A product of two Gaussians (or normal distributions) is another Gaussian. That's a valuable and useful fact! Here we use it to derive a refactoring of a common product of multivariate Gaussians: The product of a Gaussian likelihood times a Gaussian prior, where some or all of those parameters enter the likelihood only in the mean and only linearly. That is, a linear, Gaussian, Bayesian model. This product of a likelihood times a prior pdf can be refactored into a product of a marginalized likelihood (or a Bayesian evidence) times a posterior pdf, where (in this case) both of these are also Gaussian. The means and variance tensors of the refactored Gaussians are straightforward to obtain as closed-form expressions; here we deliver these expressions, with discussion. The closed-form expressions can be used to speed up and improve the precision of inferences that contain linear parameters with Gaussian priors. We connect these methods to inferences that arise frequently in physics and astronomy.
If all you want is the answer, the question is posed and answered at the beginning of Section 3. We show two toy examples, in the form of worked exercises, in Section 4. The solutions, discussion, and exercises in this Note are aimed at someone who is already familiar with the basic ideas of Bayesian inference and probability.
Bayesian methods and software for spatial data analysis are generally now well established in the scientific community. Despite the wide application of spatial models, the analysis of multivariate spatial data using R-INLA has not been widely described in the existing literature. Therefore, the main objective of this article is to demonstrate that R-INLA is a convenient toolbox to analyse different types of multivariate spatial datasets. Additionally, this will be illustrated by analysing three datasets which are publicly available. Furthermore, the details and the R code of these analyses are…
High resolution geospatial data are challenging because standard geostatistical models based on Gaussian processes are known to not scale to large data sizes. While progress has been made towards methods that can be computed more efficiently, considerably less attention has been devoted to big data methods that allow the description of complex relationships between several outcomes recorded at high resolutions by different sensors. Our Bayesian multivariate regression models based on spatial multivariate trees (SpamTrees) achieve scalability via conditional independence assumptions on latent r…
Harvey et al. (2010) extended the Bayesian estimation method by Sahu et al. (2003) to a multivariate skew-elliptical distribution with a general skewness matrix, and applied it to Bayesian portfolio optimization with higher moments. Although their method is epochal in the sense that it can handle the skewness dependency among asset returns and incorporate higher moments into portfolio optimization, it cannot identify all elements in the skewness matrix due to label switching in the Gibbs sampler. To deal with this identification issue, we propose to modify their sampling algorithm by imposing …
The acquisition function, a critical component in Bayesian optimization (BO), can often be written as the expectation of a utility function under a surrogate model. However, to ensure that acquisition functions are tractable to optimize, restrictions must be placed on the surrogate model and utility function. To extend BO to a broader class of models and utilities, we propose likelihood-free BO (LFBO), an approach based on likelihood-free inference. LFBO directly models the acquisition function without having to separately perform inference with a probabilistic surrogate model. We show that co…
In the problem of selecting variables in a multivariate linear regression model, we derive new Bayesian information criteria based on a prior mixing a smooth distribution and a delta distribution. Each of them can be interpreted as a fusion of the Akaike information criterion (AIC) and the Bayesian information criterion (BIC). Inheriting their asymptotic properties, our information criteria are consistent in variable selection in both the large-sample and the high-dimensional asymptotic frameworks. In numerical simulations, variable selection methods based on our information criteria choose th…
Multivariate functional data arise in a wide range of applications. One fundamental task is to understand the causal relationships among these functional objects of interest, which has not yet been fully explored. In this article, we develop a novel Bayesian network model for multivariate functional data where the conditional independence and causal structure are both encoded by a directed acyclic graph. Specifically, we allow the functional objects to deviate from Gaussian process, which is adopted by most existing functional data analysis models. The more reasonable non-Gaussian assumption i…
