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| Author | : Ran Wei |
| Publisher | : |
| Total Pages | : 108 |
| Release | : 2017 |
| Genre | : |
| ISBN | : |
| Author | : Hongyu Wu |
| Publisher | : |
| Total Pages | : 0 |
| Release | : 2022 |
| Genre | : Statistics |
| ISBN | : |
Variable selection methods have become an important and growing problem in Bayesian analysis. The literature on Bayesian variable selection methods tends to be applied to a single response- type, and more typically, a continuous response-type, where it is assumed that the data is Gaus- sian/symmetric. In this dissertation, we develop a novel global-local shrinkage prior in non- symmetric settings and multiple response-types settings by combining the perspectives of global- local shrinkage and the conjugate multivaraite distribution. In Chapter 2, we focus on the problem of variable selection when the data is possibly non- symmetric continuous-valued. We propose modeling continuous-valued data and the coefficient vector with the multivariate logit-beta (MLB) distribution. To perform variable selection in a Bayesian context we make use of shrinkage global-local priors to enforce sparsity. Specifically, they can be defined as a Gaussian scale mixture of a global shrinkage parameter and a local shrinkage parameter for a regression coefficient. We provide a technical discussion that illustrates that our use of the multivariate logit-beta distribution under a P ́olya-Gamma augmentation scheme has an explicit connection to a well-known global-local shrinkage method (id est, the horseshoe prior) and extends it to possibly non-symmetric data. Moreover, our method can be implemented using an efficient block Gibbs sampler. Evidence of improvements in terms of mean squared error and variable selection as compared to the standard implementation of the horseshoe prior for skewed data settings is provided in simulated and real data examples. In Chapter 3, we direct our attention to the canonical variable selection problem in multiple response-types settings, where the observed dataset consists of multiple response-types (e.g., con- tinuous, count-valued, Bernoulli trials, et cetera). We propose the same global-local shrinkage prior in Chapter 2 but for multiple response-types datasets. The implementation of our Bayesian variable selection method to such data types is straightforward given the fact that the multivariate logit-beta prior is the conjugate prior for several members from the natural exponential family of distributions, which leads to the binomial/beta and negative binomial/beta hierarchical models. Our proposed model not just allows the estimation and selection of independent regression coefficients, but also those of shared regression coefficients across-response-types, which can be used to explicitly model dependence in spatial and time-series settings. An efficient block Gibbs sampler is developed, which is found to be effective in obtaining accurate estimates and variable selection results in simulation studies and an analysis of public health and financial costs from natural disasters in the U.S.
| Author | : Xiaofan Xu |
| Publisher | : |
| Total Pages | : 76 |
| Release | : 2014 |
| Genre | : |
| ISBN | : |
The paper considers the classical Bayesian variable selection problem and an important subproblem in which grouping information of predictors is available. We propose the Half Thresholding (HT) estimator for simultaneous variable selection and estimation with shrinkage priors. Under orthogonal design matrix, variable selection consistency and asymptotic distribution of HT estimators are investigated and the oracle property is established with Three Parameter Beta Mixture of Normals (TPBN) priors. We then revisit Bayesian group lasso and use spike and slab priors for variable selection at the group level. In the process, the connection of our model with penalized regression is demonstrated, and the role of posterior median for thresholding is pointed out. We show that the posterior median estimator has the oracle property for group variable selection and estimation under orthogonal design while the group lasso has suboptimal asymptotic estimation rate when variable selection consistency is achieved. Next we consider Bayesian sparse group lasso again with spike and slab priors to select variables both at the group level and also within the group, and develop the necessary algorithm for its implementation. We demonstrate via simulation that the posterior median estimator of our spike and slab models has excellent performance for both variable selection and estimation.
| Author | : Juan Diego Vera |
| Publisher | : |
| Total Pages | : 0 |
| Release | : 2023 |
| Genre | : |
| ISBN | : |
This study investigates the effectiveness of Bayesian variable selection (BVS) procedures in dealing with missing not at random (MNAR) data for identification in selection models. Three BVS-adapted selection models, namely Bayesian LASSO, horseshoe prior, and spike-and-slab prior, were compared, along with established missing data methods such as a model that assumes a missing at random (MAR) process and full-selection model. The results indicate that the spike-and-slab prior consistently outperformed other BVS methods in terms of accuracy and bias for various parameters, including slope estimates, residual variance, and intercept. When compared with the full-selection model, the spike-and-slab model exhibited superior performance across all parameters based on mean squared error (MSE) results.Although the MAR and spike-and-slab models showed comparable performance for slope estimates, the spike-and-slab model consistently outperformed the MAR model in estimating residual variance and intercept. This comparable performance is attributed to the bias-variance tradeoff. The MAR model, while biased, demonstrated efficiency by estimating fewer parameters than selection models and obtaining robust support from the observed data. On the other hand, the spike-and-slab model outperformed the full-selection model, even when the full-selection model aligned with the true data-generating model. The adaptation of BVS to selection models, particularly through the spike-and-slab method, yielded promising results with unbiased estimates under various conditions. However, it is important to acknowledge that this study represents an initial exploration of this subject, and its scope was inherently limited. Finally, the BVS adaptations to the selection model was illustrated with data from a clinical-trial study.
