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Through refereed papers, this volume focuses on the foundations of the Bayesian paradigm; their comparison to objectivistic or frequentist Statistics counterparts; and the appropriate application of Bayesian foundations. This research in Bayesian Statistics is applicable to data analysis in biostatistics, clinical trials, law, engineering, and the social sciences. EBEB, the Brazilian Meeting on Bayesian Statistics, is held every two years by the ISBrA, the International Society for Bayesian Analysis, one of the most active chapters of the ISBA. The 12th meeting took place March 10-14, 2014 in Atibaia. Interest in foundations of inductive Statistics has grown recently in accordance with the increasing availability of Bayesian methodological alternatives. Scientists need to deal with the ever more difficult choice of the optimal method to apply to their problem. This volume shows how Bayes can be the answer. The examination and discussion on the foundations work towards the goal of proper application of Bayesian methods by the scientific community. Individual papers range in focus from posterior distributions for non-dominated models, to combining optimization and randomization approaches for the design of clinical trials, and classification of archaeological fragments with Bayesian networks.



1. What About the Posterior Distributions When the Model is Non-dominated?

Starting from the first inception of philosophical research that had subsequently led to subjective probability and Bayesian statistics, and to date the most recent developments, the probabilistic nature and the related statistical implications of Bayes theorem have been thoroughly discussed. However, the substantial contents of such a formula is very deep and new contributions are still continuing after 250 years. The simplest form of Bayes theorem is met when dominated statistical models are dealt with. This is, in a sense, comfortable, specially as far as parametric models are considered. Actually, most statistical techniques in the frame of parametric inference refer to dominated statistical models. Different problems in the applications, however, can lead to considering non-dominated models. In these cases, some complications and intriguing conclusions can arise. Concerning non-dominated statistical models, we devote this note to discussing some mathematical features that may sometimes escape the attention of statisticians. We deal with questions and results that, at a first glance, may appear of almost-exclusive measure-theoretic interest. However, they have a real statistical meaning of their own and the present note aims to stimulate some reflections about this field.
Claudio Macci, Fabio Spizzichino

2. Predictive Inference Under Exchangeability, and the Imprecise Dirichlet Multinomial Model

Coherent reasoning under uncertainty can be represented in a very general manner by coherent sets of desirable gambles. In this framework, and for a given finite category set, coherent predictive inference under exchangeability can be represented using Bernstein coherent cones of multivariate polynomials on the simplex generated by this category set. This is a powerful generalisation of de Finetti’s representation theorem allowing for both imprecision and indecision. We define an inference system as a map that associates a Bernstein coherent cone of polynomials with every finite category set. Many inference principles encountered in the literature can then be interpreted, and represented mathematically, as restrictions on such maps. We discuss two important inference principles: representation insensitivity—a strengthened version of Walley’s representation invariance—and specificity. We show that there is a infinity of inference systems that satisfy these two principles, amongst which we discuss in particular the inference systems corresponding to (a modified version of) Walley and Bernard’s imprecise Dirichlet multinomial models (IDMMs) and the Haldane inference system.
Gert de Cooman, Jasper De Bock, Márcio Diniz

3. Bayesian Learning of Material Density Function by Multiple Sequential Inversions of 2-D Images in Electron Microscopy

We discuss a novel inverse problem in which the data is generated by the sequential contractive projections of the convolution of two unknown functions, both of which we aim to learn. The method is illustrated using an application that relates to the multiple inversions of image data recorded with a scanning electron microscope, with the aim of learning the density of a given material sample and the microscopy correction function. Given the severe logistical difficulties in this application of taking multiple images at different viewing angles, a novel imaging experiment is undertaken, resulting in expansion of information. In lieu of training data, it is noted that the highly discontinuous material density function cannot be modelled using a Gaussian process (GP) as the parametrisation of the required nonstationary covariance function of such a GP cannot be achieved without training data. Consequently, we resort to estimating values of the unknown functions at chosen locations in their domain—locations at which an image data are available. Image data across a range of resolutions lead to multiscale models which we use to estimate material densities from the micrometre to nanometre length scales. We discuss applications of the method in nondestructive learning of material density using simulated metallurgical image data, as well as perform inhomogeneity detection in multicomponent composite on nanometre scales, by inverting real image data of a brick of nanoparticles.
Dalia Chakrabarty, Shashi Paul

