The structure of a Bayesian network encodes most of the information about the probability distribution of the data, which is uniquely identified given some general distributional assumptions. Therefore it's important to study the variability of its network structure, which can be used to compare the performance of different learning algorithms and to measure the strength of any arbitrary subset of arcs.
In this paper we will introduce some descriptive statistics and the corresponding parametric and Monte Carlo tests on the undirected graph underlying the structure of a Bayesian network, modeled as a multivariate Bernoulli random variable.
The structure of a Bayesian network includes a great deal of information about the probability distribution of the data, which is uniquely identified given some general distributional assumptions. Therefore it's important to study its variability, which can be used to compare the performance of different learning algorithms and to measure the strength of any arbitrary subset of arcs. In this paper we will introduce some descriptive statistics and the corresponding parametric and Monte Carlo tests on the undirected graph underlying the structure of a Bayesian network, modeled as a multivariate …
Whilst most psychological research focuses on differences in means, a growing body of literature demonstrates the value of considering differences in intra-individual variability. Compared to the number of methods available for analyzing mean differences, there is a paucity of methods available for analyzing intra-individual variability, particularly when variability is treated as a predictor. In the present article, we first reviewed methods of analyzing intra-individual variability as an outcome, including the individual standard deviation (ISD) and some recent methods. We then introduced a …
Hyperspectral unmixing is a blind source separation problem which consists in estimating the reference spectral signatures contained in a hyperspectral image, as well as their relative contribution to each pixel according to a given mixture model. In practice, the process is further complexified by the inherent spectral variability of the observed scene and the possible presence of outliers. More specifically, multi-temporal hyperspectral images, i.e., sequences of hyperspectral images acquired over the same area at different time instants, are likely to simultaneously exhibit moderate endmemb…
The reliability of detecting source variability in sparsely and irregularly sampled X-ray light curves is investigated. This is motivated by the unprecedented survey capabilities of eROSITA onboard SRG, providing light curves for many thousand sources in its final-depth equatorial deep field survey. Four methods for detecting variability are evaluated: excess variance, amplitude maximum deviations, Bayesian blocks and a new Bayesian formulation of the excess variance. We judge the false detection rate of variability based on simulated Poisson light curves of constant sources, and calibrate sig…
Bayesian Hierarchical Modelling of Noisy Gamma Processes: Model Formulation, Identifiability, Model Fitting, and Extensions to Unit-to-Unit Variability
Stochastic reduced models are an important tool in climate systems whose many spatial and temporal scales cannot be fully discretized or underlying physics may not be fully accounted for. One form of reduced model, the linear inverse model (LIM), has been widely used for regional climate predictability studies - typically focusing more on tropical or mid-latitude studies. However, most LIM fitting techniques rely on point estimation techniques deriving from fluctuation-dissipation theory. In this methodological study we explore the use of Bayesian inference techniques for LIM parameter estimat…
Photometric variability of a directly imaged exo-Earth conveys spatial information on its surface and can be used to retrieve a two-dimensional geography and axial tilt of the planet (spin-orbit tomography). In this study, we relax the assumption of the static geography and present a computationally tractable framework for dynamic spin-orbit tomography applicable to the time-varying geography. First, a Bayesian framework of static spin-orbit tomography is revisited using analytic expressions of the Bayesian inverse problem with a Gaussian prior. We then extend this analytic framework to a time…
The flexibility of the Bayesian approach to account for covariates with measurement error is combined with semiparametric regression models for a class of continuous, discrete and mixed univariate response distributions with potentially all parameters depending on a structured additive predictor. Markov chain Monte Carlo enables a modular and numerically efficient implementation of Bayesian measurement error correction based on the imputation of unobserved error-free covariate values. We allow for very general measurement errors, including correlated replicates with heterogeneous variances. Th…
Regression-based optimal fingerprinting techniques for climate change detection and attribution require the estimation of the forced signal as well as the internal variability covariance matrix in order to distinguish between their influences in the observational record. While previously developed approaches have taken into account the uncertainty linked to the estimation of the forced signal, there has been less focus on uncertainty in the covariance matrix describing natural variability, despite the fact that the specification of this covariance matrix is known to meaningfully impact the res…
Projections of future climate change rely heavily on climate models, and combining climate models through a multi-model ensemble is both more accurate than a single climate model and valuable for uncertainty quantification. However, Bayesian approaches to multi-model ensembles have been criticized for making oversimplified assumptions about bias and variability, as well as treating different models as statistically independent. This paper extends the Bayesian hierarchical approach of Sansom et al. (2017) by explicitly accounting for spatial variability and inter-model dependence. We propose a …
This paper presents three hyperspectral mixture models jointly with Bayesian algorithms for supervised hyperspectral unmixing. Based on the residual component analysis model, the proposed general formulation assumes the linear model to be corrupted by an additive term whose expression can be adapted to account for nonlinearities (NL), endmember variability (EV), or mismodelling effects (ME). The NL effect is introduced by considering a polynomial expression that is related to bilinear models. The proposed new formulation of EV accounts for shape and scale endmember changes while enforcing a sm…
