Francisco J. R. Ruiz, Susan Athey, David M. Blei.

Source: The Annals of Applied Statistics, Volume 14, Number 1, 1--27.

Abstract:
We develop SHOPPER, a sequential probabilistic model of shopping data. SHOPPER uses interpretable components to model the forces that drive how a customer chooses products; in particular, we designed SHOPPER to capture how items interact with other items. We develop an efficient posterior inference algorithm to estimate these forces from large-scale data, and we analyze a large dataset from a major chain grocery store. We are interested in answering counterfactual queries about changes in prices. We found that SHOPPER provides accurate predictions even under price interventions, and that it helps identify complementary and substitutable pairs of products.

Li Zhu, Zhiguang Huo, Tianzhou Ma, Steffi Oesterreich, George C. Tseng.

Source: The Annals of Applied Statistics, Volume 13, Number 4, 2611--2636.

Abstract:
Variable selection is a pervasive problem in modern high-dimensional data analysis where the number of features often exceeds the sample size (a.k.a. small-n-large-p problem). Incorporation of group structure knowledge to improve variable selection has been widely studied. Here, we consider prior knowledge of a hierarchical overlapping group structure to improve variable selection in regression setting. In genomics applications, for instance, a biological pathway contains tens to hundreds of genes and a gene can be mapped to multiple experimentally measured features (such as its mRNA expression, copy number variation and methylation levels of possibly multiple sites). In addition to the hierarchical structure, the groups at the same level may overlap (e.g., two pathways can share common genes). Incorporating such hierarchical overlapping groups in traditional penalized regression setting remains a difficult optimization problem. Alternatively, we propose a Bayesian indicator model that can elegantly serve the purpose. We evaluate the model in simulations and two breast cancer examples, and demonstrate its superior performance over existing models. The result not only enhances prediction accuracy but also improves variable selection and model interpretation that lead to deeper biological insight of the disease.

Silvia Metelli, Nicholas Heard.

Source: The Annals of Applied Statistics, Volume 13, Number 4, 2586--2610.

Abstract:
Monitoring computer network traffic for anomalous behaviour presents an important security challenge. Arrivals of new edges in a network graph represent connections between a client and server pair not previously observed, and in rare cases these might suggest the presence of intruders or malicious implants. We propose a Bayesian model and anomaly detection method for simultaneously characterising existing network structure and modelling likely new edge formation. The method is demonstrated on real computer network authentication data and successfully identifies some machines which are known to be compromised.

Seth Flaxman, Michael Chirico, Pau Pereira, Charles Loeffler.

Source: The Annals of Applied Statistics, Volume 13, Number 4, 2564--2585.

Abstract:
We propose a generic spatiotemporal event forecasting method which we developed for the National Institute of Justice’s (NIJ) Real-Time Crime Forecasting Challenge (National Institute of Justice (2017)). Our method is a spatiotemporal forecasting model combining scalable randomized Reproducing Kernel Hilbert Space (RKHS) methods for approximating Gaussian processes with autoregressive smoothing kernels in a regularized supervised learning framework. While the smoothing kernels capture the two main approaches in current use in the field of crime forecasting, kernel density estimation (KDE) and self-exciting point process (SEPP) models, the RKHS component of the model can be understood as an approximation to the popular log-Gaussian Cox Process model. For inference, we discretize the spatiotemporal point pattern and learn a log-intensity function using the Poisson likelihood and highly efficient gradient-based optimization methods. Model hyperparameters including quality of RKHS approximation, spatial and temporal kernel lengthscales, number of autoregressive lags and bandwidths for smoothing kernels as well as cell shape, size and rotation, were learned using cross validation. Resulting predictions significantly exceeded baseline KDE estimates and SEPP models for sparse events.

Liberty Vittert, Adrian W. Bowman, Stanislav Katina.

Source: The Annals of Applied Statistics, Volume 13, Number 4, 2539--2563.

Abstract:
One of the data structures generated by medical imaging technology is high resolution point clouds representing anatomical surfaces. Stereophotogrammetry and laser scanning are two widely available sources of this kind of data. A standardised surface representation is required to provide a meaningful correspondence across different images as a basis for statistical analysis. Point locations with anatomical definitions, referred to as landmarks, have been the traditional approach. Landmarks can also be taken as the starting point for more general surface representations, often using templates which are warped on to an observed surface by matching landmark positions and subsequent local adjustment of the surface. The aim of the present paper is to provide a new approach which places anatomical curves at the heart of the surface representation and its analysis. Curves provide intermediate structures which capture the principal features of the manifold (surface) of interest through its ridges and valleys. As landmarks are often available these are used as anchoring points, but surface curvature information is the principal guide in estimating the curve locations. The surface patches between these curves are relatively flat and can be represented in a standardised manner by appropriate surface transects to give a complete surface model. This new approach does not require the use of a template, reference sample or any external information to guide the method and, when compared with a surface based approach, the estimation of curves is shown to have improved performance. In addition, examples involving applications to mussel shells and human faces show that the analysis of curve information can deliver more targeted and effective insight than the use of full surface information.

Armin Schwartzman, Andrew J. Schork, Rong Zablocki, Wesley K. Thompson.

Source: The Annals of Applied Statistics, Volume 13, Number 4, 2509--2538.

Abstract:
Analysis of genome-wide association studies (GWAS) is characterized by a large number of univariate regressions where a quantitative trait is regressed on hundreds of thousands to millions of single-nucleotide polymorphism (SNP) allele counts, one at a time. This article proposes an estimator of the SNP heritability of the trait, defined here as the fraction of the variance of the trait explained by the SNPs in the study. The proposed GWAS heritability (GWASH) estimator is easy to compute, highly interpretable and is consistent as the number of SNPs and the sample size increase. More importantly, it can be computed from summary statistics typically reported in GWAS, not requiring access to the original data. The estimator takes full account of the linkage disequilibrium (LD) or correlation between the SNPs in the study through moments of the LD matrix, estimable from auxiliary datasets. Unlike other proposed estimators in the literature, we establish the theoretical properties of the GWASH estimator and obtain analytical estimates of the precision, allowing for power and sample size calculations for SNP heritability estimates and forming a firm foundation for future methodological development.

Kun Liang.

Source: The Annals of Applied Statistics, Volume 13, Number 4, 2452--2482.

Abstract:
Finding differentially expressed genes is a common task in high-throughput transcriptome studies. While traditional statistical methods rank the genes by their test statistics alone, we analyze an RNA sequencing dataset using the auxiliary information of gene length and the test statistics from a related microarray study. Given the auxiliary information, we propose a novel nonparametric empirical Bayes procedure to estimate the posterior probability of differential expression for each gene. We demonstrate the advantage of our procedure in extensive simulation studies and a psoriasis RNA sequencing study. The companion R package calm is available at Bioconductor.

Dale L. Zimmerman, Jun Tang, Rui Huang.