4. Problems with Constructing Tests to Accept the Null Hypothesis

Futility designs have been proposed and used by constructing classical (non-Bayesian) hypothesis tests such that the decision of therapeutic interest is to accept the null hypothesis. A consequence is that the probability of accepting (failing to reject) the null when the null is false is unknown. Reversal of the conventional null and alternative hypotheses is not required either to demonstrate futility/nonsuperiority, or to align type I and II errors with their consequences. Conventional methods to test whether the response to the investigational agent is superior to a comparative control (superiority trial) are preferable and in the case that the null hypothesis is rejected, the associated type I error is known.
André Rogatko, Steven Piantadosi

5. Cognitive-Constructivism, Quine, Dogmas of Empiricism, and Münchhausen’s Trilemma

The Bayesian research group at University of São Paulo has been exploring a specific version of cognitive constructivism (Cog-Con) that has, among its most salient features, a distinctive objective character. Cog-Con is supported by a specially designed measure of statistical significance, namely, \({\mbox{ev}}(H{\vert} X)\)—the Bayesian epistemic value of sharp hypotheses H, given the observed data X. This article explores possible parallels or contrasts between Cog-Con and the epistemological framework developed by the philosopher Willard van Orman Quine.
Julio Michael Stern

6. A Maximum Entropy Approach to Learn Bayesian Networks from Incomplete Data

This chapter addresses the problem of estimating the parameters of a Bayesian network from incomplete data. This is a hard problem, which for computational reasons cannot be effectively tackled by a full Bayesian approach. The work around is to search for the estimate with maximum posterior probability. This is usually done by selecting the highest posterior probability estimate among those found by multiple runs of Expectation-Maximization with distinct starting points. However, many local maxima characterize the posterior probability function, and several of them have similar high probability. We argue that high probability is necessary but not sufficient in order to obtain good estimates. We present an approach based on maximum entropy to address this problem and describe a simple and effective way to implement it. Experiments show that our approach produces significantly better estimates than the most commonly used method.
Giorgio Corani, Cassio P. de Campos

7. Bayesian Inference in Cumulative Distribution Fields

One approach for constructing copula functions is by multiplication. Given that products of cumulative distribution functions (CDFs) are also CDFs, an adjustment to this multiplication will result in a copula model, as discussed by Liebscher (J Mult Analysis, 2008). Parameterizing models via products of CDFs has some advantages, both from the copula perspective (e.g. it is well-defined for any dimensionality) and from general multivariate analysis (e.g. it provides models where small dimensional marginal distributions can be easily read-off from the parameters). Independently, Huang and Frey (J Mach Learn Res, 2011) showed the connection between certain sparse graphical models and products of CDFs, as well as message-passing (dynamic programming) schemes for computing the likelihood function of such models. Such schemes allow models to be estimated with likelihood-based methods. We discuss and demonstrate MCMC approaches for estimating such models in a Bayesian context, their application in copula modeling, and how message-passing can be strongly simplified. Importantly, our view of message-passing opens up possibilities to scaling up such methods, given that even dynamic programming is not a scalable solution for calculating likelihood functions in many models.
Ricardo Silva

8. MCMC-Driven Adaptive Multiple Importance Sampling

Monte Carlo (MC) methods are widely used for statistical inference and stochastic optimization. A well-known class of MC methods is composed of importance sampling (IS) and its adaptive extensions (such as adaptive multiple IS and population MC). In this work, we introduce an iterated batch importance sampler using a population of proposal densities, which are adapted according to a Markov Chain Monte Carlo (MCMC) technique over the population of location parameters. The novel algorithm provides a global estimation of the variables of interest iteratively, using all the generated samples weighted according to the so-called deterministic mixture scheme. Compared with a traditional multiple IS scheme with the same number of samples, the performance is substantially improved at the expense of a slight increase in the computational cost due to the additional MCMC steps. Moreover, the dependence on the choice of the cloud of proposals is sensibly reduced, since the proposal density in the MCMC method can be adapted in order to optimize the performance. Numerical results show the advantages of the proposed sampling scheme in terms of mean absolute error.
Luca Martino, Víctor Elvira, David Luengo, Jukka Corander

