Simulation of the Matrix Bingham–von Mises–Fisher Distribution,With Applications to Multivariate and Relational Data

Orthonormal matrices play an important role in reduced-rank matrix approximations and the analysis of matrix-valued data. A matrix Bingham–von Mises–Fisher distribution is a probability distribution on the set of orthonormal matrices that includes linear and quadratic terms in the log-density, and arises as a posterior distribution in latent factor models for multivariate and relational data. This article describes rejection and Gibbs sampling algorithms for sampling from this family of distributions, and illustrates their use in the analysis of a protein–protein interaction network. Supplemental materials, including code and data to generate all of the numerical results in this article, are available online.     

Noisy Hamiltonian Monte Carlo for Doubly Intractable Distributions

Hamiltonian Monte Carlo (HMC) has been progressively incorporated within the statistician’s toolbox as an alternative sampling method in settings when standard Metropolis–Hastings is inefficient. HMC generates a Markov chain on an augmented state space with transitions based on a deterministic differential flow derived from Hamiltonian mechanics. In practice, the evolution of Hamiltonian systems cannot be solved analytically, requiring numerical integration schemes. Under numerical integration, the resulting approximate solution no longer preserves the measure of the target distribution, therefore an accept–reject step is used to correct the bias. For doubly intractable distributions—such as posterior distributions based on Gibbs random fields—HMC suffers from some computational difficulties: computation of gradients in the differential flow and computation of the accept–reject proposals poses difficulty. In this article, we study the behavior of HMC when these quantities are replaced by Monte Carlo estimates. Supplemental codes for implementing methods used in the article are available online.     

不同先验分布下的后验分布确定土力学参数

岩土工程中各土层参数的取值是根据现场及室内试验数据,采用经典统计学方法进行确定的,但这往往忽略了先验信息的作用.与经典统计学方法不同的是,Bayes法能从考虑先验分布的角度结合样本分布去推导后验分布,为岩土参数的取值提供一种新的分析方法.岩土工程勘察可视为对总体地层的随机抽样,当抽样完成时,样本分布密度函数是确定的,故Bayes法中的后验分布取决于先验分布,因此推导出两套不同的先验分布:利用先验信息确定先验分布及共轭先验分布.通过对先验及后验分布中超参数的计算,当样本总体符合N(μ,σ²)正态分布时,对所要研究的未知参数μ和σ展开分析,综合对比不同先验分布下后验分布的区间长度,给出岩土参数Bayes推断中最佳后验分布所要选择的先验分布.结果表明:共轭情况下的后验分布总是比无信息情况下的后验区间短,概率密度函数分布更集中,取值更方便.在正态总体情形下,根据未知参数μ和σ的联合后验分布求极值方法,确定样本总体中最大概率均值μ_max和方差σ_max作为工程设计采用值,为岩土参数取值方法提供了一条新的路径,有较好的工程意义.     

Bayes theorem,information number and behavior of posteior distributions

Summary According to the sequential decision theory, one is directed to; (i) stop when the posterior distribution, having been successively watched, frist enters the domain in which by stopping he loses less, and then (ii) take that action which gives him the least loss averaged w.r.t. the current posterior distribution (Blackwell-Girshick [1], p. 254). For some reasons we don't know, however, the successive behavior of posterior distribution never draws the full attention of statisticians and probabilists, though some of them have developed the asymptotic theory. As far as the sequential decision theory is concerned, however, the asymptotic behavior tells us nothing, since the stopping rule is only related to finite time. Toe compute the posterior distribution time by time is a laborious task. But to site all possible posterior distributions sometimes facilitates our manipulation of the sequential decision problem. This paper proposes some methods. The Institute of Statistical Mathematics     