Source: The Annals of Applied Statistics, Volume 13, Number 4, 2416--2451.

Abstract:
We extend statistical shape analytic methods known as outline analysis for application to the strike zone, a central feature of the game of baseball. Although the strike zone is rigorously defined by Major League Baseball’s official rules, umpires make mistakes in calling pitches as strikes (and balls) and may even adhere to a strike zone somewhat different than that prescribed by the rule book. Our methods yield inference on geometric attributes (centroid, dimensions, orientation and shape) of this “called strike zone” (CSZ) and on the effects that years, umpires, player attributes, game situation factors and their interactions have on those attributes. The methodology consists of first using kernel discriminant analysis to determine a noisy outline representing the CSZ corresponding to each factor combination, then fitting existing elliptic Fourier and new generalized superelliptic models for closed curves to that outline and finally analyzing the fitted model coefficients using standard methods of regression analysis, factorial analysis of variance and variance component estimation. We apply these methods to PITCHf/x data comprising more than three million called pitches from the 2008–2016 Major League Baseball seasons to address numerous questions about the CSZ. We find that all geometric attributes of the CSZ, except its size, became significantly more like those of the rule-book strike zone from 2008–2016 and that several player attribute/game situation factors had statistically and practically significant effects on many of them. We also establish that the variation in the horizontal center, width and area of an individual umpire’s CSZ from pitch to pitch is smaller than their variation among CSZs from different umpires.

Susheela P. Singh, Ana-Maria Staicu, Robert R. Dunn, Noah Fierer, Brian J. Reich.

Source: The Annals of Applied Statistics, Volume 13, Number 4, 2341--2362.

Abstract:
The advent of high-throughput sequencing technologies has made data from DNA material readily available, leading to a surge of microbiome-related research establishing links between markers of microbiome health and specific outcomes. However, to harness the power of microbial communities we must understand not only how they affect us, but also how they can be influenced to improve outcomes. This area has been dominated by methods that reduce community composition to summary metrics, which can fail to fully exploit the complexity of community data. Recently, methods have been developed to model the abundance of taxa in a community, but they can be computationally intensive and do not account for spatial effects underlying microbial settlement. These spatial effects are particularly relevant in the microbiome setting because we expect communities that are close together to be more similar than those that are far apart. In this paper, we propose a flexible Bayesian spike-and-slab variable selection model for presence-absence indicators that accounts for spatial dependence and cross-dependence between taxa while reducing dimensionality in both directions. We show by simulation that in the presence of spatial dependence, popular distance-based hypothesis testing methods fail to preserve their advertised size, and the proposed method improves variable selection. Finally, we present an application of our method to an indoor fungal community found within homes across the contiguous United States.

Stephen Berg, Jun Zhu, Murray K. Clayton, Monika E. Shea, David J. Mladenoff.

Source: The Annals of Applied Statistics, Volume 13, Number 4, 2312--2340.

Abstract:
The Wisconsin Public Land Survey database describes historical forest composition at high spatial resolution and is of interest in ecological studies of forest composition in Wisconsin just prior to significant Euro-American settlement. For such studies it is useful to identify recurring subpopulations of tree species known as communities, but standard clustering approaches for subpopulation identification do not account for dependence between spatially nearby observations. Here, we develop and fit a latent discrete Markov random field model for the purpose of identifying and classifying historical forest communities based on spatially referenced multivariate tree species counts across Wisconsin. We show empirically for the actual dataset and through simulation that our latent Markov random field modeling approach improves prediction and parameter estimation performance. For model fitting we introduce a new stochastic approximation algorithm which enables computationally efficient estimation and classification of large amounts of spatial multivariate count data.

Federico Castelletti, Guido Consonni.

Source: The Annals of Applied Statistics, Volume 13, Number 4, 2289--2311.

Abstract:
A signalling pathway is a sequence of chemical reactions initiated by a stimulus which in turn affects a receptor, and then through some intermediate steps cascades down to the final cell response. Based on the technique of flow cytometry, samples of cell-by-cell measurements are collected under each experimental condition, resulting in a collection of interventional data (assuming no latent variables are involved). Usually several external interventions are applied at different points of the pathway, the ultimate aim being the structural recovery of the underlying signalling network which we model as a causal Directed Acyclic Graph (DAG) using intervention calculus. The advantage of using interventional data, rather than purely observational one, is that identifiability of the true data generating DAG is enhanced. More technically a Markov equivalence class of DAGs, whose members are statistically indistinguishable based on observational data alone, can be further decomposed, using additional interventional data, into smaller distinct Interventional Markov equivalence classes. We present a Bayesian methodology for structural learning of Interventional Markov equivalence classes based on observational and interventional samples of multivariate Gaussian observations. Our approach is objective, meaning that it is based on default parameter priors requiring no personal elicitation; some flexibility is however allowed through a tuning parameter which regulates sparsity in the prior on model space. Based on an analytical expression for the marginal likelihood of a given Interventional Essential Graph, and a suitable MCMC scheme, our analysis produces an approximate posterior distribution on the space of Interventional Markov equivalence classes, which can be used to provide uncertainty quantification for features of substantive scientific interest, such as the posterior probability of inclusion of selected edges, or paths.

Ningshan Zhang, Kyle Schmaus, Patrick O. Perry.

Source: The Annals of Applied Statistics, Volume 13, Number 4, 2260--2288.

Abstract:
We consider a particular instance of a common problem in recommender systems, using a database of book reviews to inform user-targeted recommendations. In our dataset, books are categorized into genres and subgenres. To exploit this nested taxonomy, we use a hierarchical model that enables information pooling across across similar items at many levels within the genre hierarchy. The main challenge in deploying this model is computational. The data sizes are large and fitting the model at scale using off-the-shelf maximum likelihood procedures is prohibitive. To get around this computational bottleneck, we extend a moment-based fitting procedure proposed for fitting single-level hierarchical models to the general case of arbitrarily deep hierarchies. This extension is an order of magnitude faster than standard maximum likelihood procedures. The fitting method can be deployed beyond recommender systems to general contexts with deeply nested hierarchical generalized linear mixed models.

Carolyn M. Rutter, Jonathan Ozik, Maria DeYoreo, Nicholson Collier.

Source: The Annals of Applied Statistics, Volume 13, Number 4, 2189--2212.

Abstract:
Microsimulation models (MSMs) are used to inform policy by predicting population-level outcomes under different scenarios. MSMs simulate individual-level event histories that mark the disease process (such as the development of cancer) and the effect of policy actions (such as screening) on these events. MSMs often have many unknown parameters; calibration is the process of searching the parameter space to select parameters that result in accurate MSM prediction of a wide range of targets. We develop Incremental Mixture Approximate Bayesian Computation (IMABC) for MSM calibration which results in a simulated sample from the posterior distribution of model parameters given calibration targets. IMABC begins with a rejection-based ABC step, drawing a sample of points from the prior distribution of model parameters and accepting points that result in simulated targets that are near observed targets. Next, the sample is iteratively updated by drawing additional points from a mixture of multivariate normal distributions and accepting points that result in accurate predictions. Posterior estimates are obtained by weighting the final set of accepted points to account for the adaptive sampling scheme. We demonstrate IMABC by calibrating CRC-SPIN 2.0, an updated version of a MSM for colorectal cancer (CRC) that has been used to inform national CRC screening guidelines.