9. Bayes Factors for Comparison of Restricted Simple Linear Regression Coefficients

This work compares two simple linear regression slopes that are restricted to an order constraint and to a proper subset of parameter space. Two approaches based on Bayes factors are discussed. The motivation is a practical example designed to evaluate dental plaque reduction. The results indicate that the approach that takes into account the restricted parameter space is more informative than the one with unrestricted parameter space since it allows to obtain more evidence against the null hypothesis.
Viviana Giampaoli, Carlos A. B. Pereira, Heleno Bolfarine, Julio M. Singer

10. A Spanning Tree Hierarchical Model for Land Cover Classification

Image segmentation persists as a major statistical problem, with the volume and complexity of data expanding alongside new technologies. Land cover classification, one of the largest problems in Remote Sensing, provides an important example of image segmentation whose needs transcend the choice of a particular classification method. That is, the challenges associated with land cover classification pervade the analysis process from data pre–processing to estimation of a final land cover map. Multispectral, multitemporal data with inherent spatial relationships have hardly received adequate treatment due to the large size of the data and the presence of missing values. In this chapter we propose a novel, concerted application of methods which provide a unified way to estimate model parameters, impute missing data, reduce dimensionality, and classify land cover. This comprehensive analysis adopts a Bayesian approach which incorporates prior subject matter knowledge to improve the interpretability, efficiency, and versatility of land cover classification. We explore a parsimonious parametric model whose structure allows for a natural application of principal component analysis to the isolate important spectral characteristics while preserving temporal information. Moreover, it allows us to impute missing data and estimate parameters via expectation-maximization. We employ a spanning tree approximation to a lattice Potts model prior to incorporating spatial relationships in a judiciousway and more efficiently access the posterior distribution of the pixel labels. We achieve exact inference of the labels via the centroid estimator. We demonstrate this series of analysis on a set of MODIS data centered on Montreal, Canada.
Hunter Glanz, Luis Carvalho

11. Nonparametric Bayesian Regression Under Combinations of Local Shape Constraints

A nonparametric Bayesian method for regression under combinations of local shape constraints is proposed. The shape constraints considered include monotonicity, concavity (or convexity), unimodality, and in particular, combinations of several types of range-restricted constraints. By using a B-spline basis, the support of the prior distribution is included in the set of piecewise polynomial functions. The first novelty is that, thanks to the local support property of B-splines, many combinations of constraints can easily be considered by identifying B-splines whose support intersects with each constrained region. Shape constraints are included in the coefficients prior using a truncated Gaussian distribution. However, the prior density is only known up to the normalizing constant, which does change with the dimension of coefficients. The second novelty is that we propose to simulate from the posterior distribution by using a reversible jump MCMC slice sampler, for selecting the number and the position of the knots, coupled to a simulated annealing step to project the coefficients on the constrained space. This method is valid for any combination of local shape constraints and particular attention is paid to the construction of a trans-dimensional MCMC scheme.
Khader Khadraoui

12. A Bayesian Approach to Predicting Football Match Outcomes Considering Time Effect Weight

In this chapter we propose a simulation-based method for predicting football match outcomes. We adopt a Bayesian perspective, modeling the number of goals of two opposing teams as a Poisson distribution whose mean is proportional to the relative technical level of opponents. Fédération Internationale de Football Association (FIFA) ratings were taken as the measure of technical level of teams saw well as experts’ opinions on the scores of the matches were taken in account to construct the prior distributions of the parameters. Tournament simulations were performed in order to estimate probabilities of winning the tournament assuming different values for the weight attached to the experts’ information and different choices for the sequence of weights attached to the previous observed matches. The methodology is illustrated on the 2010 Football Word Cup.
Francisco Louzada, Adriano K. Suzuki, Luis E. B. Salasar, Anderson Ara, José G. Leite

