exemple de cette utilisation est celle du modèle graphique Gaussien (lauritzen 1996, whittaker 2009) qui a été appliqué avec succès dans un grand nombre de cas.Les graphes aléatoires ont généré beaucoup d'intérêt, en particulier, sur les données relationnelles en de matrices aléatoires binaires. Une composante principale de ces modèles est la définition d'un cadre de réseau en associant les relations aux liens du réseau et les sources de données aux noeuds. La première partie de ce travail propose un cadre de modèlisation pour les graphes décomposables aléatoires et utilise des outils similaires à ceux utilisés pour les graphes aléatoires. Plutôt que de modèliser les liens entre les noeuds, le cadre modèlise les associations bipartites entre les noeuds du graphe et les noeuds des communautés latentes, à l'aide des paramètres d'affinité entre les noeuds. L'hypothèse émise étant que les communautés latentes représentent les cliques maximum des graphes décomposables. Au sein de ce cadre proposé, les règles simples de mise à jour de Markov se voient attribuées une limite basse explicite pour leur temps mélangé (temps sous convergence). La seconde partie de ce travail illustre une nouvelle application des graphes décomposables s'appuyant sur le cadre proposé. Combinatoirement, il existe un ensemble unique de sous-graphes pour toute clique maximum. En traitant chaque clique maximum en tant que communauté latente il est possible de traiter les sous-graphes des cliques maximum en tant que sous-group au sein de chaque communauté. Le cadre proposé est étendu pour incorporer une composante de sous-groupement, ce qui autorise la modélisation des graphes décomposables et simultanément la modélisation de dynamiques de sous-groupement qui se forment au sein de chaque communauté plus large.La dernière partie de ce travail traite du sujet des prédictions de lien pour les réseaux avec des données présence uniquement, où l'abscence est seulement une indication de données manquantes et non d'un lien interdit. Ce travail s'appuie sur un exemple specifique, celui de l'identification d'interactions non-documentées ou potentielles au sein d'espèces appartennant à l'ensemble des interactions documentées. L'objectif est d'aider à guider l'échantillonnage de réseaux écologiques en identifiant les relation non-documentées les plus vraisemblables. Le problème est cadré en structure bipartite de graphe où les liens représentent les interactions entre paires d'espèces. Le travail développe tout d'abord un modèle de score latent Bayésien qui ordonne les liens observés du plus probable au moins certain. Pour améliorer l'efficience du score et partant la prédiction des liens, le travail incorpore un composant de champ aléatoire de Markov utilisant lesretations phylogéniques entre espèces. Le modèle est validé en utilisant deux réseaux hôte/parasite construits à partir de deux bases de données publiées; la base globale mammifère parasiteet la base de données améliorée des maladie infectieuses, chacune contenant des milliers de paires d'interactions. Finalement, le modèle est étendu en intégrant un méchanisme de correction pour les interactions manquantes dans les données observées, qui s'avère efficace à diminuer l'incertitude dans les interactions inobservées