Economic inequalities referring to specific regions are crucial in deepening spatial heterogeneity. Income surveys are generally planned to produce reliable estimates at countries or macroregion levels, thus we implement a small area model for a set of inequality measures (Gini, Relative Theil and Atkinson indexes) to obtain microregion estimates. Considering that inequality estimators are unit-interval defined with skewed and heavy-tailed distributions, we propose a Bayesian hierarchical model at area level involving a Beta mixture. An application on EU-SILC data is carried out and a design-based simulation is performed. Our model outperforms in terms of bias, coverage and error the standard Beta regression model. Moreover, we extend the analysis of inequality estimators by deriving their approximate variance functions.
Estimation of the covariance matrix for high-dimensional multivariate datasets is a challenging and important problem in modern statistics. In this paper, we focus on high-dimensional Gaussian DAG models where sparsity is induced on the Cholesky factor L of the inverse covariance matrix. In recent work, ([Cao, Khare, and Ghosh, 2019]), we established high-dimensional sparsity selection consistency for a hierarchical Bayesian DAG model, where an Erdos-Renyi prior is placed on the sparsity pattern in the Cholesky factor L, and a DAG-Wishart prior is placed on the resulting non-zero Cholesky entr…
In recent years, multiple hypothesis testing has come to the forefront of statistical research, ostensibly in relation to applications in genomics and some other emerging fields. The false discovery rate (FDR) and its variants provide very important notions of errors in this context comparable to the role of error probabilities in classical testing problems. Accurate estimation of positive FDR (pFDR), a variant of the FDR, is essential in assessing and controlling this measure. In a recent paper, the authors proposed a model-based nonparametric Bayesian method of estimation of the pFDR functio…
Hierarchical mixtures of Unigram models for short text clustering: The role of Beta-Liouville priors
Department of Precision and Regenerative Medicine and Jonian Area (DiMePRe-J)（精准与再生医学系及乔尼亚区）；University of Bari Aldo Moro（巴里阿尔多·莫罗大学）；Sirio Astronomical Observatory（西里奥天文台）；Grotte di Castellana srl（卡斯特尔纳纳洞窟有限公司）
The seemingly disjoint problems of count and mixture modeling are united under the negative binomial (NB) process. A gamma process is employed to model the rate measure of a Poisson process, whose normalization provides a random probability measure for mixture modeling and whose marginalization leads to an NB process for count modeling. A draw from the NB process consists of a Poisson distributed finite number of distinct atoms, each of which is associated with a logarithmic distributed number of data samples. We reveal relationships between various count- and mixture-modeling distributions an…
The complete part of the earthquake frequency-magnitude distribution (FMD), above completeness magnitude mc, is well described by the Gutenberg-Richter law. The parameter mc however varies in space due to the seismic network configuration, yielding a convoluted FMD shape below max(mc). This paper investigates the shape of the generalized FMD (GFMD), which may be described as a mixture of elemental FMDs (eFMDs) defined as asymmetric Laplace distributions of mode mc [Mignan, 2012, https://doi.org/10.1029/2012JB009347]. An asymmetric Laplace mixture model (GFMD- ALMM) is thus proposed with its pa…
Sony Computer Science Laboratories, Inc.（索尼计算机科学实验室有限公司）；Tokyo（东京）
The traditional Minkowski distances are induced by the corresponding Minkowski norms in real-valued vector spaces. In this work, we propose novel statistical symmetric distances based on the Minkowski's inequality for probability densities belonging to Lebesgue spaces. These statistical Minkowski distances admit closed-form formula for Gaussian mixture models when parameterized by integer exponents. This result extends to arbitrary mixtures of exponential families with natural parameter spaces being cones: This includes the binomial, the multinomial, the zero-centered Laplacian, the Gaussian a…
The characteristics of influenza seasons varies substantially from year to year, posing challenges for public health preparation and response. Influenza forecasting is used to inform seasonal outbreak response, which can in turn potentially reduce the societal impact of an epidemic. The United States Centers for Disease Control and Prevention, in collaboration with external researchers, has run an annual prospective influenza forecasting exercise, known as the FluSight challenge. A subset of participating teams has worked together to produce a collaborative multi-model ensemble, the FluSight N…
