Recent financial disasters emphasised the need to investigate the consequence associated with the tail co-movements among institutions; episodes of contagion are frequently observed and increase the probability of large losses affecting market participants' risk capital. Commonly used risk management tools fail to account for potential spillover effects among institutions because they provide individual risk assessment. We contribute to analyse the interdependence effects of extreme events providing an estimation tool for evaluating the Conditional Value-at-Risk (CoVaR) defined as the Value-at-Risk of an institution conditioned on another institution being under distress. In particular, our approach relies on Bayesian quantile regression framework. We propose a Markov chain Monte Carlo algorithm exploiting the Asymmetric Laplace distribution and its representation as a location-scale mixture of Normals. Moreover, since risk measures are usually evaluated on time series data and returns typically change over time, we extend the CoVaR model to account for the dynamics of the tail behaviour. Application on U.S. companies belonging to different sectors of the Standard and Poor's Composite Index (S&P500) is considered to evaluate the marginal contribution to the overall systemic risk of each individual institution
Bayesian approaches for handling covariate measurement error are well established, and yet arguably are still relatively little used by researchers. For some this is likely due to unfamiliarity or disagreement with the Bayesian inferential paradigm. For others a contributory factor is the inability of standard statistical packages to perform such Bayesian analyses. In this paper we first give an overview of the Bayesian approach to handling covariate measurement error, and contrast it with regression calibration (RC), arguably the most commonly adopted approach. We then argue why the Bayesian …
A Bayesian approach is used to estimate the covariance matrix of Gaussian data. Ideas from Gaussian graphical models and model selection are used to construct a prior for the covariance matrix that is a mixture over all decomposable graphs. For this prior the probability of each graph size is specified by the user and graphs of equal size are assigned equal probability. Most previous approaches assume that all graphs are equally probable. We show empirically that the prior that assigns equal probability over graph sizes outperforms the prior that assigns equal probability over all graphs, both…
Covariance estimation and selection for multivariate datasets in a high-dimensional regime is a fundamental problem in modern statistics. Gaussian graphical models are a popular class of models used for this purpose. Current Bayesian methods for inverse covariance matrix estimation under Gaussian graphical models require the underlying graph and hence the ordering of variables to be known. However, in practice, such information on the true underlying model is often unavailable. We therefore propose a novel permutation-based Bayesian approach to tackle the unknown variable ordering issue. In pa…
We consider the problem of estimating high-dimensional covariance matrices of a particular structure, which is a summation of low rank and sparse matrices. This covariance structure has a wide range of applications including factor analysis and random effects models. We propose a Bayesian method of estimating the covariance matrices by representing the covariance model in the form of a factor model with unknown number of latent factors. We introduce binary indicators for factor selection and rank estimation for the low rank component combined with a Bayesian lasso method for the sparse compone…
Gaussian process (GP) covariance function is proposed as a matching tool in GPMatch within a full Bayesian framework under relatively weaker causal assumptions. The matching is accomplished by utilizing GP prior covariance function to define matching distance. We show that GPMatch provides a doubly robust estimate of the averaged treatment effect (ATE) much like the G-estimation, the ATE is correctly estimated when either conditions are satisfied: 1) the GP mean function correctly specifies potential outcome \(Y^{(0)}\); or 2) the GP covariance function correctly specifies matching structure. …
Missing values in covariates due to censoring by signal interference or lack of sensitivity in the measuring devices are common in industrial problems. We propose a full Bayesian solution to the prediction problem with an efficient Markov Chain Monte Carlo (MCMC) algorithm that updates all the censored covariate values jointly in a random scan Gibbs sampler. We show that the joint updating of missing covariate values can be at least two orders of magnitude more efficient than univariate updating. This increased efficiency is shown to be crucial for quickly learning the missing covariate values…
Gaussian graphical models are used for determining conditional relationships between variables. This is accomplished by identifying off-diagonal elements in the inverse-covariance matrix that are non-zero. When the ratio of variables (p) to observations (n) approaches one, the maximum likelihood estimator of the covariance matrix becomes unstable and requires shrinkage estimation. Whereas several classical (frequentist) methods have been introduced to address this issue, fully Bayesian methods remain relatively uncommon in practice and methodological literatures. Here we introduce a Bayesian m…
This paper is considered with joint estimation of state and time-varying noise covariance matrices in non-linear stochastic state space models. We present a variational Bayes and Gaussian filtering based algorithm for efficient computation of the approximate filtering posterior distributions. The Gaussian filtering based formulation of the non-linear state space model computation allows usage of efficient Gaussian integration methods such as unscented transform, cubature integration and Gauss-Hermite integration along with the classical Taylor series approximations. The performance of the algo…
In this work we present full Bayesian inference for a new flexible nonseparable class of cross-covariance functions for multivariate spatial data. A Bayesian test is proposed for separability of covariance functions which is much more interpretable than parameters related to separability. Spatial models have been increasingly applied in several areas, such as environmental science, climate science and agriculture. These data are usually available in space, time and possibly for several processes. In this context the modeling of dependence is crucial for correct uncertainty quantification and r…