Case Influence Analysis in Bayesian Inference

Abstract

We demonstrate how case influence analysis, commonly used in regression, can be applied to Bayesian hierarchical models. Draws from the joint posterior distribution of parameters are importance weighted to reflect the effect of deleting each observation in turn; the ensuing changes in the posterior distribution of each parameter are displayed graphically. The procedure is particularly useful when drawing a sample from the posterior distribution requires extensive calculations (as with a Markov Chain Monte Carlo sampler). The structure of hierarchical models, and other models with local dependence, makes the importance weights inexpensive to calculate with little additional programming. Some new alternative weighting schemes are described that extend the range of problems in which reweighting can be used to assess influence. Applications to a growth curve model and a complex hierarchical model for opinion data are described. Our focus on case influence on parameters is complementary to other work that measures influence by distances between posterior or predictive distributions.     

Bayesian inference for Rayleigh distribution under progressive censored sample

It is often the case that some information is available on the parameter of failure time distributions from previous experiments or analyses of failure time data. The Bayesian approach provides the methodology for incorporation of previous information with the current data. In this paper, given a progressively type II censored sample from a Rayleigh distribution, Bayesian estimators and credible intervals are obtained for the parameter and reliability function. We also derive the Bayes predictive estimator and highest posterior density prediction interval for future observations. Two numerical examples are presented for illustration and some simulation study and comparisons are performed. Copyright © 2006 John Wiley & Sons, Ltd.     

The log-linear model with a generalized gamma distribution for the error: A Bayesian approach

Considering a log-linear model with one covariate and a generalized gamma distribution for the error, we find the posterior densities for the parameters of interest. Since many standard survival distributions are particular cases of the generalized gamma model, the proposed bayesian method is very useful to discriminate between possible models to be used in the data analysis. The Laplace approximation for integrals (see Tierney and Kadane, 1984) is used to find the posterior distributions of the parameters involved when they cannot be obtained explicitly.     

Bayesian inference for dependent elliptical measurement error models

In this article we provide a Bayesian analysis for dependent elliptical measurement error models considering nondifferential and differential errors. In both cases we compute posterior distributions for structural parameters by using squared radial prior distributions for the precision parameters. The main result is that the posterior distribution of location parameters, for specific priors, is invariant with respect to changes in the generator function, in agreement with previous results obtained in the literature under different assumptions. Finally, although the results obtained are valid for any elliptical distribution for the error term, we illustrate those results by using the student-t distribution and a real data set.     

Single machine scheduling with two competing agents,arbitrary release dates and unit processing times

Rapid transit systems timetables are commonly designed to accommodate passenger demand in sections with the highest passenger load. However, disruptions frequently arise due to an increase in the demand, infrastructure incidences or as a consequence of fleet size reductions. All these circumstances give rise to unsupplied demand at certain stations, which generates passenger overloads in the available vehicles. The design of strategies that guarantee reasonable user waiting time with small increases of operation costs is now an important research topic. This paper proposes a tactical approach to determine optimal policies for dealing with such situations. Concretely, a short-turning strategy is analysed, where some vehicles perform short cycles in order to increase the frequency among certain stations of the lines and to equilibrate the train occupancy level. Turn-back points should be located and service offset should be determined with the objective of diminishing the passenger waiting time while preserving certain level of quality of service. Computational results and analysis for a real case study are provided.     

Bayesian Prediction via Partitioning

This article proposes a new Bayesian approach to prediction on continuous covariates. The Bayesian partition model constructs arbitrarily complex regression and classification surfaces by splitting the covariate space into an unknown number of disjoint regions. Within each region the data are assumed to be exchangeable and come from some simple distribution. Using conjugate priors, the marginal likelihoods of the models can be obtained analytically for any proposed partitioning of the space where the number and location of the regions is assumed unknown a priori. Markov chain Monte Carlo simulation techniques are used to obtain predictive distributions at the design points by averaging across posterior samples of partitions.     

