Graphical Models Research Articles

Network hub gene detection using the entire solution path information.

Gene co-expression networks typically comprise modules and their associated hub genes, which are regulating numerous downstream interactions within the network. Methods for hub screening, as well as data-driven estimation of hub co-expression networks using graphical models, can serve as useful tools for identifying these hubs. Graphical model-based penalization methods typically have one or multiple regularization terms, each of which encourages some favorable characteristics (e.g., sparsity, hubs, power-law) to the estimated complex gene network. It is common practice to find a single optimal graphical model corresponding to a specific value of the regularization parameter(s). However, instead of doing this, one could aggregate information across several graphical models, all of which depend on the same data set, along the solution path in the hub gene detection process. We propose a novel method for detecting hub genes that utilizes the information available in the solution path. Our procedure is related to stability selection, but we replace resampling with a simple statistic. This procedure amalgamates information from each node of the data-driven graphical models into a single influence statistic, similar to Cook's distance. We call this statistic the Mean Degree Squared Distance (MDSD). Our simulation and empirical studies demonstrate that the MDSD statistic maintains a good balance between false positive and true positive hubs. An R package MDSD is publicly available on GitHub under the General Public License https://github.com/markkukuismin/MDSD.

Prediction of electroconvulsive therapy outcome: A network analysis approach.

While electroconvulsive therapy (ECT) for the treatment of major depressive disorder is effective, individual response is variable and difficult to predict. These difficulties may in part result from heterogeneity at the symptom level. We aim to predict remission using baseline depression symptoms, taking the associations among symptoms into account, by using a network analysis approach. We combined individual patient data from two randomized controlled trials (total N = 161) and estimated a Mixed Graphical Model to estimate which baseline depression symptoms (corresponding to HRSD-17 items) uniquely predicted remission (defined as either HRSD≤7 or MADRS<10). We included study as moderator to evaluate study heterogeneity. For symptoms directly predictive of remission we computed odds ratios. Three baseline symptoms were uniquely predictive of remission: suicidality negatively predicted remission (OR = 0.75; bootstrapped confidence interval (bCI) = 0.44-1.00) whereas retardation (OR = 1.21; bCI = 1.00-2.02) and hypochondriasis (OR = 1.31; bCI = 1.00-2.25) positively predicted remission. The estimated effects did not differ across trials as no moderation effects were found. By using a network analysis approach this study identified that the presence of suicidal ideation predicts an overall worse treatment outcome. Psychomotor retardation and hypochondriasis, on the other hand, seem to be associated with a better outcome.

System-level Prognostics and Health Management for Complex Industrial Systems

Prognostics and health management (PHM) has become essential to guarantee aware and safe system operation and to inform economic decision-making. However, due to the nature of detection, diagnostics, and prognostic methods, applications have mainly been limited to the component level. In practice, most industrial systems consist of multiple interacting components whose partial degradation could lead to the system's failure (or subsystems).This research addresses the limitations of traditional component-level PHM techniques by proposing a novel system-level framework. By implementing a hierarchical structure of components and subsystems, we will select an optimal method for each subsystem to aggregate its component health assessments. The overall system health can then be estimated by further combining the obtained estimates. The research considers simplified and holistic modeling techniques, margin-based methods, and hybrid graphical models. This approach aims to provide reliable system health predictions and online components' sensitivity measures to enhance maintenance decision-making. We consider an application in the context of the nuclear industry, characterized by strict safety and economic requirements. Using a SIMULINK model to approximate a Pressurized Water Reactor (PWR) with real industrial inputs, we plan to add component degradation modules and use simulated sensor data and reliability information to test the proposed framework. Initial results on artificial case studies show the feasibility of integrating component-level health predictions.

Revealing the potential causality of suicidal ideation and psychopathic symptoms among survivors 10 years after the Wenchuan earthquake using network analysis.

Little is known about the interactions and causal inference between psychopathic symptoms and suicidal ideation (SI) among earthquake survivors. Using a 10-year large cohort of Chinese adolescents exposed to the Wenchuan earthquake, this study aimed to structure networks of suicidal ideation and some specific symptoms of posttraumatic symptoms and depression among survivors of the Wenchuan earthquake to clarify the interactions and causality between these symptoms. A total of 684 survivors who completed surveys at 6 months and 10 years postearthquake and had one or above psychopathic symptoms were included in the analyses. Sociodemographic, posttraumatic stress symptoms, depressive symptoms, and suicidal ideation were measured at 6 months and/or 10 years postearthquake. Network analyses were performed based on R Version 4.1.3. The Gaussian graphical model network revealed that "depressed mood" had the highest expected influence value. In addition, the directed acyclic graph network revealed that anhedonia could affect "depressed mood," which further leads to "suicidal ideation." The pathway of "anhedonia → depressed mood → suicidal ideation" was a crucial psychopathic mechanism of suicidal ideation. It informs that intervening in anhedonia and depressed mood among earthquake-exposed survivors is necessary to potentially decrease the risk of future suicidal ideation. (PsycInfo Database Record (c) 2024 APA, all rights reserved).

