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IIM Indore

Operation Management and Quantitative Techniques

World University root node

https://arxiv.org/abs/2101.

Bayesian inference in high-dimensional models

The problem of precision matrix estimation in a multivariate Gaussian model is fundamental to network estimation. Although there exist both Bayesian and frequentist approaches to this, it is difficult to obtain good Bayesian and frequentist properties under the same prior-penalty dual, complicating justification. It is well known, for example, that the Bayesian version of the popular lasso estimator has poor posterior concentration properties. To bridge this gap for the precision matrix estimation problem, our contribution is a novel prior-penalty dual that closely approximates the popular graphical horseshoe prior and penalty and performs well in both Bayesian and frequentist senses. A chief difficulty with the horseshoe prior is a lack of closed form expression of the density function, which we overcome in this article, allowing us to directly study the penalty function. In terms of theory, we establish posterior convergence rate of the precision matrix that matches the oracle rate, in addition to the frequentist consistency of the maximum a posteriori estimator. In addition, our results also provide theoretical justifications for previously developed approaches that have been unexplored so far, e.g. for the graphical horseshoe prior. Computationally efficient Expectation Conditional Maximization and Markov chain Monte Carlo algorithms are developed respectively for the penalized likelihood and fully Bayesian estimation problems, using the same latent variable framework. In numerical experiments, the horseshoe-based approaches echo their superior theoretical properties by comprehensively outperforming the competing methods. A protein-protein interaction network estimation in B-cell lymphoma is considered to validate the proposed methodology.

arXiv preprint arXiv:2104.

Precision Matrix Estimation under the Horseshoe-like Prior-Penalty Dual

https://arxiv.org/abs/2102.

Horseshoe shrinkage methods for Bayesian fusion estimation

Modeling the dynamics of COVID-19 pandemic spread is a challenging and relevant problem. Established models for the epidemic spread such as compartmental epidemiological models e.g. Susceptible-Infected-Recovered (SIR) models and its variants, have been discussed extensively in the literature and utilized to forecast the growth of the pandemic across different hot-spots in the world. The standard formulations of SIR models rely upon summary-level data, which may not be able to fully capture the complete dynamics of the pandemic growth. Since the disease spreads from carriers to susceptible individuals via some form of contact, it inherently relies upon a network of individuals for its growth, with edges established via direct interaction, such as shared physical proximity. Using individual-level COVID-19 data from the early days (January 30 to April 15, 2020) of the pandemic in India and under a network-based SIR model framework, we performed state-specific forecasting under multiple scenarios characterized by the basic reproduction number of COVID-19 across 34 Indian states and union territories. We validated our short-term projections using observed case counts and the long-term projections using national sero-survey findings. Based on healthcare availability data, we also performed projections to assess the burdens on the infrastructure along the spectrum of the pandemic growth. We have developed an interactive dashboard summarizing our results. Our predictions successfully identified the initial hot-spots of India such as Maharashtra and Delhi and those that emerged later, such as Madhya Pradesh and Kerala. These models have the potential to inform appropriate policies for isolation and mitigation strategies to contain the pandemic, through a phased approach by appropriate resource prioritization and allocation.Competing Interest StatementThe authors have declared no competing interest.Funding StatementS.B. is supported by DST INSPIRE Faculty Award Grant No. 04/2015/002165 and also by IIM Indore Young Faculty Research Chair Award grant. S.M. was partially supported by Precision Health at the University of Michigan. V.B. was supported by NIH grants R01-CA160736 and P30 CA 46592 and NSF DMS grant 1463233 and start-up funds from the University of Michigan School of Public Health. Author DeclarationsI confirm all relevant ethical guidelines have been followed and any necessary IRB and/or ethics committee approvals have been obtained.YesThe details of the IRB/oversight body that provided approval or exemption for the research described are given below:NAAll necessary patient/participant consent has been obtained and the appropriate institutional forms have been archived.YesI understand that all clinical trials and any other prospective interventional studies must be registered with an ICMJE-approved registry, such as ClinicalTrials.gov. I confirm that any such study reported in the manuscript has been registered and the trial registration ID is provided (note: if posting a prospective study registered retrospectively, please provide a statement in the trial ID field explaining why the study was not registered in advance).YesI have followed all appropriate research reporting guidelines and uploaded the relevant EQUATOR Network research reporting checklist(s) and other pertinent material as supplementary files, if applicable.YesIndividual-level, location-informed COVID-19 data for Indian states and union territories were obtained from the Kaggle database available at \urlhttps://www.kaggle.com/sudalairajkumar/covid19-in-india. Data on state-wide daily infections is available from \urlhttps://api.covid19india.org/. https://www.kaggle.com/sudalairajkumar/covid19-in-india https://api.covid19india.org/ https://bayesrx.shinyapps.io/COV-N/