This paper proposes the estimation of a smooth graphon model for network data analysis using principles of the EM algorithm. The approach considers both variability with respect to ordering the nodes of a network and smooth estimation of the graphon by nonparametric regression. To do so, (linear) B-splines are used, which allow for smooth estimation of the graphon, conditional on the node ordering. This provides the M-step. The true ordering of the nodes arising from the graphon model remains unobserved and Bayesian ideas are employed to obtain posterior samples given the network data. This yi…
We present Bayesian Spillover Graphs (BSG), a novel method for learning temporal relationships, identifying critical nodes, and quantifying uncertainty for multi-horizon spillover effects in a dynamic system. BSG leverages both an interpretable framework via forecast error variance decompositions (FEVD) and comprehensive uncertainty quantification via Bayesian time series models to contextualize temporal relationships in terms of systemic risk and prediction variability. Forecast horizon hyperparameter $h$ allows for learning both short-term and equilibrium state network behaviors. Experiments…
Samsung AI Center（三星人工智能中心）；Rutgers University（罗格斯大学）
We propose a family of novel hierarchical Bayesian deep auto-encoder models capable of identifying disentangled factors of variability in data. While many recent attempts at factor disentanglement have focused on sophisticated learning objectives within the VAE framework, their choice of a standard normal as the latent factor prior is both suboptimal and detrimental to performance. Our key observation is that the disentangled latent variables responsible for major sources of variability, the relevant factors, can be more appropriately modeled using long-tail distributions. The typical Gaussian…
The problem of the definition and the estimation of generative models based on deformable templates from raw data is of particular importance for modelling non aligned data affected by various types of geometrical variability. This is especially true in shape modelling in the computer vision community or in probabilistic atlas building for Computational Anatomy (CA). A first coherent statistical framework modelling the geometrical variability as hidden variables has been given by Allassonnière, Amit and Trouvé (JRSS 2006). Setting the problem in a Bayesian context they proved the consistency o…
Multivariate nonnegative orthant data are real vectors bounded to the left by the null vector, and they can be continuous, discrete or mixed. We first review the recent relative variability indexes for multivariate nonnegative continuous and count distributions. As a prelude, the classification of two comparable distributions having the same mean vector is done through under-, equi- and over-variability with respect to the reference distribution. Multivariate associated kernel estimators are then reviewed with new proposals that can accommodate any nonnegative orthant dataset. We focus on band…
Parameters in climate models are usually calibrated manually, exploiting only small subsets of the available data. This precludes both optimal calibration and quantification of uncertainties. Traditional Bayesian calibration methods that allow uncertainty quantification are too expensive for climate models; they are also not robust in the presence of internal climate variability. For example, Markov chain Monte Carlo (MCMC) methods typically require $O(10^5)$ model runs and are sensitive to internal variability noise, rendering them infeasible for climate models. Here we demonstrate an approac…
This paper develops a Hierarchical Bayesian Modeling (HBM) framework for uncertainty quantification of Finite Element (FE) models based on modal information. This framework uses an existing Fast Fourier Transform (FFT) approach to identify experimental modal parameters from time-history data and employs a class of maximum-entropy probability distributions to account for the mismatch between the modal parameters. It also considers a parameterized probability distribution for capturing the variability of structural parameters across multiple data sets. In this framework, the computation is addre…
Conditional heteroscedastic (CH) models are routinely used to analyze financial datasets. The classical models such as ARCH-GARCH with time-invariant coefficients are often inadequate to describe frequent changes over time due to market variability. However we can achieve significantly better insight by considering the time-varying analogues of these models. In this paper, we propose a Bayesian approach to the estimation of such models and develop computationally efficient MCMC algorithm based on Hamiltonian Monte Carlo (HMC) sampling. We also established posterior contraction rates with incre…
In many applications, smooth processes generate data that is recorded under a variety of observation regimes, such as dense, sparse or fragmented observations that are often contaminated with error. The statistical goal of registering and estimating the individual underlying functions from discrete observations has thus far been mainly approached sequentially without formal uncertainty propagation, or in an application-specific manner. We propose a unified Bayesian framework for simultaneous registration and estimation, which is flexible enough to accommodate inference on individual functions …
Two-part joint models for a longitudinal semicontinuous biomarker and a terminal event have been recently introduced based on frequentist estimation. The biomarker distribution is decomposed into a probability of positive value and the expected value among positive values. Shared random effects can represent the association structure between the biomarker and the terminal event. The computational burden increases compared to standard joint models with a single regression model for the biomarker. In this context, the frequentist estimation implemented in the R package frailtypack can be challen…
In this short technical report we model, within the Bayesian framework, the rate of positive tests reported by the the State of Indiana, accounting also for the substantial variability (and overdispeartion) in the daily count of the tests performed. The approach we take, results with a simple procedure for prediction, a posteriori, of this rate of 'positivity' and allows for an easy and a straightforward adaptation by any agency tracking daily results of COVID-19 tests. The numerical results provided herein were obtained via an updatable R Markdown document.