Quantum algorithms for weighted constrained sampling and weighted model counting

AbstractWe consider the problems of weighted constrained sampling and weighted model counting, where we are given a propositional formula and a weight for each world. The first problem consists of sampling worlds with a probability proportional to their weight given that the formula is satisfied. The latter is the problem of computing the sum of the weights of the models of the formula. Both have applications in many fields such as probabilistic reasoning, graphical models, statistical physics, statistics, and hardware verification. In this article, we propose quantum weighted constrained sampling (QWCS) and quantum weighted model counting (QWMC), two quantum algorithms for performing weighted constrained sampling and weighted model counting, respectively. Both are based on the quantum search/quantum model counting algorithms that are modified to take into account the weights. In the black box model of computation, where we can only query an oracle for evaluating the Boolean function given an assignment, QWCS requires $$O(2^{\frac{n}{2}}+1/\sqrt{\text {WMC}})$$ O ( 2 n 2 + 1 / WMC ) oracle calls, where n is the number of Boolean variables and $$\text {WMC}$$ WMC is the normalized between 0 and 1 weighted model count of the formula, while a classical algorithm has a complexity of $$\Omega (1/\text {WMC})$$ Ω ( 1 / WMC ) . QWMC takes $$\Theta (2^{\frac{n}{2}})$$ Θ ( 2 n 2 ) oracle calss, while classically the best complexity is $$\Theta (2^n)$$ Θ ( 2 n ) , thus achieving a quadratic speedup.

CURATE: Scaling-Up Differentially Private Causal Graph Discovery

Causal graph discovery (CGD) is the process of estimating the underlying probabilistic graphical model that represents the joint distribution of features of a dataset. CGD algorithms are broadly classified into two categories: (i) constraint-based algorithms, where the outcome depends on conditional independence (CI) tests, and (ii) score-based algorithms, where the outcome depends on optimized score function. Because sensitive features of observational data are prone to privacy leakage, differential privacy (DP) has been adopted to ensure user privacy in CGD. Adding the same amount of noise in this sequential-type estimation process affects the predictive performance of algorithms. Initial CI tests in constraint-based algorithms and later iterations of the optimization process of score-based algorithms are crucial; thus, they need to be more accurate and less noisy. Based on this key observation, we present CURATE (CaUsal gRaph AdapTivE privacy), a DP-CGD framework with adaptive privacy budgeting. In contrast to existing DP-CGD algorithms with uniform privacy budgeting across all iterations, CURATE allows for adaptive privacy budgeting by minimizing error probability (constraint-based), maximizing iterations of the optimization problem (score-based) while keeping the cumulative leakage bounded. To validate our framework, we present a comprehensive set of experiments on several datasets and show that CURATE achieves higher utility compared to existing DP-CGD algorithms with less privacy leakage.

Scaling Terrain-Aware Spatial Machine Learning for Flood Mapping on Large Scale Earth Imagery Data

The accurate and prompt mapping of flood-affected regions is important for effective disaster management, including damage assessment and relief efforts. While high-resolution optical imagery from satellites during disasters presents an opportunity for automated flood inundation mapping, existing segmentation models face challenges due to noises like cloud cover and tree canopies. Thanks to the digital elevation model (DEM) data readily available from sources such as United States Geological Survey (USGS), terrain guidance was utilized by recent graphical models such as hidden Markov trees (HMTs) to improve segmentation quality. Unfortunately, these methods either can only handle a small area where water levels at different locations are assumed to be consistent, or require restricted assumptions such as there is only one river channel. This paper presents an algorithm for flood extent mapping on large-scale Earth imagery, applicable to a large geographic area with multiple river channels. Since water level can vary a lot from upstream to downstream, we propose to detect river pixels in order to partition the remaining pixels into localized zones, each with a unique water level. In each zone, water at all locations flow to the same river entry point. Pixels in each zone are organized by an HMT to capture water flow directions guided by elevations. Moreover, a novel regularization scheme is designed to enforce inter-zone consistency by penalizing pixel-pairs of adjacent zones that violate terrain guidance. Efficient parallelization is made possible by coloring the zone adjacency graph to identify zones and zone-pairs that have no dependency and hence can be processed in parallel, and incremental one-pass terrain-guided scanning is conducted wherever applicable to reuse computations. Experiments demonstrate that our solution is more accurate than existing solutions, and can efficiently and accurately map out flooding pixels in a giant area of size 24,805 × 40,129. Despite the imbalanced workloads caused by a few large zonal HMTs dominating the serial computing time, our parallelization approach is effective and manages to achieve up to 14.3 × speedup on a machine with Intel Xeon Gold 6126 CPU @ 2.60GHz (24 cores, 48 threads) using 32 threads.