medRxiv

Network-based Modeling of COVID-19 Dynamics: Early Pandemic Spread in India

We study the graph signal denoising problem by estimating a piecewise constant signal over an undirected graph. We propose a new Bayesian approach that first converts a general graph to a chain graph via the depth first search algorithm and then imposes a heavy-tailed t-shrinkage prior on the differences between consecutive signals over the induced chain graph. We show that the posterior computation can be conveniently conducted by fully exploring the conjugacy structure in the model. We also derive the posterior contraction rate for the proposed estimator and show that this rate is optimal up to a logarithmic factor, besides automatically adapting to the unknown edge sparsity level of the graph. We demonstrate the excellent empirical performance of the proposed method via extensive simulation studies and applications to stock market data.MSC Codes 62F15

arXiv

Graph signal denoising using t-shrinkage priors

We consider the problem of learning the structure of a high dimensional precision matrix under sparsity assumptions. We propose to use a shrinkage prior, called the DL-graphical prior based on the Dirichlet-Laplace prior used for the Gaussian mean problem. A posterior sampling scheme based on Gibbs sampling is also provided along with theoretical guarantees of the method by obtaining the posterior convergence rate of the precision matrix.

ARXIV.ORG

Bayesian Structure Learning in Graphical Models using Shrinkage priors

Personalized (patient-specific) approaches have recently emerged with a precision medicine paradigm that acknowledges the fact that molecular pathway structures and activity might be considerably different within and across tumors. The functional cancer genome and proteome provide rich sources of information to identify patient-specific variations in signaling pathways and activities within and across tumors; however, current analytic methods lack the ability to exploit the diverse and multi-layered architecture of these complex biological networks. We assessed pan-cancer pathway activities for &gt;7700 patients across 32 tumor types from The Cancer Proteome Atlas by developing a personalized cancer-specific integrated network estimation (PRECISE) model. PRECISE is a general Bayesian framework for integrating existing interaction databases, data-driven de novo causal structures, and upstream molecular profiling data to estimate cancer-specific integrated networks, infer patient-specific networks and elicit interpretable pathway-level signatures. PRECISE-based pathway signatures, can delineate pan-cancer commonalities and differences in proteomic network biology within and across tumors, demonstrates robust tumor stratification that is both biologically and clinically informative and superior prognostic power compared to existing approaches. Towards establishing the translational relevance of the functional proteome in research and clinical settings, we provide an online, publicly available, comprehensive database and visualization repository of our findings (https://mjha.shinyapps.io/PRECISE/).

Fulltext

Scientific Reports

Personalized Integrated Network Modeling of the Cancer Proteome Atlas

Clustering methods for multivariate data exploiting the underlying geometry of the graphical structure between variables are presented. As opposed to standard approaches for graph clustering that assume known graph structures, the edge structure of the unknown graph is first estimated using sparse regression based approaches for sparse graph structure learning. Subsequently, graph clustering on the lower dimensional projections of the graph is performed based on Laplacian embeddings using a penalized k-means approach, motivated by Dirichlet process mixture models in Bayesian nonparametrics. In contrast to standard algorithmic approaches for known graphs, the proposed method allows estimation and inference for both graph structure learning and clustering. More importantly, the arguments for Laplacian embeddings as suitable projections for graph clustering are formalized by providing theoretical support for the consistency of the eigenspace of the estimated graph Laplacians. Fast computational algorithms are proposed to scale the method to large number of nodes. Extensive simulations are presented to compare the clustering performance with standard methods. The methods are applied to a novel pan-cancer proteomic data set, and protein networks and clusters are evaluated across multiple different cancer types.