Predicting extreme events is important in many applications in risk analysis. The extreme-value theory suggests modelling extremes by max-stable distributions. The Bayesian approach provides a natural framework for statistical prediction. Although various Bayesian inferential procedures have been proposed in the literature of univariate extremes and some for multivariate extremes, the study of their asymptotic properties has been left largely untouched. In this paper we focus on a semiparatric Bayesian method for estimating max-stable distributions in arbitrary dimension. We establish consiste…
Efficient Bayesian Inference for Multivariate Factor Stochastic Volatility Models
通过因子随机波动模型讨论了多元时间序列中动态协方差阵的有效贝叶斯估计。特别地，我们提出了两种交织策略（Yu和Meng，Journal of Computational and Graphical Statistics，20（3），531-570，2011）以显著加速标准MCMC方法的收敛和混合。类似于边际数据增强技术，建议的加速程序利用非可识别性问题，经常出现在因素模型。我们的新交织策略很容易实现，几乎没有额外的计算成本，然而，它们可以提高估计效率的几个数量级，如在广泛的模拟研究中所示。最后，我们的算法应用到26维汇率数据集说明了优越的性能，新的方法对现实世界的数据。
With rapid development of techniques to measure brain activity and structure, statistical methods for analyzing modern brain-imaging play an important role in the advancement of science. Imaging data that measure brain function are usually multivariate time series and are heterogeneous across both imaging sources and subjects, which lead to various statistical and computational challenges. In this paper, we propose a group-based method to cluster a collection of multivariate time series via a Bayesian mixture of smoothing splines. Our method assumes each multivariate time series is a mixture o…
A Bayesian changepoint methodology for high dimensional multivariate time series and space-time data: A study of structural change using remotely sensed data
The Dirichlet-multinomial (DM) distribution plays a fundamental role in modern statistical methodology development and application. Recently, the DM distribution and its variants have been used extensively to model multivariate count data generated by high-throughput sequencing technology in omics research due to its ability to accommodate the compositional structure of the data as well as overdispersion. A major limitation of the DM distribution is that it is unable to handle excess zeros typically found in practice which may bias inference. To fill this gap, we propose a novel Bayesian zero-…
Online Bayesian system identification in multivariate autoregressive models via message passing
Eindhoven Artificial Intelligence Systems Institute（埃因霍温人工智能系统研究所）；Department of Electrical Engineering, Eindhoven University of Technology（电子工程系，埃因霍温理工大学）
Joint modeling of spatially-oriented dependent variables is commonplace in the environmental sciences, where scientists seek to estimate the relationships among a set of environmental outcomes accounting for dependence among these outcomes and the spatial dependence for each outcome. Such modeling is now sought for massive data sets with variables measured at a very large number of locations. Bayesian inference, while attractive for accommodating uncertainties through hierarchical structures, can become computationally onerous for modeling massive spatial data sets because of its reliance on i…
A Bayesian multivariate extreme value mixture model
School of Mathematics and Statistics, University of Glasgow（格拉斯哥大学数学与统计学院）；School of Mathematics and Statistics, University of St Andrews（圣安德鲁斯大学数学与统计学院）
We consider the problem of multivariate density deconvolution where the distribution of a random vector needs to be estimated from replicates contaminated with conditionally heteroscedastic measurement errors. We propose a conceptually straightforward yet fundamentally novel and highly robust approach to multivariate density deconvolution by stochastically rotating the replicates toward the corresponding true latent values. We also address the additionally significantly challenging problem of accommodating conditionally heteroscedastic measurement errors in this newly introduced framework. We …
Objective Bayesian inference procedures are derived for the parameters of the multivariate random effects model generalized to elliptically contoured distributions. The posterior for the overall mean vector and the between-study covariance matrix is deduced by assigning two noninformative priors to the model parameter, namely the Berger and Bernardo reference prior and the Jeffreys prior, whose analytical expressions are obtained under weak distributional assumptions. It is shown that the only condition needed for the posterior to be proper is that the sample size is larger than the dimension …