Infinite Dirichlet mixture models learning via expectation propagation

In this article, we propose a novel Bayesian nonparametric clustering algorithm based on a Dirichlet process mixture of Dirichlet distributions which have been shown to be very flexible for modeling proportional data. The idea is to let the number of mixture components increases as new data to cluster arrive in such a manner that the model selection problem (i.e. determination of the number of clusters) can be answered without recourse to classic selection criteria. Thus, the proposed model can be considered as an infinite Dirichlet mixture model. An expectation propagation inference framework is developed to learn this model by obtaining a full posterior distribution on its parameters. Within this learning framework, the model complexity and all the involved parameters are evaluated simultaneously. To show the practical relevance and efficiency of our model, we perform a detailed analysis using extensive simulations based on both synthetic and real data. In particular, real data are generated from three challenging applications namely images categorization, anomaly intrusion detection and videos summarization.     

Quantifying expert opinion for modelling fauna habitat distributions

Summary  The practical elicitation of expert beliefs about logistic regression models is considered. An experiment is reported in which ecologists quantified their prior beliefs about the relationship between various environmental attributes and the habitat distribution of certain rare and endangered fauna. Prior distributions were elicited from the ecologists and combined with sample data to form posterior distributions. The elicitation method was proposed by Garthwaite and Al-Awadhi (2004) and is implemented through an interactive graphical computer program. Classical stepwise logistic regression and alternative forms of prior distribution are compared using cross validation. Data on the environmental attributes have been mapped and stored in a GIS database and the posterior distributions can be used to predict the probability of a species' presence/absence at any site in the database.     

Modeling the dynamics of interest rate volatility with skewed fat-tailed distributions

This paper proposes generalized parametric models of the short-term interest rate that nest one-factor CEV and discrete time GARCH models. The paper estimates the generalized and nested models with skewed fat-tailed distributions to determine the correct specification of the conditional distribution of interest rates. The results indicate that the discrete time models that incorporate the level and GARCH effects into the diffusion function and that accommodate the tail-thickness of the interest rate distribution perform much better than the CEV model in forecasting the future volatility of interest rates. The results also show that the significance of nonlinearity in the drift function relies crucially on the specification of the volatility function.     

Bartlett′s Decomposition of the Posterior Distribution of the Covariance for Normal Monotone Ignorable Missing Data

This paper presents a decomposition for the posterior distribution of the covarianee matrix of normal models under a family of prior distributions when missing data are ignorable and monotone. This decomposition is an extension of Bartlett′s decomposition of the Wishart distribution to monotone missing data. It is not only theoretically interesting but also practically useful. First, with monotone missing data, it allows more efficient drawing of parameters from the posterior distribution than the factorized likelihood approach. Furthermore, with nonmonotone missing data, it allows for a very efficient monotone date augmentation algorithm and thereby multiple imputation or the missing data needed to create a monotone pattern.     

Modelling conflicting information using subexponential distributions and related classes

In the Bayesian modelling the data and the prior information concerning a certain parameter of interest may conflict, in the sense that the information carried by them disagree. The most common form of conflict is the presence of outlying information in the data, which may potentially lead to wrong posterior conclusions. To prevent this problem we use robust models which aim to control the influence of the atypical information in the posterior distribution. Roughly speaking, we conveniently use heavy-tailed distributions in the model in order to resolve conflicts in favour of those sources of information which we believe is more credible. The class of heavy-tailed distributions is quite wide and the literature have been concerned in establishing conditions on the data and prior distributions in order to reject the outlying information. In this work we focus on the subexponential and $\mathfrak L $ classes of heavy-tailed distributions, in which we establish sufficient conditions under which the posterior distribution automatically rejects the conflicting information.     

Conditional laplace transforms for bayesian nonparametric inference in reliabity theory

In order to apply nonparametric meyhods to reliability problems, it is desibrable to have available priors over a broad class of survival distributions.In the paper, this is achieved by taking the failure rate function to be the sum oof a nonnegative stochastic process with increasing sample patjhs and a process with decreasing sample paths. This approach produces a prior which chooses an absolutely survival distribution that can have an IFR, DFR, or U-shapped failure rate. Posterior Laplace transforms of the failure rate are obtained based on survival data allows censoring. Bayes estimates of the failure rate as well as the lifetime distribution are then calculated from these posterior Laplace transforms. This approach is also applied to a competing risks model and the proportional hazards model of Cox.     