13. Homogeneity Tests for 2×2 Contingency Tables

Using the likelihood ratio statistic, we develop a significance index, called P-value, to test the hypothesis of homogeneity in 2×2 contingency tables. The P-value does not depend on asymptotic distributions, and is based on the elimination of the nuisance parameter. Therefore, we obtain the exact distribution of the likelihood ratio statistic in a way that is, moreover, compatible with the likelihood principle. For a better understanding of significance indices to test homogeneity, we perform a study comparing the P-value with some indices (likelihood ratio test (LRT), chi-square test) and with the full Bayesian significance test (FBST). This comparative study shows an interesting relation between all the analyzed indices, Bayesian and frequentist.
Natalia Oliveira, Marcio Diniz, Adriano Polpo

14. Combining Optimization and Randomization Approaches for the Design of Clinical Trials

Intentional sampling methods are non-randomized procedures that select a group of individuals for a sample with the purpose of meeting specific prescribed criteria. In this paper, we extend previous works related to intentional sampling, and address the problem of sequential allocation for clinical trials with few patients. Roughly speaking, patients are enrolled sequentially, according to the order in which they start the treatment at the clinic or hospital. The allocation problem consists in assigning each new patient to one, and only one, of the alternative treatment arms. The main requisite is that the profiles in the alternative arms remain similar with respect to some relevant patients’ attributes (age, gender, disease, symptom severity and others). We perform numerical experiments based on a real case study and discuss how to conveniently set up perturbation parameters, in order to yield a suitable balance between optimality—the similarity among the relative frequencies of patients in the several categories for both arms, and decoupling—the absence of a tendency to allocate each pair of patients consistently to the same arm.
Victor Fossaluza, Marcelo de Souza Lauretto, Carlos Alberto de Bragança Pereira, Julio Michael Stern

15. Factor Analysis with Mixture Modeling to Evaluate Coherent Patterns in Microarray Data

The computational advances over the last decades have allowed the use of complex models to analyze large data sets. The development of simulation-based methods, such as the Markov chain Monte Carlo (MCMC) method, has contributed to an increased interest in the Bayesian framework as an alternative to work with factor models. Many studies have applied the factor analysis to explore gene expression data with results often outperforming traditional methods for estimating and identifying patterns and metagene groups related to the underlying biology. In this chapter, we present a sparse latent factor model (SLFM) using a mixture prior (sparsity prior) to evaluate the significance of each factor loading; when the loading is significant, the effect of the corresponding factor is detected through patterns displayed along the samples. The SLFM is applied to investigate simulated and real microarray data. The real data sets represent the gene expression for different types of cancer; these include breast, brain, ovarian, and lung tumors. The proposed model can indicate how strong is the observed expression pattern allowing the measurement of the evidence of presence/absence of the gene activity. Finally, we compare the SLFM with two simpler gene detection methods available in the literature. The results suggest that the SLFM outperforms the traditional methods.
Joao Daniel Nunes Duarte, Vinicius Diniz Mayrink

16. Bayesian Hypothesis Testing in Finite Populations: Bernoulli Multivariate Variables

Bayesian hypothesis testing for the (operational) parameter of interest in a Bernoulli (multivariate) process observed in a finite population is the focus of this study. We introduce statistical test procedures for the relevant parameter under the predictivistic perspective of Bruno de Finetti in contrast with the usual superpopulation models. The comparison between these approaches, exemplified in a simple scenario of majority elections, shows considerable differences between the corresponding results for the case of observed large sampling fractions.
Brian Alvarez R. de Melo, Luis Gustavo Esteves