Computational Statistics & Data Analysis

Spectral clustering via sparse graph structure learning with application to proteomic signaling networks in cancer

I congratulate Professor Caron and Professor Fox for a well-written paper that establishes a novel statistical network modeling framework, which uses completely random measures to model graphs with various levels of sparsity. Although it has been made clear in the paper that the characteristics of the underlying completely random measure, e.g., the discount parameter $σ$ < 1 of the generalized gamma process, play a crucial role in determining the sparsity levels of the generated graphs, I should like to call attention to the potentially important role played by the link function f(x) = 1 − e−x, in generating unweighted undirected sparse graphs with zij |wi, wj ∼ Bernoulli[f(2wiwj)] for i 6= j. With this link function, the contribution of zij to the negative log-likelihood of the model can be expressed aswhich quickly explodes towards ∞ as the product wiwj approaches zero when zij = 1. Thus the choice of this link function implies an inductive bias towards fitting nonzero edges zij = 1, while not strongly penalizing zero edges zij = 0 even if their corresponding products wiwj are large. The same link function, which is referred to as the Bernoulli-Poisson link, has also been used in Zhou (2015), which constructs nonparametric Bayesian network models for overlapping community detection and missing link prediction, allowing the computation to scale linearly with the number of edges, rather than quadratically with the number of nodes. It would be of interest to articulate the role of this specific link function in supporting sparse graphs under the proposed framework.In addition to controlling for sparse graphs how the number of edges increases as a function of the number of nodes, another topic worth further investigation is how to introduce structured sparsity patterns to the graph adjacency matrices, including modeling dense on-diagonal but sparse off-diagonal blocks, sparse on-diagonal but dense off-diagonal blocks, or a mixture of both. It would be interesting to find out whether the new notion of exchangeability could be maintained while achieving these network modeling goals.

Discussion on “Sparse graphs using exchangeable random measures” by François Caron and Emily B. Fox

We study the posterior convergence behaviour of a precision matrix corresponding to a Gaussian graphical model in the high‐dimensional set‐up under sparsity assumptions. Recent works include studying posterior convergence rates of precision matrices assuming an approximate banding structure, and extension of such result to arbitrary decomposable graphical models using a transformation to Cholesky factor of the precision matrices. In this paper, we study the same for the wider class of arbitrary decomposable graphical models under similar sparsity assumptions using a G‐Wishart prior, but without the complications of using a Cholesky factor, and arrive at identical posterior convergence rates. Copyright © 2017 John Wiley &amp; Sons, Ltd.

Stat

Posterior convergence rates for high-dimensional precision matrix estimation using G-Wishart priors

Tumor heterogeneity is a crucial area of cancer research wherein inter- and intra-tumor differences are investigated to assess and monitor disease development and progression, especially in cancer. The proliferation of imaging and linked genomic data has enabled us to evaluate tumor heterogeneity on multiple levels. In this work, we examine magnetic resonance imaging (MRI) in patients with brain cancer to assess image-based tumor heterogeneity. Standard approaches to this problem use scalar summary measures (e.g., intensity-based histogram statistics) that do not adequately capture the complete and finer scale information in the voxel-level data. In this paper, we introduce a novel technique, DEMARCATE (DEnsity-based MAgnetic Resonance image Clustering for Assessing Tumor hEterogeneity) to explore the entire tumor heterogeneity density profiles (THDPs) obtained from the full tumor voxel space. THDPs are smoothed representations of the probability density function of the tumor images. We develop tools for analyzing such objects under the Fisher–Rao Riemannian framework that allows us to construct metrics for THDP comparisons across patients, which can be used in conjunction with standard clustering approaches. Our analyses of The Cancer Genome Atlas (TCGA) based Glioblastoma dataset reveal two significant clusters of patients with marked differences in tumor morphology, genomic characteristics and prognostic clinical outcomes. In addition, we see enrichment of image-based clusters with known molecular subtypes of glioblastoma multiforme, which further validates our representation of tumor heterogeneity and subsequent clustering techniques.

NeuroImage: Clinical

DEMARCATE: Density-based magnetic resonance image clustering for assessing tumor heterogeneity in cancer

We consider the problem of estimating a sparse precision matrix of a multivariate Gaussian distribution, where the dimension p may be large. Gaussian graphical models provide an important tool in describing conditional independence through presence or absence of edges in the underlying graph. A popular non-Bayesian method of estimating a graphical structure is given by the graphical lasso. In this paper, we consider a Bayesian approach to the problem. We use priors which put a mixture of a point mass at zero and certain absolutely continuous distribution on off-diagonal elements of the precision matrix. Hence the resulting posterior distribution can be used for graphical structure learning. The posterior convergence rate of the precision matrix is obtained and is shown to match the oracle rate. The posterior distribution on the model space is extremely cumbersome to compute using the commonly used reversible jump Markov chain Monte Carlo methods. However, the posterior mode in each graph can be easily identified as the graphical lasso restricted to each model. We propose a fast computational method for approximating the posterior probabilities of various graphs using the Laplace approximation approach by expanding the posterior density around the posterior mode. We also provide estimates of the accuracy in the approximation.