Jason Xu, Samson Koelle, Peter Guttorp, Chuanfeng Wu, Cynthia Dunbar, Janis L. Abkowitz, Vladimir N. Minin.

Source: The Annals of Applied Statistics, Volume 13, Number 4, 2091--2119.

Abstract:
Single-cell lineage tracking strategies enabled by recent experimental technologies have produced significant insights into cell fate decisions, but lack the quantitative framework necessary for rigorous statistical analysis of mechanistic models describing cell division and differentiation. In this paper, we develop such a framework with corresponding moment-based parameter estimation techniques for continuous-time, multi-type branching processes. Such processes provide a probabilistic model of how cells divide and differentiate, and we apply our method to study hematopoiesis , the mechanism of blood cell production. We derive closed-form expressions for higher moments in a general class of such models. These analytical results allow us to efficiently estimate parameters of much richer statistical models of hematopoiesis than those used in previous statistical studies. To our knowledge, the method provides the first rate inference procedure for fitting such models to time series data generated from cellular barcoding experiments. After validating the methodology in simulation studies, we apply our estimator to hematopoietic lineage tracking data from rhesus macaques. Our analysis provides a more complete understanding of cell fate decisions during hematopoiesis in nonhuman primates, which may be more relevant to human biology and clinical strategies than previous findings from murine studies. For example, in addition to previously estimated hematopoietic stem cell self-renewal rate, we are able to estimate fate decision probabilities and to compare structurally distinct models of hematopoiesis using cross validation. These estimates of fate decision probabilities and our model selection results should help biologists compare competing hypotheses about how progenitor cells differentiate. The methodology is transferrable to a large class of stochastic compartmental and multi-type branching models, commonly used in studies of cancer progression, epidemiology and many other fields.

Gabriela V. Cohen Freue, David Kepplinger, Matías Salibián-Barrera, Ezequiel Smucler.

Source: The Annals of Applied Statistics, Volume 13, Number 4, 2065--2090.

Abstract:
In large-scale quantitative proteomic studies, scientists measure the abundance of thousands of proteins from the human proteome in search of novel biomarkers for a given disease. Penalized regression estimators can be used to identify potential biomarkers among a large set of molecular features measured. Yet, the performance and statistical properties of these estimators depend on the loss and penalty functions used to define them. Motivated by a real plasma proteomic biomarkers study, we propose a new class of penalized robust estimators based on the elastic net penalty, which can be tuned to keep groups of correlated variables together in the selected model and maintain robustness against possible outliers. We also propose an efficient algorithm to compute our robust penalized estimators and derive a data-driven method to select the penalty term. Our robust penalized estimators have very good robustness properties and are also consistent under certain regularity conditions. Numerical results show that our robust estimators compare favorably to other robust penalized estimators. Using our proposed methodology for the analysis of the proteomics data, we identify new potentially relevant biomarkers of cardiac allograft vasculopathy that are not found with nonrobust alternatives. The selected model is validated in a new set of 52 test samples and achieves an area under the receiver operating characteristic (AUC) of 0.85.

Hannah Worthington, Rachel McCrea, Ruth King, Richard Griffiths.

Source: The Annals of Applied Statistics, Volume 13, Number 4, 2043--2064.

Abstract:
Capture-recapture studies often involve collecting data on numerous capture occasions over a relatively short period of time. For many study species this process is repeated, for example, annually, resulting in capture information spanning multiple sampling periods. To account for the different temporal scales, the robust design class of models have traditionally been applied providing a framework in which to analyse all of the available capture data in a single likelihood expression. However, these models typically require strong constraints, either the assumption of closure within a sampling period (the closed robust design) or conditioning on the number of individuals captured within a sampling period (the open robust design). For real datasets these assumptions may not be appropriate. We develop a general modelling structure that requires neither assumption by explicitly modelling the movement of individuals into the population both within and between the sampling periods, which in turn permits the estimation of abundance within a single consistent framework. The flexibility of the novel model structure is further demonstrated by including the computationally challenging case of multi-state data where there is individual time-varying discrete covariate information. We derive an efficient likelihood expression for the new multi-state multi-period stopover model using the hidden Markov model framework. We demonstrate the significant improvement in parameter estimation using our new modelling approach in terms of both the multi-period and multi-state components through both a simulation study and a real dataset relating to the protected species of great crested newts, Triturus cristatus .

Ye Zhang, Zhigang Yao, Patrik Forssén, Torgny Fornstedt.

Source: The Annals of Applied Statistics, Volume 13, Number 4, 2011--2042.

Abstract:
The means to obtain the rate constants of a chemical reaction is a fundamental open problem in both science and the industry. Traditional techniques for finding rate constants require either chemical modifications of the reactants or indirect measurements. The rate constant map method is a modern technique to study binding equilibrium and kinetics in chemical reactions. Finding a rate constant map from biosensor data is an ill-posed inverse problem that is usually solved by regularization. In this work, rather than finding a deterministic regularized rate constant map that does not provide uncertainty quantification of the solution, we develop an adaptive variational Bayesian approach to estimate the distribution of the rate constant map, from which some intrinsic properties of a chemical reaction can be explored, including information about rate constants. Our new approach is more realistic than the existing approaches used for biosensors and allows us to estimate the dynamics of the interactions, which are usually hidden in a deterministic approximate solution. We verify the performance of the new proposed method by numerical simulations, and compare it with the Markov chain Monte Carlo algorithm. The results illustrate that the variational method can reliably capture the posterior distribution in a computationally efficient way. Finally, the developed method is also tested on the real biosensor data (parathyroid hormone), where we provide two novel analysis tools—the thresholding contour map and the high order moment map—to estimate the number of interactions as well as their rate constants.

Bret Zeldow, Vincent Lo Re III, Jason Roy.

Source: The Annals of Applied Statistics, Volume 13, Number 3, 1989--2010.

Abstract:
Bayesian Additive Regression Trees (BART) is a flexible machine learning algorithm capable of capturing nonlinearities between an outcome and covariates and interactions among covariates. We extend BART to a semiparametric regression framework in which the conditional expectation of an outcome is a function of treatment, its effect modifiers, and confounders. The confounders are allowed to have unspecified functional form, while treatment and effect modifiers that are directly related to the research question are given a linear form. The result is a Bayesian semiparametric linear regression model where the posterior distribution of the parameters of the linear part can be interpreted as in parametric Bayesian regression. This is useful in situations where a subset of the variables are of substantive interest and the others are nuisance variables that we would like to control for. An example of this occurs in causal modeling with the structural mean model (SMM). Under certain causal assumptions, our method can be used as a Bayesian SMM. Our methods are demonstrated with simulation studies and an application to dataset involving adults with HIV/Hepatitis C coinfection who newly initiate antiretroviral therapy. The methods are available in an R package called semibart.