CaClust: linking genotype to transcriptional heterogeneity of follicular lymphoma using BCR and exomic variants.

Tumours exhibit high genotypic and transcriptional heterogeneity. Both affect cancer progression and treatment, but have been predominantly studied separately in follicular lymphoma. To comprehensively investigate the evolution and genotype-to-phenotype maps in follicular lymphoma, we introduce CaClust, a probabilistic graphical model integrating deep whole exome, single-cell RNA and B-cell receptor sequencing data to infer clone genotypes, cell-to-clone mapping, and single-cell genotyping. CaClust outperforms a state-of-the-art model on simulated and patient data. In-depth analyses of single cells from four samples showcase effects of driver mutations, follicular lymphoma evolution, possible therapeutic targets, and single-cell genotyping that agrees with an independent targeted resequencing experiment.

A new formal approach to safe response time upper bound evaluation for distributed control systems using dioid algebra

This paper presents an approach to computing a safe upper bound of the response time of Distributed Control System (DCS). From a graphical model of live autonomous Timed Event Graphs (TEGs) with asynchronous links and weights, we draw algebraic representations of the time/event dynamics using the Minax〚γ,δ〛 dioid. By introducing scalers and two operators for handling batching or multiplying events, we are able to calculate the trajectories and transfer functions. With deterministic and worst-case hypotheses, the worst delays induced by the model is then computed, composing an upper bound for the response time. We validate the method through a numerical application, confronted to experimental measurements. The method is modular, can be automated and is easily used to any DCS modelled with TEGs.

Bayesian deep learning: An enhanced AI framework for legal reasoning alignment

The integration of artificial intelligence into the field of law has penetrated the underlying logic of legal operations. Currently, legal AI systems face difficulties in representing legal knowledge, exhibit insufficient legal reasoning capabilities, have poor explainability, and are inefficient in handling causal inference and uncertainty. In legal practice, various legal reasoning methods (deductive reasoning, inductive reasoning, abductive reasoning, etc.) are often intertwined and used comprehensively. However, the reasoning modes employed by current legal AI systems are inadequate. Identifying AI models that are more suitable for legal reasoning is crucial for advancing the development of legal AI systems.Distinguished from the current high-profile large language models, we believe that Bayesian reasoning is highly compatible with legal reasoning, as it can perferm abductive reasoning, excel at causal inference, and admits the "defeasibility" of reasoning conclusions, which is consistent with the cognitive development pattern of legal professionals from apriori to posteriori. AI models based on Bayesian methods can also become the main technological support for legal AI systems. Bayesian neural networks have advantages in uncertainty modeling, avoiding overfitting, and explainability. Legal AI systems based on Bayesian deep learning frameworks can combine the advantages of deep learning and probabilistic graphical models, facilitating the exchange and supplementation of information between perception tasks and reasoning tasks. In this paper, we take perpetrator prediction systems and legal judegment prediction systems as examples to discuss the construction and basic operation modes of the Bayesian deep learning framework. Bayesian deep learning can enhance reasoning ability, improve the explainability of models, and make the reasoning process more transparent and visualizable. Furthermore, Bayesian deep learning framework is well-suited for human-machine collaborative tasks, enabling the complementary strengths of humans and machines.

