Bayesian analysis of robust Poisson geometric process model using heavy-tailed distributions

We propose a robust Poisson geometric process model with heavy-tailed distributions to cope with the problem of outliers as it may lead to an overestimation of mean and variance resulting in inaccurate interpretations of the situations. Two heavy-tailed distributions namely Student’s t and exponential power distributions with different tailednesses and kurtoses are used and they are represented in scale mixture of normal and scale mixture of uniform respectively. The proposed model is capable of describing the trend and meanwhile the mixing parameters in the scale mixture representations can detect the outlying observations. Simulations and real data analysis are performed to investigate the properties of the models.     

Bayesian nearest-neighbor analysis via record value statistics and nonhomogeneous spatial Poisson processes

This article proposes record value statistics and nonhomogeneous spatial Poisson processes as new nearest-neighbor approaches for detecting spatial patterns. A Markov chain Monte Carlo method with data augmentation was developed to compute the Bayes estimates of posterior quantities of interest. Simulation studies showed that the new approaches yield high detection rates and low false positive rates. We applied the new approaches to detect localized clusters of specific trees and to outline seismic faults in the study space.     

A Bayesian analysis of moving average processes with time-varying parameters

A new Bayesian method is proposed for estimation and forecasting with Gaussian moving average (MA) processes with time-varying parameters. The focus is placed on MA models of order one, but a general result is given for an MA process of an arbitrary known order. A multiplicative model for the evolution of the squares of the parameters is introduced following Bayesian conjugacy through beta and truncated gamma distributions and a discount factor. Two new distributions are proposed providing the prior and posterior distributions of the parameters of the model and the one-step forecast distribution of the process. Several well-known distributional results are extended by replacing the gamma distribution with the truncated gamma distribution. The proposed methodology is illustrated with two examples consisting of simulated data and of aluminium spot prices of the London metal exchange.     

Bayesian estimation of stimulus responses in Poisson spike trains

A Bayesian method is developed for estimating neural responses to stimuli, using likelihood functions incorporating the assumption that spike trains follow either pure Poisson statistics or Poisson statistics with a refractory period. The Bayesian and standard estimates of the mean and variance of responses are similar and asymptotically converge as the size of the data sample increases. However, the Bayesian estimate of the variance of the variance is much lower. This allows the Bayesian method to provide more precise interval estimates of responses. Sensitivity of the Bayesian method to the Poisson assumption was tested by conducting simulations perturbing the Poisson spike trains with noise. This did not affect Bayesian estimates of mean and variance to a significant degree, indicating that the Bayesian method is robust. The Bayesian estimates were less affected by the presence of noise than estimates provided by the standard method.     

Point estimates for variance-structure parameters in Bayesian analysis of hierarchical models

Markov chain Monte Carlo (MCMC) made Bayesian analysis feasible for hierarchical models, but the literature about their variance parameters is sparse. This is particularly so for point estimators of variance-structure parameters, which are useful for simplifying tables and sample-size calculations, and as “plug-in” estimators in complex calculations. This paper uses simulation experiments to compare three such point estimators, the posterior mode, median, and mean, for three parameterizations of the variance structure, as precisions, standard deviations, and variances. We first consider simple linear regression, where fairly explicit expressions are possible, and then three more complex models: crossed random effects, smoothed analysis of variance (SANOVA), and the conditional autoregressive (CAR) model with two classes of neighbor relations. We illustrate the latter results using periodontal data. The posterior mean often performs poorly in terms of bias and mean-squared error, and should be avoided. The posterior median never performs worse than the mean and often performs far better. The surprise is that, on the whole, the posterior mode performs best regardless of the variance structure's parameterization, although the potential for multi-modality may make it unattractive for general use.     

A SAS-macro for estimation of the cumulative incidence using Poisson regression

In survival analyses, we often estimate the hazard rate of a specific cause. Sometimes the main focus is not the hazard rates but the cumulative incidences, i.e., the probability of having failed from a specific cause prior to a given time. The cumulative incidences may be calculated using the hazard rates, and the hazard rates are often estimated by the Cox regression. This procedure may not be suitable for large studies due to limited computer resources. Instead one uses Poisson regression, which approximates the Cox regression. Rosth?j et al. presented a SAS-macro for the estimation of the cumulative incidences based on the Cox regression. I present the functional form of the probabilities and variances when using piecewise constant hazard rates and a SAS-macro for the estimation using Poisson regression. The use of the macro is demonstrated through examples and compared to the macro presented by Rosth?j et al.     

Empirical analysis of an on-line adaptive system using a mixture of Bayesian networks

An on-line reinforcement learning system that adapts to environmental changes using a mixture of Bayesian networks is described. Building intelligent systems able to adapt to dynamic environments is important for deploying real-world applications. Machine learning approaches, such as those using reinforcement learning methods and stochastic models, have been used to acquire behavior appropriate to environments characterized by uncertainty. However, efficient hybrid architectures based on these approaches have not yet been developed. The results of several experiments demonstrated that an agent using the proposed system can flexibly adapt to various kinds of environmental changes.     