Journal of Multivariate Analysis

Bayesian structure learning in graphical models

The development of parsimonious models for reliable inference and prediction of responses in high-dimensional regression settings is often challenging due to relatively small sample sizes and the presence of complex interaction patterns between a large number of covariates. We propose an efficient, nonparametric framework for simultaneous variable selection, clustering and prediction in high-throughput regression settings with continuous outcomes. The proposed model utilizes the sparsity induced by Poisson-Dirichlet processes (PDPs) to group the covariates into lower-dimensional latent clusters consisting of covariates with similar patterns among the samples. The data are permitted to direct the choice of a suitable cluster allocation scheme, choosing between PDPs and their special case, a Dirichlet process. Subsequently, the latent clusters are used to build a nonlinear prediction model for the responses using an adaptive mixture of linear and nonlinear elements, thus achieving a balance between model parsimony and flexibility. Through analyses of gene expression microarray datasets we demonstrate the reliability of the proposed method's clustering mechanism and show that the technique compares favorably to, and often outperforms, existing methodologies in terms of the prediction accuracies of the subject-specific clinical outcomes.

Nonparametric Bayesian Inference in Biostatistics

Nonparametric Variable Selection, Clustering and Prediction for Large Biological Datasets

Variable selection in regression models has been well studied in the literature, with many non-Bayesian and Bayesian methods available in this regard. An important class of regression models is generalized linear models, which involve situations where the response variable is discrete. To add more flexibility, generalized additive partial linear models can be considered, where some predictors can have a non-linear effect while some predictors have a strictly linear effect. We consider Bayesian variable selection in these models. The functions in the non-parametric additive part of the model are expanded in a B-spline basis and multivariate Laplace prior put on the coefficients with point mass at zero. The coefficients corresponding to the strictly linear components are assigned a univariate Laplace prior with point mass at zero. The prior times the likelihood is mathematically intractable, but we find an approximation by expansion around the posterior mode, which is the group lasso solution in generalized linear model setting for the choice of prior. We thus completely avoid Markov chain Monte Carlo methods, which are extremely slow and unreliable in high-dimensional models. We evaluate the performance of the Bayesian method by conducting simulation studies and real data analyses.

Bayesian variable selection in generalized additive partial linear models

We consider Bayesian estimation of a p × p precision matrix, when p can be much larger than the available sample size n. It is well known that consistent estimation in such ultra-high dimensional situations requires regularization such as banding, tapering or thresholding. We consider a banding structure in the model and induce a prior distribution on a banded precision matrix through a Gaussian graphical model, where an edge is present only when two vertices are within a given distance. For a proper choice of the order of graph, we obtain the convergence rate of the posterior distribution and Bayes estimators based on the graphical model in the L∞-operator norm uniformly over a class of precision matrices, even if the true precision matrix may not have a banded structure. Along the way to the proof, we also compute the convergence rate of the maximum likelihood estimator (MLE) under the same set of condition, which is of independent interest. The graphical model based MLE and Bayes estimators are automatically positive definite, which is a desirable property not possessed by some other estimators in the literature. We also conduct a simulation study to compare finite sample performance of the Bayes estimators and the MLE based on the graphical model with that obtained by using a Cholesky decomposition of the precision matrix. Finally, we discuss a practical method of choosing the order of the graphical model using the marginal likelihood function.

Electronic Journal of Statistics

Posterior convergence rates for estimating large precision matrices using graphical models

Bayesian Inference for High Dimensional Models: Convergence Properties and Computational Issues.

Variable selection techniques for the classical linear regression model have been widely investigated. Variable selection in fully nonparametric and additive regression models has been studied more recently. A Bayesian approach for nonparametric additive regression models is considered, where the functions in the additive model are expanded in a B-spline basis and a multivariate Laplace prior is put on the coefficients. Posterior probabilities of models defined by selection of predictors in the working model are computed, using a Laplace approximation method. The prior times the likelihood is expanded around the posterior mode, which can be identified with the group LASSO, for which a fast computing algorithm exists. Thus Markov chain Monte-Carlo or any other time consuming sampling based methods are completely avoided, leading to quick assessment of various posterior model probabilities. This technique is applied to the high-dimensional situation where the number of parameters exceeds the number of observations. extcopyright 2013 Elsevier B.V. All rights reserved.