Youyi Zhang, Jeffrey S. Morris, Shivali Narang Aerry, Arvind U. K. Rao, Veerabhadran Baladandayuthapani.

Source: The Annals of Applied Statistics, Volume 13, Number 3, 1957--1988.

Abstract:
Technological innovations have produced large multi-modal datasets that include imaging and multi-platform genomics data. Integrative analyses of such data have the potential to reveal important biological and clinical insights into complex diseases like cancer. In this paper, we present Bayesian approaches for integrative analysis of radiological imaging and multi-platform genomic data, where-in our goals are to simultaneously identify genomic and radiomic, that is, radiology-based imaging markers, along with the latent associations between these two modalities, and to detect the overall prognostic relevance of the combined markers. For this task, we propose Radio-iBAG: Radiomics-based Integrative Bayesian Analysis of Multiplatform Genomic Data , a multi-scale Bayesian hierarchical model that involves several innovative strategies: it incorporates integrative analysis of multi-platform genomic data sets to capture fundamental biological relationships; explores the associations between radiomic markers accompanying genomic information with clinical outcomes; and detects genomic and radiomic markers associated with clinical prognosis. We also introduce the use of sparse Principal Component Analysis (sPCA) to extract a sparse set of approximately orthogonal meta-features each containing information from a set of related individual radiomic features, reducing dimensionality and combining like features. Our methods are motivated by and applied to The Cancer Genome Atlas glioblastoma multiforme data set, where-in we integrate magnetic resonance imaging-based biomarkers along with genomic, epigenomic and transcriptomic data. Our model identifies important magnetic resonance imaging features and the associated genomic platforms that are related with patient survival times.

Chanmin Kim, Michael J. Daniels, Joseph W. Hogan, Christine Choirat, Corwin M. Zigler.

Source: The Annals of Applied Statistics, Volume 13, Number 3, 1927--1956.

Abstract:
Emission control technologies installed on power plants are a key feature of many air pollution regulations in the US. While such regulations are predicated on the presumed relationships between emissions, ambient air pollution and human health, many of these relationships have never been empirically verified. The goal of this paper is to develop new statistical methods to quantify these relationships. We frame this problem as one of mediation analysis to evaluate the extent to which the effect of a particular control technology on ambient pollution is mediated through causal effects on power plant emissions. Since power plants emit various compounds that contribute to ambient pollution, we develop new methods for multiple intermediate variables that are measured contemporaneously, may interact with one another, and may exhibit joint mediating effects. Specifically, we propose new methods leveraging two related frameworks for causal inference in the presence of mediating variables: principal stratification and causal mediation analysis. We define principal effects based on multiple mediators, and also introduce a new decomposition of the total effect of an intervention on ambient pollution into the natural direct effect and natural indirect effects for all combinations of mediators. Both approaches are anchored to the same observed-data models, which we specify with Bayesian nonparametric techniques. We provide assumptions for estimating principal causal effects, then augment these with an additional assumption required for causal mediation analysis. The two analyses, interpreted in tandem, provide the first empirical investigation of the presumed causal pathways that motivate important air quality regulatory policies.

Byron C. Jaeger, D. Leann Long, Dustin M. Long, Mario Sims, Jeff M. Szychowski, Yuan-I Min, Leslie A. Mcclure, George Howard, Noah Simon.

Source: The Annals of Applied Statistics, Volume 13, Number 3, 1847--1883.

Abstract:
We introduce and evaluate the oblique random survival forest (ORSF). The ORSF is an ensemble method for right-censored survival data that uses linear combinations of input variables to recursively partition a set of training data. Regularized Cox proportional hazard models are used to identify linear combinations of input variables in each recursive partitioning step. Benchmark results using simulated and real data indicate that the ORSF’s predicted risk function has high prognostic value in comparison to random survival forests, conditional inference forests, regression and boosting. In an application to data from the Jackson Heart Study, we demonstrate variable and partial dependence using the ORSF and highlight characteristics of its ten-year predicted risk function for atherosclerotic cardiovascular disease events (ASCVD; stroke, coronary heart disease). We present visualizations comparing variable and partial effect estimation according to the ORSF, the conditional inference forest, and the Pooled Cohort Risk equations. The obliqueRSF R package, which provides functions to fit the ORSF and create variable and partial dependence plots, is available on the comprehensive R archive network (CRAN).

Arkaprava Roy, Subhashis Ghosal, Jeffrey Prescott, Kingshuk Roy Choudhury.

Source: The Annals of Applied Statistics, Volume 13, Number 3, 1791--1816.

Abstract:
We study possible relations between Alzheimer’s disease progression and the structure of the connectome which is white matter connecting different regions of the brain. Regression models in covariates including age, gender and disease status for the extent of white matter connecting each pair of regions of the brain are proposed. Subject inhomogeneity is also incorporated in the model through random effects with an unknown distribution. As there is a large number of pairs of regions, we also adopt a dimension reduction technique through graphon ( J. Combin. Theory Ser. B 96 (2006) 933–957) functions which reduces the functions of pairs of regions to functions of regions. The connecting graphon functions are considered unknown but the assumed smoothness allows putting priors of low complexity on these functions. We pursue a nonparametric Bayesian approach by assigning a Dirichlet process scale mixture of zero to mean normal prior on the distributions of the random effects and finite random series of tensor products of B-splines priors on the underlying graphon functions. We develop efficient Markov chain Monte Carlo techniques for drawing samples for the posterior distributions using Hamiltonian Monte Carlo (HMC). The proposed Bayesian method overwhelmingly outperforms a competing method based on ANCOVA models in the simulation setup. The proposed Bayesian approach is applied on a dataset of 100 subjects and 83 brain regions and key regions implicated in the changing connectome are identified.

Huiping Xu, Xiaochun Li, Changyu Shen, Siu L. Hui, Shaun Grannis.

Source: The Annals of Applied Statistics, Volume 13, Number 3, 1753--1790.

Abstract:
The conditional independence assumption of the Felligi and Sunter (FS) model in probabilistic record linkage is often violated when matching real-world data. Ignoring conditional dependence has been shown to seriously bias parameter estimates. However, in record linkage, the ultimate goal is to inform the match status of record pairs and therefore, record linkage algorithms should be evaluated in terms of matching accuracy. In the literature, more flexible models have been proposed to relax the conditional independence assumption, but few studies have assessed whether such accommodations improve matching accuracy. In this paper, we show that incorporating the conditional dependence appropriately yields comparable or improved matching accuracy than the FS model using three real-world data linkage examples. Through a simulation study, we further investigate when conditional dependence models provide improved matching accuracy. Our study shows that the FS model is generally robust to the conditional independence assumption and provides comparable matching accuracy as the more complex conditional dependence models. However, when the match prevalence approaches 0% or 100% and conditional dependence exists in the dominating class, it is necessary to address conditional dependence as the FS model produces suboptimal matching accuracy. The need to address conditional dependence becomes less important when highly discriminating fields are used. Our simulation study also shows that conditional dependence models with misspecified dependence structure could produce less accurate record matching than the FS model and therefore we caution against the blind use of conditional dependence models.