17. Bayesian Ridge-Regularized Covariance Selection with Community Behavior in Latent Gaussian Graphical Models

Gaussian graphical models have been extensively used to model conditional independence via the concentration matrix of a random vector. They are particularly relevant to incorporate structure when the length of the vector is large and naive methods lead to unstable estimation of the concentration matrix. In covariance selection, we have a latent network among vector components such that two components are not connected if they are conditionally independent, that is, if their corresponding entry in the concentration matrix is zero. In this work, we expect that, in addition, vector components show a block dependency structure that represents community behavior in the context of biological and social applications, that is, connections between nodes from different blocks are sparse while connections within nodes of the same block are dense. Thus, to identify the latent network and detect communities, we propose a Bayesian approach with a hierarchical prior in two levels: a spike-and-slab prior on each off-diagonal entry of the concentration matrix for variable selection; and a degree-corrected stochastic blockmodel (SBM) to capture the community behavior. To conduct inference, we develop an efficient routine based on ridge regularization and maximum a posteriori (MAP) estimation. Finally, we demonstrate the proposed approach in a meta-genomic dataset of complex microbial biofilms from dental plaque and show how bacterial communities can be identified.
Lijun Peng, Luis E. Carvalho

18. Bayesian Inference of Deterministic Population Growth Models

Deterministic mathematical models play an important role in our understanding of population growth dynamics. In particular, the effect of temperature on the growth of disease-carrying insects is of great interest. In this chapter we propose a modified Verhulst—logistic growth—equation with temperature-dependent parameters. Namely, the growth rate r and the carrying capacity K are given by thermodynamic functions of temperature T, r(T) and K(T). Our main concern is with the problem of learning about unknown parameters of these deterministic functions from observations of population time series P(t, T). We propose a strategy to estimate the parameters of r(T) and K(T) by treating the model output P(t,T) as a realization of a Gaussian process (GP) with fixed variance and mean function given by the analytic solution to the modified Verhulst equation. We use Hamiltonian Monte Carlo (HMC), implemented using the recently developed rstan package of the R statistical computing environment, to approximate the posterior distribution of the parameters of interest. In order to evaluate the performance of our algorithm, we perform a Monte Carlo study on a simulated example, calculating bias and nominal coverage of credibility intervals. We then proceed to apply this approach to laboratory data on the temperature-dependent growth of a Chagas disease arthropod vector, Rhodnius prolixus. Analysis of this data shows that the growth rate for the insect population under study achieves its maximum around 26◦C and the carrying capacity is maximum around 25◦C, suggesting that R. prolixus populations may thrive even in non-tropical climates.
Luiz Max Carvalho, Claudio J. Struchiner, Leonardo S. Bastos

19. A Weibull Mixture Model for the Votes of a Brazilian Political Party

Statistical modeling in political analysis is used recently to describe electoral behaviour of political party. In this chapter we propose a Weibull mixture model that describes the votes obtained by a political party in Brazilian presidential elections. We considered the votes obtained by the Workers’ Party in five presidential elections from 1994 to 2010. A Bayesian approach was considered and a random walk Metropolis algorithm within Gibbs sampling was implemented. Next, Bayes factor was considered to the choice of the number of components in the mixture. In addition the probability of obtain 50 % of the votes in the first round was estimated. The results show that only few components are needed to describe the votes obtained in this election. Finally, we found that the probability of obtaining 50 % of the votes in the first ballot is increasing along time. Future developments are discussed.
Rosineide F. da Paz, Ricardo S. Ehlers, Jorge L. Bazán

20. An Alternative Operational Risk Methodology for Regulatory Capital Calculation

The main objective of this work is to suggest a new method for calculation of regulatory capital required for operational risk as an alternative to the corresponding version advocated by the Basel Committee of Banking Supervision. Our method takes into account genuine dependence among the losses of possible risk units within a financial institution. Our proposal reduces the amount of regulatory capital suggested by Basel Committee, where the risk units are assumed to be perfectly positive-dependent. A simulation study is performed to compare both approaches. Finally, we discuss when Bayesian methods are preferable to the classical ones.
Guaraci Requena, Débora Delbem, Carlos Diniz