Texas Rural Land Market Integration: A Causal Analysis using Machine Learning Applications

Texas rural land markets have several special features that makes it unique from other rural land markets in the United States. In 2021, Texas agricultural land, including buildings, is valued at $299.88 billion, which is almost 10% of the nation's total agricultural real estate value and 83% of the state's land is categorized as rural. In addition, due to its size and geologic features, Texas’ diverse landscape creates complex and widely divergent conditions affecting ownership and marketing of the land. Despite this complexity, lack of granular level and reliable transactional data on land sales has prevented thorough investigation into Texas land markets to uncover various interdependencies. Using quarterly transactional land value data from 1966-2017, this study uses cutting-edge machine learning algorithms and probabilistic graphical models to uncover causal interaction patterns of different land markets in Texas. The results reveal that Texas rural land markets are interdependent. Current and potential landholders and lenders can use the results from this work to aid strategic decision making. Financial institutions and investment groups could be made aware of the trend of one land market relative to other markets and adjust their holdings accordingly. Landowners may better understand changes in net wealth, which affect their ability to borrow capital and operate efficiently. Moreover, lenders may also benefit from the information to manage collateral and thus maintain the stability of their operation.

A special contribution from spatial ability to math word problem solving: Evidence from structural equation modelling and network analysis

There is a growing body of research into the factors contributing to math word problem solving. However, these studies usually use limited number of potential predictors (precluding assessing of their contribution in comparison with other factors or “g” general intelligence) and some predictors (such as analogical and hypothetical reasoning) are largely omitted. Thus, the aim of the current study was to explore contributions of different types of reasoning to math word problem solving and whether these contributions have added value compared with each other and general cognitive ability. Chinese schoolchildren in Grades 3 (N = 199; Mage = 102.4 months), 4 (N = 162; Mage = 114.6), 5 (N = 174; Mage = 126.1) and 6 (N = 180; Mage = 138.6) completed 8 tasks tapping into spatial, mechanical, verbal, mathematic, hypothetical and analogical reasoning. Our data showed that when 6 general cognitive factors load onto General cognitive ability factor in a Structural Equation Model (SEM), only spatial visualization has additional contribution to Word problem solving factor. Gaussian Graphical models (GGMs) showed that 2 verbal tasks and spatial visualization showed stable (present in at least 3 out of 4 grades) contributions to both word problem solving tasks. Analogical reasoning showed contribution to Process of word problem solving only. To sum up, both SEM and GGMs converged on the importance of spatial ability for mathematical word problems solving. Our results call for verbal and spatial ability to be routinely assessed and targeted by educational interventions within maths curriculum.

Assessment of the oil and gas potential of the eastern edge of the Northern Ustyurt using new geophysical data

Purpose. Detailed substantiation of the geological structure of the eastern edge of the Northern Ustyurt, clarification of modern geological and geophysical data, as well as highlighting the mainstages of evaluating the results of exploration work to confirm the oil and gas potential and subsequent exploitation of the fields. Methodology. The authors used the following scientific methods: generalization – to systematize scientific provisions; analogy and comparison – to characterize deflection in different periods; analysis and synthesis – to substantiate geological and geophysical data; algorithmmization – to determine conceptual provisions regarding the assessment of the oil and gas bearing capacity of the trough. Findings. In order to achieve the goals, the indicators of oil and gas capacity in the areas of the eastern side of the Northern Ustyurt of different geological ages were substantiated. The geological and geophysical features of the basin dating back to 2010 compared to modern refinement of geological data were characterized. The productive and unproductive structures of the eastern side of the Northern Ustyurt are characterized, taking into account the showing of oil and gas based on deposits of different geological ages. The importance of conducting geological exploration and creating corresponding cartographic materials is justified, followed by the design of predictive graphical models, geological sections, and a detailed plan of the territory. Originality. The influence of high hydro carbon generation rates on the further identification of “oil windows” has been substantiated for the first time. The hierarchical levels of predicting the oil and gas content of the trough are identified. The necessity of improving the mechanisms for assessing oil and gas content is substantiated. Practical value. The substantiation of the features of the assessment of the oil and gas content of the of the eastern edge of the Northern Ustyurt, which should be based on forecasting promising areas of prospecting, is the key to distinguishing between productive and unproductive objects, determining their real scale and directing investment flow precisely to promising areas with minimal environmental and economic losses.

On the application of Gaussian graphical models to paired data problems

Gaussian graphical models are nowadays commonly applied to the comparison of groups sharing the same variables, by jointly learning their independence structures. We consider the case where there are exactly two dependent groups and the association structure is represented by a family of coloured Gaussian graphical models suited to deal with paired data problems. To learn the two dependent graphs, together with their across-graph association structure, we implement a fused graphical lasso penalty. We carry out a comprehensive analysis of this approach, with special attention to the role played by some relevant submodel classes. In this way, we provide a broad set of tools for the application of Gaussian graphical models to paired data problems. These include results useful for the specification of penalty values in order to obtain a path of lasso solutions and an ADMM algorithm that solves the fused graphical lasso optimization problem. Finally, we carry out a simulation study to compare our method with the traditional graphical lasso, and present an application of our method to cancer genomics where it is of interest to compare cancer cells with a control sample from histologically normal tissues adjacent to the tumor. All the methods described in this article are implemented in the R package pdglasso available at https://github.com/savranciati/pdglasso.