Yiyi Liu, Joshua L. Warren, Hongyu Zhao.

Source: The Annals of Applied Statistics, Volume 13, Number 3, 1733--1752.

Abstract:
Understanding the heterogeneity of cells is an important biological question. The development of single-cell RNA-sequencing (scRNA-seq) technology provides high resolution data for such inquiry. A key challenge in scRNA-seq analysis is the high variability of measured RNA expression levels and frequent dropouts (missing values) due to limited input RNA compared to bulk RNA-seq measurement. Existing clustering methods do not perform well for these noisy and zero-inflated scRNA-seq data. In this manuscript we propose a Bayesian hierarchical model, called BasClu, to appropriately characterize important features of scRNA-seq data in order to more accurately cluster cells. We demonstrate the effectiveness of our method with extensive simulation studies and applications to three real scRNA-seq datasets.

Qiwei Li, Xinlei Wang, Faming Liang, Guanghua Xiao.

Source: The Annals of Applied Statistics, Volume 13, Number 3, 1708--1732.

Abstract:
With the advance of imaging technology, digital pathology imaging of tumor tissue slides is becoming a routine clinical procedure for cancer diagnosis. This process produces massive imaging data that capture histological details in high resolution. Recent developments in deep-learning methods have enabled us to identify and classify individual cells from digital pathology images at large scale. Reliable statistical approaches to model the spatial pattern of cells can provide new insight into tumor progression and shed light on the biological mechanisms of cancer. We consider the problem of modeling spatial correlations among three commonly seen cells observed in tumor pathology images. A novel geostatistical marking model with interpretable underlying parameters is proposed in a Bayesian framework. We use auxiliary variable MCMC algorithms to sample from the posterior distribution with an intractable normalizing constant. We demonstrate how this model-based analysis can lead to sharper inferences than ordinary exploratory analyses, by means of application to three benchmark datasets and a case study on the pathology images of $188$ lung cancer patients. The case study shows that the spatial correlation between tumor and stromal cells predicts patient prognosis. This statistical methodology not only presents a new model for characterizing spatial correlations in a multitype spatial point pattern conditioning on the locations of the points, but also provides a new perspective for understanding the role of cell–cell interactions in cancer progression.

Jesús D. Arroyo Relión, Daniel Kessler, Elizaveta Levina, Stephan F. Taylor.

Source: The Annals of Applied Statistics, Volume 13, Number 3, 1648--1677.

Abstract:
While statistical analysis of a single network has received a lot of attention in recent years, with a focus on social networks, analysis of a sample of networks presents its own challenges which require a different set of analytic tools. Here we study the problem of classification of networks with labeled nodes, motivated by applications in neuroimaging. Brain networks are constructed from imaging data to represent functional connectivity between regions of the brain, and previous work has shown the potential of such networks to distinguish between various brain disorders, giving rise to a network classification problem. Existing approaches tend to either treat all edge weights as a long vector, ignoring the network structure, or focus on graph topology as represented by summary measures while ignoring the edge weights. Our goal is to design a classification method that uses both the individual edge information and the network structure of the data in a computationally efficient way, and that can produce a parsimonious and interpretable representation of differences in brain connectivity patterns between classes. We propose a graph classification method that uses edge weights as predictors but incorporates the network nature of the data via penalties that promote sparsity in the number of nodes, in addition to the usual sparsity penalties that encourage selection of edges. We implement the method via efficient convex optimization and provide a detailed analysis of data from two fMRI studies of schizophrenia.

Lisa Schlosser, Torsten Hothorn, Reto Stauffer, Achim Zeileis.

Source: The Annals of Applied Statistics, Volume 13, Number 3, 1564--1589.

Abstract:
To obtain a probabilistic model for a dependent variable based on some set of explanatory variables, a distributional approach is often adopted where the parameters of the distribution are linked to regressors. In many classical models this only captures the location of the distribution but over the last decade there has been increasing interest in distributional regression approaches modeling all parameters including location, scale and shape. Notably, so-called nonhomogeneous Gaussian regression (NGR) models both mean and variance of a Gaussian response and is particularly popular in weather forecasting. Moreover, generalized additive models for location, scale and shape (GAMLSS) provide a framework where each distribution parameter is modeled separately capturing smooth linear or nonlinear effects. However, when variable selection is required and/or there are nonsmooth dependencies or interactions (especially unknown or of high-order), it is challenging to establish a good GAMLSS. A natural alternative in these situations would be the application of regression trees or random forests but, so far, no general distributional framework is available for these. Therefore, a framework for distributional regression trees and forests is proposed that blends regression trees and random forests with classical distributions from the GAMLSS framework as well as their censored or truncated counterparts. To illustrate these novel approaches in practice, they are employed to obtain probabilistic precipitation forecasts at numerous sites in a mountainous region (Tyrol, Austria) based on a large number of numerical weather prediction quantities. It is shown that the novel distributional regression forests automatically select variables and interactions, performing on par or often even better than GAMLSS specified either through prior meteorological knowledge or a computationally more demanding boosting approach.

Yanjun Qian, Jianhua Z. Huang, Chiwoo Park, Yu Ding.

Source: The Annals of Applied Statistics, Volume 13, Number 3, 1537--1563.

Abstract:
In situ transmission electron microscope (TEM) adds a promising instrument to the exploration of the nanoscale world, allowing motion pictures to be taken while nano objects are initiating, crystalizing and morphing into different sizes and shapes. To enable in-process control of nanocrystal production, this technology innovation hinges upon a solution addressing a statistical problem, which is the capability of online tracking a dynamic, time-varying probability distribution reflecting the nanocrystal growth. Because no known parametric density functions can adequately describe the evolving distribution, a nonparametric approach is inevitable. Towards this objective, we propose to incorporate the dynamic evolution of the normalized particle size distribution into a state space model, in which the density function is represented by a linear combination of B-splines and the spline coefficients are treated as states. The closed-form algorithm runs online updates faster than the frame rate of the in situ TEM video, making it suitable for in-process control purpose. Imposing the constraints of curve smoothness and temporal continuity improves the accuracy and robustness while tracking the probability distribution. We test our method on three published TEM videos. For all of them, the proposed method is able to outperform several alternative approaches.

Ahmed Aziz Ezzat, Mikyoung Jun, Yu Ding.

Source: The Annals of Applied Statistics, Volume 13, Number 3, 1484--1510.