Bayesian model comparison relies upon the model evidence, yet for many models of interest the model evidence is unavailable in closed form and must be approximated. Many of the estimators for evidence that have been proposed in the Monte Carlo literature suffer from high variability. This paper considers the reduction of variance that can be achieved by exploiting control variates in this setting. Our methodology is based on thermodynamic integration and applies whenever the gradient of both the log-likelihood and the log-prior with respect to the parameters can be efficiently evaluated. Resul…
Bayesian inference is a popular and widely-used approach to infer phylogenies (evolutionary trees). However, despite decades of widespread application, it remains difficult to judge how well a given Bayesian Markov chain Monte Carlo (MCMC) run explores the space of phylogenetic trees. In this paper, we investigate the Monte Carlo error of phylogenies, focusing on high-dimensional summaries of the posterior distribution, including variability in estimated edge/branch (known in phylogenetics as "split") probabilities and tree probabilities, and variability in the estimated summary tree. Specific…
Soil Texture Prediction with Bayesian Generalized Additive Models for Spatial Compositional Data
Universitat Politècnica de València（瓦伦西亚理工大学）；Biogipuzkoa Health Research Institute（比奥吉普斯夸健康研究所）；Deusto Physical TherapIker, Faculty of Health Sciences, University of Deusto（德乌斯托大学健康科学学院德乌斯托物理治疗Iker）；School of Science and Technology, IE University（IE大学科学与技术学院）
Bayesian model comparison is often based on the posterior distribution over the set of compared models. This distribution is often observed to concentrate on a single model even when other measures of model fit or forecasting ability indicate no strong preference. Furthermore, a moderate change in the data sample can easily shift the posterior model probabilities to concentrate on another model. We document overconfidence in two high-profile applications in economics and neuroscience. To shed more light on the sources of overconfidence we derive the sampling variance of the Bayes factor in uni…
This paper presents a hierarchical Bayesian modeling framework for the uncertainty quantification in modal identification of linear dynamical systems using multiple vibration data sets. This novel framework integrates the state-of-the-art Bayesian formulations into a hierarchical setting aiming to capture both the identification precision and the ensemble variability prompted due to modeling errors. Such cutting-edge developments have been absent from the modal identification literature, sustained as a long-standing problem at the research spotlight. Central to this framework is a Gaussian hyp…
Max Planck Institute for Intelligent Systems（马克斯·普朗克智能系统研究所）；Microsoft Research Cambridge（微软研究院剑桥实验室）
Computer vision is hard because of a large variability in lighting, shape, and texture; in addition the image signal is non-additive due to occlusion. Generative models promised to account for this variability by accurately modelling the image formation process as a function of latent variables with prior beliefs. Bayesian posterior inference could then, in principle, explain the observation. While intuitively appealing, generative models for computer vision have largely failed to deliver on that promise due to the difficulty of posterior inference. As a result the community has favoured effic…
In the presence of modeling errors, the mainstream Bayesian methods seldom give a realistic account of uncertainties as they commonly underestimate the inherent variability of parameters. This problem is not due to any misconception in the Bayesian framework since it is absolutely robust with respect to the modeling assumptions and the observed data. Rather, this issue has deep roots in users' inability to develop an appropriate class of probabilistic models. This paper bridges this significant gap, introducing a novel Bayesian hierarchical setting, which breaks time-history vibrational respon…
Correlated component analysis as proposed by Dmochowski et al. (2012) is a tool for investigating brain process similarity in the responses to multiple views of a given stimulus. Correlated components are identified under the assumption that the involved spatial networks are identical. Here we propose a hierarchical probabilistic model that can infer the level of universality in such multi-view data, from completely unrelated representations, corresponding to canonical correlation analysis, to identical representations as in correlated component analysis. This new model, which we denote Bayesi…