21. Bayesian Approach of the Exponential Poisson Logarithmic Model

Recently, a new three-parameter lifetime distribution motivated mainly by lifetime issues has been proposed by the authors. In this chapter, we consider the Bayesian analysis for this new distribution and compare its performance with the classic ones. The approximate Bayes estimators obtained by Markov chain Monte Carlo (MCMC) methods under the assumption of noninformative priors are compared with the maximum likelihood estimators by simulation. Finally, the model is fitted to a real data set and it is compared with several models.
José Augusto Fioruci, Bao Yiqi, Francisco Louzada, Vicente G. Cancho

22. Bayesian Estimation of Birnbaum–Saunders Log-Linear Model

The Birnbaum–Saunders (BS) distribution was derived to model failure times of materials subjected to fluctuating stresses and strains. Motivated by applications in the characterizations of materials, in 1991 Rieck and Nedelman proposed a log-linear model for the BS distribution. This model has many applications, for instance, to compare the median time life of several populations or to assess the effect of covariates on accelerated life testing. In addition to the model studied under the classical approach, we considered Markov chain Monte Carlo (MCMC) and we made an implementation in WinBUGS to get a Bayesian approach under noninformative priori distribution. Similar results for both classical and Bayesian approaches were obtained. This implementation was also adapted for censoring and we assessed the influence of different percentages of censored data.
Elizabeth González Patiño

23. Bayesian Weighted Information Measures

Following Ebrahimi et al. (J Stat Res Iran 3:113–137, 2006), we study weighted information measure in univariate case. In particular, we address the concept of comparison models based on information measure and, in our case, specially Kullback–Leibler discrimination measure. The main result is presenting the relationship of weighted mutual information measure and weighted entropy. Indeed, the importance of Weibull distribution family in weighted Kullback–Leibler information and Kullback–Leibler information has been carefully examined, which is useful in comparison models. As a notable application of the result, we study normal distributions, which can prove the expected motivation.
Salimeh Yasaei Sekeh

24. Classifying the Origin of Archeological Fragments with Bayesian Networks

Classification of archeological fragments is the focus of the present chapter. The fragments were collected from various archeological sites in the state of Mato Grosso do Sul at Lalima village. They are thought to have originated from three Indian tribes: the Guarani (66 %), the Jacadigo (22 %), and the Kadiwéu (12 %).
We use information contained in an archeological researcher’s database. It contains qualitative and quantitative observations obtained from the characteristics of the pieces. The researcher’s expertise provided a precise classification of about 760 pieces. A supervised model of classification was created to infer the Indian technological traditions of 2300 pieces of fragments collected from the same sites. Bayesian nets were the basis for building the model. Bayesian nets are directed acyclic graphs (DAG) that properly represent the dependency within a set of random covariates. This kind of network represents the joint probability distribution of these variables and a particular factorization of it. Our approach provides a robust classification: it is based on the probabilities of fragment being originated from each one of the three archeological communities. Also, if the probability of technological tradition indicates “low probabilities” for all three groups, there could be an indication of the presence of an additional community. Comparison with alternative methods to build the networks was also presented.
Melaine Cristina de Oliveira, Andressa Soreira, Victor Fossaluza

25. A Note on Bayesian Inference for Long-Range Dependence of a Stationary Two-State Process

In this work we propose a Bayesian approach for selecting the range of a stationary process with two states. The analysis is based on approximate posterior distributions of the Hurst index obtained from a likelihood-free method. Our empirical study shows that a main advantage of our approach, along with its of simplicity, is the possibility of obtaining an approximate sample of the posterior distribution on the Hurst index, thus providing better estimates. Furthermore, there is no need for Gaussian nor asymptotic assumptions.
Plinio L. D. Andrade, Laura L. R. Rifo

26. Bayesian Partition for Variable Selection in the Power Series Cure Rate Model

In this chapter we present a model of survival with a cure fraction where a feature of the model is that variable selection is performed by Bayesian partition model. To this end we consider a orthogonal hyperplane tessellation to obtain a local structure on space covariates. The proposed model is based on the promotion time where the number of competitive causes follows a power series distribution.
Jhon F. B. Gonzales, Vera. L. D. Tomazella, Mário de Castro