Abstract:
Accurate short-term forecasts are indispensable for the integration of wind energy in power grids. On a wind farm, local wind conditions exhibit sizeable variations at a fine temporal resolution. Existing statistical models may capture the in-sample variations in wind behavior, but are often shortsighted to those occurring in the near future, that is, in the forecast horizon. The calibrated regime-switching method proposed in this paper introduces an action of regime dependent calibration on the predictand (here the wind speed variable), which helps correct the bias resulting from out-of-sample variations in wind behavior. This is achieved by modeling the calibration as a function of two elements: the wind regime at the time of the forecast (and the calibration is therefore regime dependent), and the runlength, which is the time elapsed since the last observed regime change. In addition to regime-switching dynamics, the proposed model also accounts for other features of wind fields: spatio-temporal dependencies, transport effect of wind and nonstationarity. Using one year of turbine-specific wind data, we show that the calibrated regime-switching method can offer a wide margin of improvement over existing forecasting methods in terms of both wind speed and power.

Johann Gagnon-Bartsch, Yotam Shem-Tov.

Source: The Annals of Applied Statistics, Volume 13, Number 3, 1464--1483.

Abstract:
The gold standard for identifying causal relationships is a randomized controlled experiment. In many applications in the social sciences and medicine, the researcher does not control the assignment mechanism and instead may rely upon natural experiments or matching methods as a substitute to experimental randomization. The standard testable implication of random assignment is covariate balance between the treated and control units. Covariate balance is commonly used to validate the claim of as good as random assignment. We propose a new nonparametric test of covariate balance. Our Classification Permutation Test (CPT) is based on a combination of classification methods (e.g., random forests) with Fisherian permutation inference. We revisit four real data examples and present Monte Carlo power simulations to demonstrate the applicability of the CPT relative to other nonparametric tests of equality of multivariate distributions.

Lekha Patel, Nils Gustafsson, Yu Lin, Raimund Ober, Ricardo Henriques, Edward Cohen.

Source: The Annals of Applied Statistics, Volume 13, Number 3, 1397--1429.

Abstract:
Fluorescing molecules (fluorophores) that stochastically switch between photon-emitting and dark states underpin some of the most celebrated advancements in super-resolution microscopy. While this stochastic behavior has been heavily exploited, full characterization of the underlying models can potentially drive forward further imaging methodologies. Under the assumption that fluorophores move between fluorescing and dark states as continuous time Markov processes, the goal is to use a sequence of images to select a model and estimate the transition rates. We use a hidden Markov model to relate the observed discrete time signal to the hidden continuous time process. With imaging involving several repeat exposures of the fluorophore, we show the observed signal depends on both the current and past states of the hidden process, producing emission probabilities that depend on the transition rate parameters to be estimated. To tackle this unusual coupling of the transition and emission probabilities, we conceive transmission (transition-emission) matrices that capture all dependencies of the model. We provide a scheme of computing these matrices and adapt the forward-backward algorithm to compute a likelihood which is readily optimized to provide rate estimates. When confronted with several model proposals, combining this procedure with the Bayesian Information Criterion provides accurate model selection.

Jorge A. León.

Source: Bernoulli, Volume 26, Number 3, 2436--2462.

Abstract:
Let $B^{H}$ be a fractional Brownian motion with Hurst parameter $Hin (0,1/2)$ and $p:mathbb{R} ightarrow mathbb{R}$ a polynomial function. The main purpose of this paper is to introduce a Stratonovich type stochastic integral with respect to $B^{H}$, whose domain includes the process $p(B^{H})$. That is, an integral that allows us to integrate $p(B^{H})$ with respect to $B^{H}$, which does not happen with the symmetric integral given by Russo and Vallois ( Probab. Theory Related Fields 97 (1993) 403–421) in general. Towards this end, we combine the approaches utilized by León and Nualart ( Stochastic Process. Appl. 115 (2005) 481–492), and Russo and Vallois ( Probab. Theory Related Fields 97 (1993) 403–421), whose aims are to extend the domain of the divergence operator for Gaussian processes and to define some stochastic integrals, respectively. Then, we study the relation between this Stratonovich integral and the extension of the divergence operator (see León and Nualart ( Stochastic Process. Appl. 115 (2005) 481–492)), an Itô formula and the existence of a unique solution of some Stratonovich stochastic differential equations. These last results have been analyzed by Alòs, León and Nualart ( Taiwanese J. Math. 5 (2001) 609–632), where the Hurst paramert $H$ belongs to the interval $(1/4,1/2)$.

Jong Yun Hwang, Ji Oon Lee, Wooseok Yang.

Source: Bernoulli, Volume 26, Number 3, 2400--2435.

Abstract:
We consider the spectral properties of sparse stochastic block models, where $N$ vertices are partitioned into $K$ balanced communities. Under an assumption that the intra-community probability and inter-community probability are of similar order, we prove a local semicircle law up to the spectral edges, with an explicit formula on the deterministic shift of the spectral edge. We also prove that the fluctuation of the extremal eigenvalues is given by the GOE Tracy–Widom law after rescaling and centering the entries of sparse stochastic block models. Applying the result to sparse stochastic block models, we rigorously prove that there is a large gap between the outliers and the spectral edge without centering.

Bo Ning, Seonghyun Jeong, Subhashis Ghosal.

Source: Bernoulli, Volume 26, Number 3, 2353--2382.

Abstract:
We study frequentist properties of a Bayesian high-dimensional multivariate linear regression model with correlated responses. The predictors are separated into many groups and the group structure is pre-determined. Two features of the model are unique: (i) group sparsity is imposed on the predictors; (ii) the covariance matrix is unknown and its dimensions can also be high. We choose a product of independent spike-and-slab priors on the regression coefficients and a new prior on the covariance matrix based on its eigendecomposition. Each spike-and-slab prior is a mixture of a point mass at zero and a multivariate density involving the $ell_{2,1}$-norm. We first obtain the posterior contraction rate, the bounds on the effective dimension of the model with high posterior probabilities. We then show that the multivariate regression coefficients can be recovered under certain compatibility conditions. Finally, we quantify the uncertainty for the regression coefficients with frequentist validity through a Bernstein–von Mises type theorem. The result leads to selection consistency for the Bayesian method. We derive the posterior contraction rate using the general theory by constructing a suitable test from the first principle using moment bounds for certain likelihood ratios. This leads to posterior concentration around the truth with respect to the average Rényi divergence of order $1/2$. This technique of obtaining the required tests for posterior contraction rate could be useful in many other problems.

Sreenivasa Rao Jammalamadaka, Simos Meintanis, Thomas Verdebout.

Source: Bernoulli, Volume 26, Number 3, 2226--2252.