Multi-task Learning for Gaussian Graphical Regressions with High Dimensional Covariates

Gaussian graphical regression is a powerful approach for regressing the precision matrix of a Gaussian graphical model on covariates, which permits the response variables and covariates to outnumber the sample size. However, traditional approaches of fitting the model via separate node-wise lasso regressions overlook the network-induced structure among these regressions, leading to high error rates, particularly when the number of nodes is large. To address this issue, we propose a multi-task learning estimator for fitting Gaussian graphical regression models, which incorporates a cross-task group sparsity penalty and a within-task element-wise sparsity penalty to govern the sparsity of active covariates and their effects on the graph, respectively. We also develop an efficient augmented Lagrangian algorithm for computation, which solves subproblems with a semi-smooth Newton method. We further prove that our multi-task learning estimator has considerably lower error rates than the separate node-wise regression estimates, as the cross-task penalty enables borrowing information across tasks. We examine the utility of our method through simulations and an application to a gene co-expression network study with brain cancer patients.

Application of artificial intelligence to study the causality in public health: a systematic review

Abstract Introduction In epidemiology, establishing causation requires meeting rigorous criteria beyond mere association. Traditional methodologies often struggle to provide definitive causal conclusions due to confounding variables. However, leveraging artificial Intelligence (AI) and machine Learning (ML) offers a promising solution by modeling complex interactions among relevant factors. This systematic review explores the AI techniques utilized to uncover causal relationships in public health applications. Methods A literature search was conducted in Pubmed, Web of Science, and Scopus, employing the following search strategy: (Causalit*[Title/abstract] OR Causation*[Title/abstract] OR Causal[Title/abstract]) AND (‘Machine Learning’[Title/abstract] OR “Deep Learning” [Title/abstract] OR ‘Artificial Intelligence’ [Title/abstract] OR Algorithm* [Title/abstract]). The search aimed to identify studies encompassing causal inference techniques utilizing ML and AI methodologies in public health. Results From a total of 28,230 articles published up to August 2023, after removing duplicates (11,158) and articles that did not meet the inclusion criteria (16,168 post title/abstract screening, and 856 post full-text screening), the systematic review included 48 articles. Bayesian additive regression trees, Bayesian network models, causal forests analysis, and graphical causal models emerged as the prevailing causal methodologies. AI-driven causal methodologies have been investigated across various domains within public health, encompassing disease prediction, evaluation of treatment efficacy, identification of risk factors, analysis of health behaviors, precision Public Health initiatives, and environmental health assessment. Discussion While our systematic review affirms the potential of AI-driven causal approaches across several public health domains, further research is warranted before these emerging techniques can be effectively integrated into real settings. Key messages • In epidemiology, establishing causation requires meeting rigorous criteria beyond mere association and traditional methodologies often struggle to provide definitive causal conclusions. • This systematic review explores the AI techniques utilized to uncover causal relationships in public health applications.

Construction and validation of a line chart for gestational diabetes mellitus based on clinical indicators

BackgroundGestational diabetes mellitus (GDM) is a common complication of mid–to-late pregnancy. Here, we constructed a predictive model for GDM based on a combination of clinical characteristics and relevant serum markers.MethodsData from full-term singleton vaginal deliveries from January 2022 to January 2023 were retrospectively collected from the obstetrics department. The data collected were segregated and assigned to training, validation, and external test sets. Maternal demographic characteristics, living and working habits, and haematological indicators, such as liver function and lipids were collected using a questionnaire designed for the study. The “rms” package in R was used to explore GDM-associated factors through stepwise regression at P < 0.05. A predictive model was developed based on the results of multifactorial logistic regression analysis. We then evaluated the differentiation of the column-line graphical model and performed internal and external validation. To assess the accuracy of the bar graphical model, we plotted calibration and decision curves.ResultsData from 265 pregnant women were included in the training and internal validation sets, and data from 113 pregnant women were included in the external validation set. The logistic regression algorithm screened 8 indicators as predictors. A prediction model was constructed with ALT, TBA, TC, and TG levels while considering whether GDM affects appetite, the husband– wife relationship, family history, and parental relationships as predictors. The Hosmer–Lemeshow goodness-of-fit test revealed that the chi-square values for the modelling, internal validation, and external validation groups (χ2 = 5.964, 3.249, and 12.182, respectively) were all P > 0.05. The ROC curve AUCs for the three groups were 0.93 (95% CI: 0.89–0.97), 0.72 (95% CI: 0.62–0.81), and 0.68 (95% CI: 0.53–0.83), respectively.ConclusionIn this study, a GDM prediction model was constructed to achieve high performance in GDM risk prediction based on routine obstetric tests and information.