27. Bayesian Semiparametric Symmetric Models for Binary Data

This work proposes a general Bayesian semiparametric model for binary data. Symmetric prior probability curves as an extension for discussed ideas from Basu and Mukhopadhyay (Generalized Linear Models: A Bayesian Perspective, pp. 231–241, 1998) are considered using the blocked Gibbs sampler, which is more general than the Polya urn Gibbs sampler. The Bayesian semiparametric approach allows us to incorporate uncertainty around the F distribution of the latent data and to model heavy-tailed or light-tailed distributions. In particular, the Bayesian semiparametric logistic model is introduced, which enables one to elicit prior distributions for regression coefficients from information about odds ratios; this is quite interesting in applied research. Then, this framework opens several possibilities to deal with binary data in the Bayesian perspective.
Marcio Augusto Diniz, Carlos Alberto de Bragança Pereira, Adriano Polpo

28. Assessing a Spatial Boost Model for Quantitative Trait GWAS

Bayesian variable selection provides a principled framework for incorporating prior information to regularize parameters in high-dimensional large-p-small-n regression models such as genomewide association studies (GWAS). Although these models produce more informative results, researchers often disregard them in favor of simpler models because of their high computational cost. We explore our recently proposed spatial boost model for GWAS on quantitative traits to assess the computational efficiency of a more representative model. The spatial boost model is a Bayesian hierarchical model that exploits spatial information on the genome to uniquely define prior probabilities of association of genetic markers based on their proximities to relevant genes. We propose analyzing large data sets by first applying an expectation–maximization filter to reduce the dimensionality of the space and then applying an efficient Gibbs sampler on the remaining markers. Finally we conduct a thorough simulation study based on real genotypes provided by the Wellcome Trust Case Control Consortium and compare our model to single association tests.
Ian Johnston, Yang Jin, Luis Carvalho

29. The Exponential-Poisson Regression Model for Recurrent Events: A Bayesian Approach

In this chapter, we introduce a new regression model for recurrent event data, in which the time of each recurrence is associated to one or multiple latent causes and no information is provided about the cause responsible for the event occurrence. This model is characterized by a fully parametric rate function and it is based on the exponential-Poisson distribution. The time of each recurrence is then given by the minimum lifetime value among all latent causes. Inference aspects of the proposed model are discussed via Bayesian inference by using Markov Chain Monte Carlo (MCMC) method. A simulation study investigates the frequentist properties of the posterior estimators for different sample sizes. A real-data application demonstrates the use of the proposed model.
Márcia A. C. Macera, Francisco Louzada, Vicente G. Cancho

30. Conditional Predictive Inference for Beta Regression Model with Autoregressive Errors

In this chapter, we study a partially linear model with autoregressive beta distributed errors [6] from the Bayesian point of view. Our proposal also provides a useful method to determine the optimal order of the autoregressive processes through an adaptive procedure using the conditional predictive ordinate (CPO) statistic [9].
In this context, the linear predictor of the beta regression model \(g(\mu_{t})\) incorporates an unknown smooth function for the auxiliary time covariate t and a sequence of autoregressive errors ϵ t , i.e.,
$$\begin{aligned} g(\mu_t)=x_{t}^{\top}\beta+f(t)+ \varepsilon_t,\end{aligned}$$
for \(t=1,\ldots,T\), where x t is a \(k\times1\) vector of nonstochastic explanatory variable values and β is a \(k\times1\) fixed parameter vector. Furthermore, these models have a convenient hierarchical representation allowing to us an easily implementation of a Markov chain Monte Carlo (MCMC) scheme. We also propose to modify the traditional conditional predictive ordinate (CPO) to obtain what we call the autoregressive CPO, which is computed for each new observation using only the data from previous time periods.
Guillermo Ferreira, Jean Paul Navarrete, Luis M. Castro, Mário de Castro
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