The University of Electro-Communications（电气通信大学）；National Police Agency Info-Communications Bureau（警察厅信息通信局）
In this paper, we propose a Bayesian Graphical LASSO for correlated countable data and apply it to spatial crime data. In the proposed model, we assume a Gaussian Graphical Model for the latent variables which dominate the potential risks of crimes. To evaluate the proposed model, we determine optimal hyperparameters which represent samples better. We apply the proposed model for estimation of the sparse inverse covariance of the latent variable and evaluate the partial correlation coefficients. Finally, we illustrate the results on crime spots data and consider the estimated latent variables …
Data in the form of ranking lists are frequently encountered, and combining ranking results from different sources can potentially generate a better ranking list and help understand behaviors of the rankers. Of interest here are the rank data under the following settings: (i) covariate information available for the ranked entities; (ii) rankers of varying qualities or having different opinions; and (iii) incomplete ranking lists for non-overlapping subgroups. We review some key ideas built around the Thurstone model family by researchers in the past few decades and provide a unifying approach …
Historical data about disease outcomes can be integrated into the analysis of clinical trials in many ways. We build on existing literature that uses prognostic scores from a predictive model to increase the efficiency of treatment effect estimates via covariate adjustment. Here we go further, utilizing a Bayesian framework that combines prognostic covariate adjustment with an empirical prior distribution learned from the predictive performances of the prognostic model on past trials. The Bayesian approach interpolates between prognostic covariate adjustment with strict type I error control wh…
In regional economics research, a problem of interest is to detect similarities between regions, and estimate their shared coefficients in economics models. In this article, we propose a mixture of finite mixtures (MFM) clustered regression model with auxiliary covariates that account for similarities in demographic or economic characteristics over a spatial domain. Our Bayesian construction provides both inference for number of clusters and clustering configurations, and estimation for parameters for each cluster. Empirical performance of the proposed model is illustrated through simulation e…
This article introduces a flexible and adaptive nonparametric method for estimating the association between multiple covariates and power spectra of multiple time series. The proposed approach uses a Bayesian sum of trees model to capture complex dependencies and interactions between covariates and the power spectrum, which are often observed in studies of biomedical time series. Local power spectra corresponding to terminal nodes within trees are estimated nonparametrically using Bayesian penalized linear splines. The trees are considered to be random and fit using a Bayesian backfitting Mark…
Covariance matrix estimation arises in multivariate problems including multivariate normal sampling models and regression models where random effects are jointly modeled, e.g. random-intercept, random-slope models. A Bayesian analysis of these problems requires a prior on the covariance matrix. Here we assess, through a simulation study and a real data set, the impact this prior choice has on posterior inference of the covariance matrix. Inverse Wishart distribution is the natural choice for a covariance matrix prior because its conjugacy on normal model and simplicity, is usually available in…
We obtain the optimal Bayesian minimax rate for the unconstrained large covariance matrix of multivariate normal sample with mean zero, when both the sample size, n, and the dimension, p, of the covariance matrix tend to infinity. Traditionally the posterior convergence rate is used to compare the frequentist asymptotic performance of priors, but defining the optimality with it is elusive. We propose a new decision theoretic framework for prior selection and define Bayesian minimax rate. Under the proposed framework, we obtain the optimal Bayesian minimax rate for the spectral norm for all rat…
In many applications, survey data are collected from different survey centers in different regions. It happens that in some circumstances, response variables are completely observed while the covariates have missing values. In this paper, we propose a joint spatial regression model for the response variable and missing covariates via a sequence of one-dimensional conditional spatial regression models. We further construct a joint spatial model for missing covariate data mechanisms. The properties of the proposed models are examined and a Markov chain Monte Carlo sampling algorithm is used to s…
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…
Approximate Bayesian inference for neural networks is considered a robust alternative to standard training, often providing good performance on out-of-distribution data. However, Bayesian neural networks (BNNs) with high-fidelity approximate inference via full-batch Hamiltonian Monte Carlo achieve poor generalization under covariate shift, even underperforming classical estimation. We explain this surprising result, showing how a Bayesian model average can in fact be problematic under covariate shift, particularly in cases where linear dependencies in the input features cause a lack of posteri…
Modeling correlation (and covariance) matrices can be challenging due to the positive-definiteness constraint and potential high-dimensionality. Our approach is to decompose the covariance matrix into the correlation and variance matrices and propose a novel Bayesian framework based on modeling the correlations as products of unit vectors. By specifying a wide range of distributions on a sphere (e.g. the squared-Dirichlet distribution), the proposed approach induces flexible prior distributions for covariance matrices (that go beyond the commonly used inverse-Wishart prior). For modeling real-…
The paper addresses joint sparsity selection in the regression coefficient matrix and the error precision (inverse covariance) matrix for high-dimensional multivariate regression models in the Bayesian paradigm. The selected sparsity patterns are crucial to help understand the network of relationships between the predictor and response variables, as well as the conditional relationships among the latter. While Bayesian methods have the advantage of providing natural uncertainty quantification through posterior inclusion probabilities and credible intervals, current Bayesian approaches either r…