Shrinkage estimation of gene interaction networks in single-cell RNA sequencing data

BackgroundGene interaction networks are graphs in which nodes represent genes and edges represent functional interactions between them. These interactions can be at multiple levels, for instance, gene regulation, protein-protein interaction, or metabolic pathways. To analyse gene interaction networks at a large scale, gene co-expression network analysis is often applied on high-throughput gene expression data such as RNA sequencing data. With the advance in sequencing technology, expression of genes can be measured in individual cells. Single-cell RNA sequencing (scRNAseq) provides insights of cellular development, differentiation and characteristics at the transcriptomic level. High sparsity and high-dimensional data structures pose challenges in scRNAseq data analysis.ResultsIn this study, a sparse inverse covariance matrix estimation framework for scRNAseq data is developed to capture direct functional interactions between genes. Comparative analyses highlight high performance and fast computation of Stein-type shrinkage in high-dimensional data using simulated scRNAseq data. Data transformation approaches also show improvement in performance of shrinkage methods in non-Gaussian distributed data. Zero-inflated modelling of scRNAseq data based on a negative binomial distribution enhances shrinkage performance in zero-inflated data without interference on non zero-inflated count data.ConclusionThe proposed framework broadens application of graphical model in scRNAseq analysis with flexibility in sparsity of count data resulting from dropout events, high performance, and fast computational time. Implementation of the framework is in a reproducible Snakemake workflow https://github.com/calathea24/ZINBGraphicalModel and R package ZINBStein https://github.com/calathea24/ZINBStein.

Bayesian Multilayered Mediation Analysis for Cancer Pharmacogenomics

ABSTRACTMultiomic data from multilevel biological systems are becoming common and motivate integrative modelling approaches to decipher within‐ and cross‐platform dependencies. Mediation analysis aims to identify mediating mechanisms that regulate the effect of an exposure on an outcome. In multiomic contexts, identification of genomic mediators of disease outcomes provides a deeper understanding of mechanisms of disease and corresponding therapeutic targets. While there has been significant work on joint modelling of high‐dimensional potential mediators, approaches that can identify individual mediators in presence of high‐dimensional potential mediators are lacking. We posit that the multiomic data are interrelated following multilayered Gaussian graphical models that include undirected and directed acyclic graphs as special cases. We develop a Bayesian inferential framework for multilayered mediation analysis with continuous, binary, and ordinal outcomes using probit models. As opposed to existing approaches focusing on identifying joint mediation effects, we decompose the joint effect into effects attributable to individual mediators in the framework of interventional mediation analysis. Simulations demonstrate our method outperforms other existing approaches to identify mediators that have nonzero indirect effects to the outcome. We apply our method to multiomic analysis on drug sensitivity outcomes of palbociclib and agents for endocrine therapy, standard care for breast cancer.

Concentration of a Sparse Bayesian Model With Horseshoe Prior in Estimating High‐Dimensional Precision Matrix

ABSTRACTPrecision matrices are crucial in many fields such as social networks, neuroscience and economics, representing the edge structure of Gaussian graphical models (GGMs), where a zero in an off‐diagonal position of the precision matrix indicates conditional independence between nodes. In high‐dimensional settings where the dimension of the precision matrix exceeds the sample size and the matrix is sparse, methods like graphical Lasso, graphical SCAD and CLIME are popular for estimating GGMs. While frequentist methods are well‐studied, Bayesian approaches for (unstructured) sparse precision matrices are less explored. The graphical horseshoe estimate, applying the global‐local horseshoe prior, shows superior empirical performance, but theoretical work for sparse precision matrix estimations using shrinkage priors is limited. This paper addresses these gaps by providing concentration results for the tempered posterior with the fully specified horseshoe prior in high‐dimensional settings. Moreover, we also provide novel theoretical results for model misspecification, offering a general oracle inequality for the posterior. A concise set of simulations is performed to validate our theoretical findings.