The INLAMSM package for the R programming language provides a collection of multivariate spatial models for lattice data that can be used with package INLA for Bayesian inference. The multivariate spatial models include different structures to model the spatial variation of the variables and the between-variables variability. In this way, fitting multivariate spatial models becomes faster and easier. The use of the different models included in the package is illustrated using two different datasets: the well-known North Carolina SIDS data and mortality by three causes of death in Comunidad Val…
Functional connectivity fingerprints are among today's best choices to obtain a faithful sampling of an individual's brain and cognition in health and disease. Here we make a case for key advantages of analyzing such connectome profiles using Bayesian analysis strategies. They (i) afford full probability estimates of the studied neurocognitive phenomenon (ii) provide analytical machinery to separate methodological uncertainty and biological variability in a coherent manner (iii) usher towards avenues to go beyond classical null-hypothesis significance testing and (iv) enable estimates of credi…
Being in the era of big data, modeling and prediction of count data have become significantly important in many fields including health, finance, social, etc. Although linear Poisson regression has been widely used to model count and rate data, it might not be always suitable as it cannot capture some inherent variability within complex data. In this study, we introduce a probabilistically driven nonlinear Poisson regression model with Bayesian artificial neural networks (ANN) to model count or rate data. This new nonlinear Poisson regression model developed with Bayesian ANN provides higher p…
Central to many inferential situations is the estimation of rational functions of parameters. The mainstream in statistics and econometrics estimates these quantities based on the plug-in approach without consideration of the main objective of the inferential situation. We propose the Bayesian Minimum Expected Loss (MELO) approach focusing explicitly on the function of interest, and calculating its frequentist variability. Asymptotic properties of the MELO estimator are similar to the plug-in approach. Nevertheless, simulation exercises show that our proposal is better in situations characteri…
RNA-Seq data characteristically exhibits large variances, which need to be appropriately accounted for in the model. We first explore the effects of this variability on the maximum likelihood estimator (MLE) of the overdispersion parameter of the negative binomial distribution, and propose instead the use an estimator obtained via maximization of the marginal likelihood in a conjugate Bayesian framework. We show, via simulation studies, that the marginal MLE can better control this variation and produce a more stable and reliable estimator. We then formulate a conjugate Bayesian hierarchical m…
Bogazici University（博阿齐奇大学）；Brno University of Technology（布尔诺理工大学）；IT4I Center of Excellence（IT4I卓越中心）
In this paper we describe our submission to the Zerospeech 2020 challenge, where the participants are required to discover latent representations from unannotated speech, and to use those representations to perform speech synthesis, with synthesis quality used as a proxy metric for the unit quality. In our system, we use the Bayesian Subspace Hidden Markov Model (SHMM) for unit discovery. The SHMM models each unit as an HMM whose parameters are constrained to lie in a low dimensional subspace of the total parameter space which is trained to model phonetic variability. Our system compares favor…
Uncertainty quantification in mechanics: A unified Bayesian perspective
Courant Institute of Mathematical Sciences, New York University（纽约大学柯朗数学科学研究所）；Division of Applied Mathematics, Brown University（布朗大学应用数学系）；Institute of Applied Mechanics, Friedrich-Alexander-Universität Erlangen-Nürnberg（埃尔朗根-纽伦堡大学应用力学研究所）；Institute of Biomechanics, Graz University of Technology（格拉茨工业大学生物力学研究所）；Department of Structural Engineering, Norwegian University of Science and Technology (NTNU)（挪威科技大学结构工程系）；Department of Mechanical Engineering, Stanford University（斯坦福大学机械工程系）
Assessing variability according to distinct factors in data is a fundamental technique of statistics. The method commonly regarded to as analysis of variance (ANOVA) is, however, typically confined to the case where all levels of a factor are present in the data (i.e. the population of factor levels has been exhausted). Random and mixed effects models are used for more elaborate cases, but require distinct nomenclature, concepts and theory, as well as distinct inferential procedures. Following a hierarchical Bayesian approach, a comprehensive ANOVA framework is shown, which unifies the above s…