Online Bayesian inference for the parameters of PRISM programs

This paper presents a method for approximating posterior distributions over the parameters of a given PRISM program. A sequential approach is taken where the distribution is updated one datapoint at a time. This makes it applicable to online learning situations where data arrives over time. The method is applicable whenever the prior is a mixture of products of Dirichlet distributions. In this case the true posterior will be a mixture of very many such products. An approximation is effected by merging products of Dirichlet distributions. An analysis of the quality of the approximation is presented. Due to the heavy computational burden of this approach, the method has been implemented in the Mercury logic programming language. Initial results using a hidden Markov model and a probabilistic graph are presented.     

Structure learning of Bayesian networks by genetic algorithms: aperformance analysis of control parameters

We present a new approach to structure learning in the field of Bayesian networks. We tackle the problem of the search for the best Bayesian network structure, given a database of cases, using the genetic algorithm philosophy for searching among alternative structures. We start by assuming an ordering between the nodes of the network structures. This assumption is necessary to guarantee that the networks that are created by the genetic algorithms are legal Bayesian network structures. Next, we release the ordering assumption by using a “repair operator” which converts illegal structures into legal ones. We present empirical results and analyze them statistically. The best results are obtained with an elitist genetic algorithm that contains a local optimizer     

Video Denoising via Empirical Bayesian Estimation of Space-Time Patches

In this paper we present a new patch-based empirical Bayesian video denoising algorithm. The method builds a Bayesian model for each group of similar space-time patches. These patches are not motion-compensated, and therefore avoid the risk of inaccuracies caused by motion estimation errors. The high dimensionality of spatiotemporal patches together with a limited number of available samples poses challenges when estimating the statistics needed for an empirical Bayesian method. We therefore assume that groups of similar patches have a low intrinsic dimensionality, leading to a spiked covariance model. Based on theoretical results about the estimation of spiked covariance matrices, we propose estimators of the eigenvalues of the a priori covariance in high-dimensional spaces as simple corrections of the eigenvalues of the sample covariance matrix. We demonstrate empirically that these estimators lead to better empirical Wiener filters. A comparison on classic benchmark videos demonstrates improved visual quality and an increased PSNR with respect to state-of-the-art video denoising methods.     

Moment equations of a stochastic system with generalized Poisson parameters†

Differential equations for the moments of a system with generalized Poisson parameters are derived by finding the rate of change of the expected value of an arbitrary function of the state variables. Results are very similar to the equations for the moments of tin Ito stochastic differential equation.     

Empirical Analysis of Software Fault Content and Fault Proneness Using Bayesian Methods

We present a methodology for Bayesian analysis of software quality. We cast our research in the broader context of constructing a causal framework that can include process, product, and other diverse sources of information regarding fault introduction during the software development process. In this paper, we discuss the aspect of relating internal product metrics to external quality metrics. Specifically, we build a Bayesian network (BN) model to relate object-oriented software metrics to software fault content and fault proneness. Assuming that the relationship can be described as a generalized linear model, we derive parametric functional forms for the target node conditional distributions in the BN. These functional forms are shown to be able to represent linear, Poisson, and binomial logistic regression. The models are empirically evaluated using a public domain data set from a software subsystem. The results show that our approach produces statistically significant estimations and that our overall modeling method performs no worse than existing techniques.     

Bayesian analysis of GUHA hypotheses

The LISp-Miner system for data mining and knowledge discovery uses the GUHA method to comb through a large data base and finds 2 × 2 contingency tables that satisfy a certain condition given by generalised quantifiers and thereby suggest the existence of possible relations between attributes. In this paper, we show how a more detailed interpretation of the data in the tables that were found by GUHA can be obtained using Bayesian statistical methods. Using a multinomial sampling model and Dirichlet prior, we derive posterior distributions for parameters that correspond to GUHA generalised quantifiers. Examples are presented illustrating the new Bayesian post-processing tools implemented in LISp-Miner. A statistical model for the analysis of contingency tables for data from two subpopulations is also presented.     

A Bayesian analysis of the thermal challenge problem

A major question for the application of computer models is Does the computer model adequately represent reality? Viewing the computer models as a potentially biased representation of reality, Bayarri et al. [M. Bayarri, J. Berger, R. Paulo, J. Sacks, J. Cafeo, J. Cavendish, C. Lin, J. Tu, A framework for validation of computer models, Technometrics 49 (2) (2007) 138–154] develop the simulator assessment and validation engine (SAVE) method as a general framework for answering this question. In this paper, we apply the SAVE method to the challenge problem which involves a thermal computer model designed for certain devices. We develop a statement of confidence that the devices modeled can be applied in intended situations.     