Abstract:
Circular and spherical data arise in many applications, especially in biology, Earth sciences and astronomy. In dealing with such data, one of the preliminary steps before any further inference, is to test if such data is isotropic, that is, uniformly distributed around the circle or the sphere. In view of its importance, there is a considerable literature on the topic. In the present work, we provide new tests of uniformity on the circle based on original asymptotic results. Our tests are motivated by the shape of locally and asymptotically maximin tests of uniformity against generalized von Mises distributions. We show that they are uniformly consistent. Empirical power comparisons with several competing procedures are presented via simulations. The new tests detect particularly well multimodal alternatives such as mixtures of von Mises distributions. A practically-oriented combination of the new tests with already existing Sobolev tests is proposed. An extension to testing uniformity on the sphere, along with some simulations, is included. The procedures are illustrated on a real dataset.

Anton Thalmaier, James Thompson.

Source: Bernoulli, Volume 26, Number 3, 2202--2225.

Abstract:
In this article, we derive moment estimates, exponential integrability, concentration inequalities and exit times estimates for canonical diffusions firstly on sub-Riemannian limits of Riemannian foliations and secondly in the nonsmooth setting of $operatorname{RCD}^{*}(K,N)$ spaces. In each case, the necessary ingredients are Itô’s formula and a comparison theorem for the Laplacian, for which we refer to the recent literature. As an application, we derive pointwise Carmona-type estimates on eigenfunctions of Schrödinger operators.

Javier Cárcamo, Antonio Cuevas, Luis-Alberto Rodríguez.

Source: Bernoulli, Volume 26, Number 3, 2143--2175.

Abstract:
We show that various functionals related to the supremum of a real function defined on an arbitrary set or a measure space are Hadamard directionally differentiable. We specifically consider the supremum norm, the supremum, the infimum, and the amplitude of a function. The (usually non-linear) derivatives of these maps adopt simple expressions under suitable assumptions on the underlying space. As an application, we improve and extend to the multidimensional case the results in Raghavachari ( Ann. Statist. 1 (1973) 67–73) regarding the limiting distributions of Kolmogorov–Smirnov type statistics under the alternative hypothesis. Similar results are obtained for analogous statistics associated with copulas. We additionally solve an open problem about the Berk–Jones statistic proposed by Jager and Wellner (In A Festschrift for Herman Rubin (2004) 319–331 IMS). Finally, the asymptotic distribution of maximum mean discrepancies over Donsker classes of functions is derived.

Akio Fujiwara, Koichi Yamagata.

Source: Bernoulli, Volume 26, Number 3, 2105--2142.

Abstract:
We herein develop a theory of contiguity in the quantum domain based upon a novel quantum analogue of the Lebesgue decomposition. The theory thus formulated is pertinent to the weak quantum local asymptotic normality introduced in the previous paper [Yamagata, Fujiwara, and Gill, Ann. Statist. 41 (2013) 2197–2217], yielding substantial enlargement of the scope of quantum statistics.

Sarat B. Moka, Dirk P. Kroese.

Source: Bernoulli, Volume 26, Number 3, 2082--2104.

Abstract:
We present a perfect sampling algorithm for Gibbs point processes, based on the partial rejection sampling of Guo, Jerrum and Liu (In STOC’17 – Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing (2017) 342–355 ACM). Our particular focus is on pairwise interaction processes, penetrable spheres mixture models and area-interaction processes, with a finite interaction range. For an interaction range $2r$ of the target process, the proposed algorithm can generate a perfect sample with $O(log(1/r))$ expected running time complexity, provided that the intensity of the points is not too high and $Theta(1/r^{d})$ parallel processor units are available.

Tom Alberts, Firas Rassoul-Agha, Mackenzie Simper.

Source: Bernoulli, Volume 26, Number 3, 1927--1955.

Abstract:
We prove that given any fixed asymptotic velocity, the finite length O’Connell–Yor polymer has an infinite length limit satisfying the law of large numbers with this velocity. By a Markovian property of the quenched polymer this reduces to showing the existence of Busemann functions : almost sure limits of ratios of random point-to-point partition functions. The key ingredients are the Burke property of the O’Connell–Yor polymer and a comparison lemma for the ratios of partition functions. We also show the existence of infinite length limits in the Brownian last passage percolation model.

Adam Osękowski.

Source: Bernoulli, Volume 26, Number 3, 1912--1926.

Abstract:
Let $X=(X_{t})_{tgeq 0}in H^{1}$ and $Y=(Y_{t})_{tgeq 0}in{mathrm{BMO}} $ be arbitrary continuous-path martingales. The paper contains the proof of the inequality egin{equation*}mathbb{E}int _{0}^{infty }iglvert dlangle X,Y angle_{t}igrvert leq sqrt{2}Vert XVert _{H^{1}}Vert YVert _{mathrm{BMO}_{2}},end{equation*} and the constant $sqrt{2}$ is shown to be the best possible. The proof rests on the construction of a certain special function, enjoying appropriate size and concavity conditions.

Holger Sambale, Arthur Sinulis.

Source: Bernoulli, Volume 26, Number 3, 1863--1890.

Abstract:
We derive sufficient conditions for a probability measure on a finite product space (a spin system ) to satisfy a (modified) logarithmic Sobolev inequality. We establish these conditions for various examples, such as the (vertex-weighted) exponential random graph model, the random coloring and the hard-core model with fugacity. This leads to two separate branches of applications. The first branch is given by mixing time estimates of the Glauber dynamics. The proofs do not rely on coupling arguments, but instead use functional inequalities. As a byproduct, this also yields exponential decay of the relative entropy along the Glauber semigroup. Secondly, we investigate the concentration of measure phenomenon (particularly of higher order) for these spin systems. We show the effect of better concentration properties by centering not around the mean, but around a stochastic term in the exponential random graph model. From there, one can deduce a central limit theorem for the number of triangles from the CLT of the edge count. In the Erdős–Rényi model the first-order approximation leads to a quantification and a proof of a central limit theorem for subgraph counts.

Xin Bing, Florentina Bunea, Marten Wegkamp.

Source: Bernoulli, Volume 26, Number 3, 1765--1796.