| Author | : Andrew Gelman |
| Publisher | : CRC Press |
| Total Pages | : 663 |
| Release | : 2013-11-27 |
| Genre | : Mathematics |
| ISBN | : 1439898200 |
Winner of the 2016 De Groot Prize from the International Society for Bayesian AnalysisNow in its third edition, this classic book is widely considered the leading text on Bayesian methods, lauded for its accessible, practical approach to analyzing data and solving research problems. Bayesian Data Analysis, Third Edition continues to take an applied
| Author | : Anjali Agarwal |
| Publisher | : |
| Total Pages | : 90 |
| Release | : 2016 |
| Genre | : |
| ISBN | : |
A major focus of intensive methodological research in recent times has been on knowledge extraction from high-dimensional datasets made available by advances in research technologies. Coupled with the growing popularity of Bayesian methods in statistical analysis, a range of new techniques have evolved that allow innovative model-building and inference in high-dimensional settings – an important one among these being Bayesian variable selection (BVS). The broad goal of this thesis is to explore different BVS methods and demonstrate their application in high-dimensional psychological data analysis. In particular, the focus will be on a class of sparsity-enforcing priors called 'spike-and-slab' priors which are mixture priors on regression coefficients with density functions that are peaked at zero (the 'spike') and also have large probability mass for a wide range of non-zero values (the 'slab'). It is demonstrated that BVS with spike-and-slab priors achieved a reasonable degree of dimensionality-reduction when applied to a psychiatric dataset in a logistic regression setup. BVS performance was also compared to that of LASSO (least absolute shrinkage and selection operator), a popular machine-learning technique, as reported in Ahn et al.(2016). The findings indicate that BVS with a spike-and-slab prior provides a competitive alternative to machine-learning methods, with the additional advantages of ease of interpretation and potential to handle more complex models. In conclusion, this thesis serves to add a new cutting-edge technique to the lab’s tool-shed and helps introduce Bayesian variable-selection to researchers in Cognitive Psychology where it still remains relatively unexplored as a dimensionality-reduction tool.
| Author | : Edward Jerrold Cripps |
| Publisher | : |
| Total Pages | : 0 |
| Release | : 2008 |
| Genre | : |
| ISBN | : |
This article proposes a new data-based prior for the error variance in a Gaussian linear regression model, when the model is used for Bayesian variable selection and model averaging. For a given subset of variables in the model, this prior has a mode that is an unbiased estimator of the error variance but is suitably dispersed to make it uninformative relative to the marginal likelihood. The advantage of this empirical Bayes prior for the error variance is that it is centered and dispersed sensibly and avoids the arbitrary specification of hyperparameters. The performance of the new prior is compared to that of a prior proposed previously in the literature using several simulated examples and two loss functions. For each example our paper also reports results for the model that orthogonalizes the predictor variables before performing subset selection. A real example is also investigated. The empirical results suggest that for both the simulated and real data, the performance of the estimators based on the prior proposed in our article compares favourably with that of a prior used previously in the literature.
| Author | : Peter Müller |
| Publisher | : Springer |
| Total Pages | : 203 |
| Release | : 2015-06-17 |
| Genre | : Mathematics |
| ISBN | : 3319189689 |
This book reviews nonparametric Bayesian methods and models that have proven useful in the context of data analysis. Rather than providing an encyclopedic review of probability models, the book’s structure follows a data analysis perspective. As such, the chapters are organized by traditional data analysis problems. In selecting specific nonparametric models, simpler and more traditional models are favored over specialized ones. The discussed methods are illustrated with a wealth of examples, including applications ranging from stylized examples to case studies from recent literature. The book also includes an extensive discussion of computational methods and details on their implementation. R code for many examples is included in online software pages.
| Author | : Xiahan Tang |
| Publisher | : |
| Total Pages | : 94 |
| Release | : 2017 |
| Genre | : Bayesian statistical decision theory |
| ISBN | : |
This thesis first describes the general idea behind Bayes Inference, various sampling methods based on Bayes theorem and many examples. Then a Bayes approach to model selection, called Stochastic Search Variable Selection (SSVS) is discussed. It was originally proposed by George and McCulloch (1993). In a normal regression model where the number of covariates is large, only a small subset tend to be significant most of the times. This Bayes procedure specifies a mixture prior for each of the unknown regression coefficient, the mixture prior was originally proposed by Geweke (1996). This mixture prior will be updated as data becomes available to generate a posterior distribution that assigns higher posterior probabilities to coefficients that are significant in explaining the response. Spatial modeling method is described in this thesis. Prior distribution for all unknown parameters and latent variables are specified. Simulated studies under different models have been implemented to test the efficiency of SSVS. A real dataset taken by choosing a small region from the Cape Floristic Region in South Africa is used to analyze the plants distribution in that region. The original multi-cateogory response is transformed into a presence and absence (binary) response for simpler analysis. First, SSVS is used on this dataset to select the subset of significant covariates. Then a spatial model is fitted using the chosen covariates and, post-estimation, predictive map of posterior probabilities of presence and absence are obtained for the study region. Posterior estimates for the true regression coefficients are also provided along with map for spatial random effects.
| Author | : |
| Publisher | : World Bank Publications |
| Total Pages | : 17 |
| Release | : |
| Genre | : |
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