Bayesian inference in spherical linear models: robustness and conjugate analysis

The early work of Zellner on the multivariate Student-t linear model has been extended to Bayesian inference for linear models with dependent non-normal error terms, particularly through various papers by Osiewalski, Steel and coworkers. This article provides a full Bayesian analysis for a spherical linear model. The density generator of the spherical distribution is here allowed to depend both on the precision parameter φ and on the regression coefficients β. Another distinctive aspect of this paper is that proper priors for the precision parameter are discussed.The normal-chi-squared family of prior distributions is extended to a new class, which allows the posterior analysis to be carried out analytically. On the other hand, a direct joint modelling of the data vector and of the parameters leads to conjugate distributions for the regression and the precision parameters, both individually and jointly. It is shown that some model specifications lead to Bayes estimators that do not depend on the choice of the density generator, in agreement with previous results obtained in the literature under different assumptions. Finally, the distribution theory developed to tackle the main problem is useful on its own right.     

Pyramid Quantile Regression

We describe a Bayesian model for simultaneous linear quantile regression at several specified quantile levels. More specifically, we propose to model the conditional distributions by using random probability measures, known as quantile pyramids, introduced by Hjort and Walker. Unlike many existing approaches, this framework allows us to specify meaningful priors on the conditional distributions, while retaining the flexibility afforded by the nonparametric error distribution formulation. Simulation studies demonstrate the flexibility of the proposed approach in estimating diverse scenarios, generally outperforming other competitive methods. We also provide conditions for posterior consistency. The method is particularly promising for modeling the extremal quantiles. Applications to extreme value analysis and in higher dimensions are also explored through data examples. Supplemental material for this article is available online.     

Bayesian robustness under a skew-normal class of prior distribution

We develop a global sensitivity analysis to measure the robustness of the Bayesian estimators with respect to a class of prior distributions. This class arises when we consider multiplicative contamination of a base prior distribution. A similar structure was presented by van der Linde [12]. Some particular specifications for this multiplicative contamination class coincide with well known families of skewed distributions. In this paper, we explore the skew-normal multiplicative contamination class for the prior distribution of the location parameter of a normal model. Results of a Bayesian conjugation and expressions for some measures of distance between posterior means and posterior variance are obtained. We also elaborate on the behavior of the posterior means and of the posterior variances through a simulation study.     

An informational distance for estimating the faithfulness of a possibility distribution,viewed as a family of probability distributions,with respect to data

An acknowledged interpretation of possibility distributions in quantitative possibility theory is in terms of families of probabilities that are upper and lower bounded by the associated possibility and necessity measures. This paper proposes an informational distance function for possibility distributions that agrees with the above-mentioned view of possibility theory in the continuous and in the discrete cases. Especially, we show that, given a set of data following a probability distribution, the optimal possibility distribution with respect to our informational distance is the distribution obtained as the result of the probability-possibility transformation that agrees with the maximal specificity principle. It is also shown that when the optimal distribution is not available due to representation bias, maximizing this possibilistic informational distance provides more faithful results than approximating the probability distribution and then applying the probability-possibility transformation. We show that maximizing the possibilistic informational distance is equivalent to minimizing the squared distance to the unknown optimal possibility distribution. Two advantages of the proposed informational distance function is that (i) it does not require the knowledge of the shape of the probability distribution that underlies the data, and (ii) it amounts to sum up the elementary terms corresponding to the informational distance between the considered possibility distribution and each piece of data. We detail the particular case of triangular and trapezoidal possibility distributions and we show that any unimodal unknown probability distribution can be faithfully upper approximated by a triangular distribution obtained by optimizing the possibilistic informational distance.