A Bayesian mixture model to quantify parameters of spatial clustering

A new Bayesian approach for quantifying spatial clustering is proposed that employs a mixture of gamma distributions to model the squared distance of points to their second nearest neighbors. The method is designed to answer questions arising in biophysical research on nanoclusters of Ras proteins. It takes into account the presence of disturbing metacluster structures as well as non-clustering objects, both common among Ras clusters. Its focus lies on estimating the proportion of points lying in clusters, the mean cluster size and the mean cluster radius without depending on prior knowledge of the parameters. The performance of the model compared to other cluster methods is demonstrated in a comprehensive simulation study, employing a specific new class of spatial point processes, the double Matérn cluster process. Further results and arguments as well as data and code are available as supplementary material.     

Approach to the correlation discovery of Chinese linguistic parameters based on Bayesian method

Bayesian approach is an important method in statistics.The Bayesian belief network is a powerful knowledge representation and reasoning tool under the conditions of uncertainty.Paper,an approach to Bayesian network construction is given for discovering the Chinese Linguistic parameter relationship in the corpus.     

Bayesian smoothing spline analysis of variance

Smoothing spline ANOVA (SSANOVA) provides an approach to semiparametric function estimation based on an ANOVA type of decomposition. Wahba et al. (1995) decomposed the regression function based on a tensor sum decomposition of inner product spaces into orthogonal subspaces, so the effects of the estimated functions from each subspace can be viewed independently. Recent research related to smoothing spline ANOVA focuses on either frequentist approaches or a Bayesian framework for variable selection and prediction. In our approach, we seek “objective” priors especially suited to estimation. The prior for linear terms including level effects is a variant of the Zellner–Siow prior (Zellner and Siow, 1980), and the prior for a smooth effect is specified in terms of effective degrees of freedom. We study this fully Bayesian SSANOVA model for Gaussian response variables, and the method is illustrated with a real data set.     

Poisson modes and general nonlinear constitutive models in the large displacement analysis of beams

Most existing formulations for structural elements such as beams, plates and shells do not allow for the use of general nonlinear constitutive models in a straightforward manner. Furthermore, such structural element models, due to the nature of the generalized coordinates used, do not capture some Poisson modes such as the ones that couple the deformation of the cross section of the structural element and stretch and bending. In this paper, beam models that employ general nonlinear constitutive equations are presented using finite elements based on the nonlinear absolute nodal coordinate formulation. This formulation relaxes the assumptions of the Euler–Bernoulli and Timoshenko beam theories, and allows for the use of general nonlinear constitutive models. The finite elements based on the absolute nodal coordinate formulation also allow for the rotation as well as the deformation of the cross section, thereby capturing Poisson modes which can not be captured using other beam models. In this investigation, three different nonlinear constitutive models based on the hyper-elasticity theory are considered. These three models are based on the Neo–Hookean constitutive law for compressible materials, the Neo–Hookean constitutive law for incompressible materials, and the Mooney–Rivlin constitutive law in which the material is assumed to be incompressible. These models, which allow capturing Poisson modes, are suitable for many materials and applications, including rubber-like materials and biological tissues which are governed by nonlinear elastic behavior. Numerical examples that demonstrate the implementation of these nonlinear constitutive models in the absolute nodal coordinate formulation are presented. The results obtained using the nonlinear and linear constitutive models are compared in this study. These results show that the use of nonlinear constitutive models can significantly enhance the performance and improve the computational efficiency of the finite element models based on the absolute nodal coordinate formulation. The results also show that when linear constitutive models are used in the large deformation analysis, singular configurations are encountered and basic formulas such as Nanson’s formula are no longer valid. These singular deformation configurations are not encountered when the nonlinear constitutive models are used.     

Empirical hardness of finding optimal Bayesian network structures: algorithm selection and runtime prediction

Various algorithms have been proposed for finding a Bayesian network structure that is guaranteed to maximize a given scoring function. Implementations of state-of-the-art algorithms, solvers, for this Bayesian network structure learning problem rely on adaptive search strategies, such as branch-and-bound and integer linear programming techniques. Thus, the time requirements of the solvers are not well characterized by simple functions of the instance size. Furthermore, no single solver dominates the others in speed. Given a problem instance, it is thus a priori unclear which solver will perform best and how fast it will solve the instance. We show that for a given solver the hardness of a problem instance can be efficiently predicted based on a collection of non-trivial features which go beyond the basic parameters of instance size. Specifically, we train and test statistical models on empirical data, based on the largest evaluation of state-of-the-art exact solvers to date. We demonstrate that we can predict the runtimes to a reasonable degree of accuracy. These predictions enable effective selection of solvers that perform well in terms of runtimes on a particular instance. Thus, this work contributes a highly efficient portfolio solver that makes use of several individual solvers.     

Bayesian analysis of mixtures of factor analyzers

For Bayesian inference on the mixture of factor analyzers, natural conjugate priors on the parameters are introduced, and then a Gibbs sampler that generates parameter samples following the posterior is constructed. In addition, a deterministic estimation algorithm is derived by taking modes instead of samples from the conditional posteriors used in the Gibbs sampler. This is regarded as a maximum a posteriori estimation algorithm with hyperparameter search. The behaviors of the Gibbs sampler and the deterministic algorithm are compared on a simulation experiment.