Abstract:
Topic models have become popular for the analysis of data that consists in a collection of n independent multinomial observations, with parameters $N_{i}inmathbb{N}$ and $Pi_{i}in[0,1]^{p}$ for $i=1,ldots,n$. The model links all cell probabilities, collected in a $p imes n$ matrix $Pi$, via the assumption that $Pi$ can be factorized as the product of two nonnegative matrices $Ain[0,1]^{p imes K}$ and $Win[0,1]^{K imes n}$. Topic models have been originally developed in text mining, when one browses through $n$ documents, based on a dictionary of $p$ words, and covering $K$ topics. In this terminology, the matrix $A$ is called the word-topic matrix, and is the main target of estimation. It can be viewed as a matrix of conditional probabilities, and it is uniquely defined, under appropriate separability assumptions, discussed in detail in this work. Notably, the unique $A$ is required to satisfy what is commonly known as the anchor word assumption, under which $A$ has an unknown number of rows respectively proportional to the canonical basis vectors in $mathbb{R}^{K}$. The indices of such rows are referred to as anchor words. Recent computationally feasible algorithms, with theoretical guarantees, utilize constructively this assumption by linking the estimation of the set of anchor words with that of estimating the $K$ vertices of a simplex. This crucial step in the estimation of $A$ requires $K$ to be known, and cannot be easily extended to the more realistic set-up when $K$ is unknown. This work takes a different view on anchor word estimation, and on the estimation of $A$. We propose a new method of estimation in topic models, that is not a variation on the existing simplex finding algorithms, and that estimates $K$ from the observed data. We derive new finite sample minimax lower bounds for the estimation of $A$, as well as new upper bounds for our proposed estimator. We describe the scenarios where our estimator is minimax adaptive. Our finite sample analysis is valid for any $n,N_{i},p$ and $K$, and both $p$ and $K$ are allowed to increase with $n$, a situation not handled well by previous analyses. We complement our theoretical results with a detailed simulation study. We illustrate that the new algorithm is faster and more accurate than the current ones, although we start out with a computational and theoretical disadvantage of not knowing the correct number of topics $K$, while we provide the competing methods with the correct value in our simulations.

Cristina Butucea, Amandine Dubois, Martin Kroll, Adrien Saumard.

Source: Bernoulli, Volume 26, Number 3, 1727--1764.

Abstract:
We address the problem of non-parametric density estimation under the additional constraint that only privatised data are allowed to be published and available for inference. For this purpose, we adopt a recent generalisation of classical minimax theory to the framework of local $alpha$-differential privacy and provide a lower bound on the rate of convergence over Besov spaces $mathcal{B}^{s}_{pq}$ under mean integrated $mathbb{L}^{r}$-risk. This lower bound is deteriorated compared to the standard setup without privacy, and reveals a twofold elbow effect. In order to fulfill the privacy requirement, we suggest adding suitably scaled Laplace noise to empirical wavelet coefficients. Upper bounds within (at most) a logarithmic factor are derived under the assumption that $alpha$ stays bounded as $n$ increases: A linear but non-adaptive wavelet estimator is shown to attain the lower bound whenever $pgeq r$ but provides a slower rate of convergence otherwise. An adaptive non-linear wavelet estimator with appropriately chosen smoothing parameters and thresholding is shown to attain the lower bound within a logarithmic factor for all cases.

Pavel Chigansky, Marina Kleptsyna, Dmytro Marushkevych.

Source: Bernoulli, Volume 26, Number 3, 1706--1726.

Abstract:
Spectral decomposition of the covariance operator is one of the main building blocks in the theory and applications of Gaussian processes. Unfortunately, it is notoriously hard to derive in a closed form. In this paper, we consider the eigenproblem for Gaussian bridges. Given a base process, its bridge is obtained by conditioning the trajectories to start and terminate at the given points. What can be said about the spectrum of a bridge, given the spectrum of its base process? We show how this question can be answered asymptotically for a family of processes, including the fractional Brownian motion.

Jason M. Klusowski, Yihong Wu.

Source: Bernoulli, Volume 26, Number 3, 1635--1664.

Abstract:
Learning properties of large graphs from samples has been an important problem in statistical network analysis since the early work of Goodman ( Ann. Math. Stat. 20 (1949) 572–579) and Frank ( Scand. J. Stat. 5 (1978) 177–188). We revisit a problem formulated by Frank ( Scand. J. Stat. 5 (1978) 177–188) of estimating the number of connected components in a large graph based on the subgraph sampling model, in which we randomly sample a subset of the vertices and observe the induced subgraph. The key question is whether accurate estimation is achievable in the sublinear regime where only a vanishing fraction of the vertices are sampled. We show that it is impossible if the parent graph is allowed to contain high-degree vertices or long induced cycles. For the class of chordal graphs, where induced cycles of length four or above are forbidden, we characterize the optimal sample complexity within constant factors and construct linear-time estimators that provably achieve these bounds. This significantly expands the scope of previous results which have focused on unbiased estimators and special classes of graphs such as forests or cliques. Both the construction and the analysis of the proposed methodology rely on combinatorial properties of chordal graphs and identities of induced subgraph counts. They, in turn, also play a key role in proving minimax lower bounds based on construction of random instances of graphs with matching structures of small subgraphs.

Ivan Nourdin, Giovanni Peccati, Stéphane Seuret.

Source: Bernoulli, Volume 26, Number 3, 1619--1634.

Abstract:
We describe the size of the sets of sojourn times $E_{gamma }={tgeq 0:|B_{t}|leq t^{gamma }}$ associated with a fractional Brownian motion $B$ in terms of various large scale dimensions.

Amir Najafi, Seyed Abolfazl Motahari, Hamid R. Rabiee.

Source: Bernoulli, Volume 26, Number 2, 1535--1559.

Abstract:
A Bernoulli Mixture Model (BMM) is a finite mixture of random binary vectors with independent dimensions. The problem of clustering BMM data arises in a variety of real-world applications, ranging from population genetics to activity analysis in social networks. In this paper, we analyze the clusterability of BMMs from a theoretical perspective, when the number of clusters is unknown. In particular, we stipulate a set of conditions on the sample complexity and dimension of the model in order to guarantee the Probably Approximately Correct (PAC)-clusterability of a dataset. To the best of our knowledge, these findings are the first non-asymptotic bounds on the sample complexity of learning or clustering BMMs.

Kamran Kalbasi, Thomas Mountford.

Source: Bernoulli, Volume 26, Number 2, 1504--1534.

Abstract:
In this paper, we study the local times of vector-valued Gaussian fields that are ‘diagonally operator-self-similar’ and whose increments are stationary. Denoting the local time of such a Gaussian field around the spatial origin and over the temporal unit hypercube by $Z$, we show that there exists $lambdain(0,1)$ such that under some quite weak conditions, $lim_{n ightarrow+infty}frac{sqrt[n]{mathbb{E}(Z^{n})}}{n^{lambda}}$ and $lim_{x ightarrow+infty}frac{-logmathbb{P}(Z>x)}{x^{frac{1}{lambda}}}$ both exist and are strictly positive (possibly $+infty$). Moreover, we show that if the underlying Gaussian field is ‘strongly locally nondeterministic’, the above limits will be finite as well. These results are then applied to establish similar statements for the intersection local times of diagonally operator-self-similar Gaussian fields with stationary increments.

Wensheng Wang, Zhonggen Su, Yimin Xiao.

Source: Bernoulli, Volume 26, Number 2, 1410--1430.

Abstract:
We establish the exact moduli of non-differentiability of Gaussian random fields with stationary increments. As an application of the result, we prove that the uniform Hölder condition for the maximum local times of Gaussian random fields with stationary increments obtained in Xiao (1997) is optimal. These results are applicable to fractional Riesz–Bessel processes and stationary Gaussian random fields in the Matérn and Cauchy classes.