The paper presents a novel approach to direct covariance function learning for Bayesian optimisation, with particular emphasis on experimental design problems where an existing corpus of condensed knowledge is present. The method presented borrows techniques from reproducing kernel Banach space theory (specifically m-kernels) and leverages them to convert (or re-weight) existing covariance functions into new, problem-specific covariance functions. The key advantage of this approach is that rather than relying on the user to manually select (with some hyperparameter tuning and experimentation) …
Although there is a rich literature on methods for allowing the variance in a univariate regression model to vary with predictors, time and other factors, relatively little has been done in the multivariate case. Our focus is on developing a class of nonparametric covariance regression models, which allow an unknown p x p covariance matrix to change flexibly with predictors. The proposed modeling framework induces a prior on a collection of covariance matrices indexed by predictors through priors for predictor-dependent loadings matrices in a factor model. In particular, the predictor-dependen…
Functional data, with basic observational units being functions (e.g., curves, surfaces) varying over a continuum, are frequently encountered in various applications. While many statistical tools have been developed for functional data analysis, the issue of smoothing all functional observations simultaneously is less studied. Existing methods often focus on smoothing each individual function separately, at the risk of removing important systematic patterns common across functions. We propose a nonparametric Bayesian approach to smooth all functional observations simultaneously and nonparametr…
Accurate diagnosis of disease is of fundamental importance in clinical practice and medical research. Before a medical diagnostic test is routinely used in practice, its ability to distinguish between diseased and nondiseased states must be rigorously assessed through statistical analysis. The receiver operating characteristic (ROC) curve is the most popular used tool for evaluating the discriminatory ability of continuous-outcome diagnostic tests. It has been acknowledged that several factors (e.g., subject-specific characteristics, such as age and/or gender) can affect the test's accuracy be…
In an empirical Bayesian setting, we provide a new multiple testing method, useful when an additional covariate is available, that influences the probability of each null hypothesis being true. We measure the posterior significance of each test conditionally on the covariate and the data, leading to greater power. Using covariate-based prior information in an unsupervised fashion, we produce a list of significant hypotheses which differs in length and order from the list obtained by methods not taking covariate-information into account. Covariate-modulated posterior probabilities of each null …
The functional generalized additive model (FGAM) was recently proposed in McLean et al. (2013) as a more flexible alternative to the common functional linear model (FLM) for regressing a scalar on functional covariates. In this paper, we develop a Bayesian version of FGAM for the case of Gaussian errors with identity link function. Our approach allows the functional covariates to be sparsely observed and measured with error, whereas the estimation procedure of McLean et al. (2013) required that they be noiselessly observed on a regular grid. We consider both Monte Carlo and variational Bayes m…
We consider Bayesian shrinkage predictions for the Normal regression problem under the frequentist Kullback-Leibler risk function. Firstly, we consider the multivariate Normal model with an unknown mean and a known covariance. While the unknown mean is fixed, the covariance of future samples can be different from training samples. We show that the Bayesian predictive distribution based on the uniform prior is dominated by that based on a class of priors if the prior distributions for the covariance and future covariance matrices are rotation invariant. Then, we consider a class of priors for t…
We consider the joint sparse estimation of regression coefficients and the covariance matrix for covariates in a high-dimensional regression model, where the predictors are both relevant to a response variable of interest and functionally related to one another via a Gaussian directed acyclic graph (DAG) model. Gaussian DAG models introduce sparsity in the Cholesky factor of the inverse covariance matrix, and the sparsity pattern in turn corresponds to specific conditional independence assumptions on the underlying predictors. A variety of methods have been developed in recent years for Bayesi…
This paper focuses on Bayesian shrinkage for covariance matrix estimation. We examine posterior properties and frequentist risks of Bayesian estimators based on new hierarchical inverse-Wishart priors. More precisely, we give the existence conditions of the posterior distributions. Advantages in terms of numerical simulations of posteriors are shown. A simulation study illustrates the performance of the estimation procedures under three loss functions for relevant sample sizes and various covariance structures.
We consider Bayesian inference in inverse regression problems where the objective is to infer about unobserved covariates from observed responses and covariates. We establish posterior consistency of such unobserved covariates in Bayesian inverse regression problemsunder appropriate priors in a leave-one-out cross-validation setup. We relate this to posterior consistency of inverse reference distributions (Bhattacharya (2013)) for assessing model adequacy. We illustrate our theory and methods with various examples of Bayesian inverse regression, along with adequate simulation experiments.
Bayesian Varying-Effects Vector Autoregressive Models for Inference of Brain Connectivity Networks and Covariate Effects in Pediatric Traumatic Brain Injury
We address the problem of providing inference from a Bayesian perspective for parameters selected after viewing the data. We present a Bayesian framework for providing inference for selected parameters, based on the observation that providing Bayesian inference for selected parameters is a truncated data problem. We show that if the prior for the parameter is non-informative, or if the parameter is a "fixed" unknown constant, then it is necessary to adjust the Bayesian inference for selection. Our second contribution is the introduction of Bayesian False Discovery Rate controlling methodology,…