We introduce a Bayesian approach to predictive density calibration and combination that accounts for parameter uncertainty and model set incompleteness through the use of random calibration functionals and random combination weights. Building on the work of Ranjan, R. and Gneiting, T. (2010) and Gneiting, T. and Ranjan, R. (2013), we use infinite beta mixtures for the calibration. The proposed Bayesian nonparametric approach takes advantage of the flexibility of Dirichlet process mixtures to achieve any continuous deformation of linearly combined predictive distributions. The inference procedu…
We introduce a mixture-model of beta distributions to identify significant correlations among $P$ predictors when $P$ is large. The method relies on theorems in convex geometry, which we use to show how to control the error rate of edge detection in graphical models. Our `betaMix' method does not require any assumptions about the network structure, nor does it assume that the network is sparse. The results in this article hold for a wide class of data generating distributions that include light-tailed and heavy-tailed spherically symmetric distributions.
The posterior distribution of the number of components k in a finite mixture satisfies a set of inequality constraints. The result holds irrespective of the parametric form of the mixture components and under assumptions on the prior distribution weaker than those routinely made in the literature on Bayesian analysis of finite mixtures. The inequality constraints can be used to perform an ``internal'' consistency check of MCMC estimates of the posterior distribution of k and to provide improved estimates which are required to satisfy the constraints. Bounds on the posterior probability of k co…
In this study, we consider unsupervised clustering of categorical vectors that can be of different size using mixture. We use likelihood maximization to estimate the parameters of the underlying mixture model and a penalization technique to select the number of mixture components. Regardless of the true distribution that generated the data, we show that an explicit penalty, known up to a multiplicative constant, leads to a non-asymptotic oracle inequality with the Kullback-Leibler divergence on the two sides of the inequality. This theoretical result is illustrated by a document clustering app…
Time series data may exhibit clustering over time and, in a multiple time series context, the clustering behavior may differ across the series. This paper is motivated by the Bayesian non--parametric modeling of the dependence between the clustering structures and the distributions of different time series. We follow a Dirichlet process mixture approach and introduce a new class of multivariate dependent Dirichlet processes (DDP). The proposed DDP are represented in terms of vector of stick-breaking processes with dependent weights. The weights are beta random vectors that determine different …
We consider a novel paradigm for Bayesian testing of hypotheses and Bayesian model comparison. Our alternative to the traditional construction of posterior probabilities that a given hypothesis is true or that the data originates from a specific model is to consider the models under comparison as components of a mixture model. We therefore replace the original testing problem with an estimation one that focus on the probability weight of a given model within a mixture model. We analyze the sensitivity on the resulting posterior distribution on the weights of various prior modeling on the weigh…
University of Michigan（密歇根大学）；Technische Universität Berlin（柏林工业大学）
Recent research has established sufficient conditions for finite mixture models to be identifiable from grouped observations. These conditions allow the mixture components to be nonparametric and have substantial (or even total) overlap. This work proposes an algorithm that consistently estimates any identifiable mixture model from grouped observations. Our analysis leverages an oracle inequality for weighted kernel density estimators of the distribution on groups, together with a general result showing that consistent estimation of the distribution on groups implies consistent estimation of m…
Mixtures of Zellner's g-priors have been studied extensively in linear models and have been shown to have numerous desirable properties for Bayesian variable selection and model averaging. Several extensions of g-priors to Generalized Linear Models (GLMs) have been proposed in the literature; however, the choice of prior distribution of g and resulting properties for inference have received considerably less attention. In this paper, we unify mixtures of g-priors in GLMs by assigning the truncated Compound Confluent Hypergeometric (tCCH) distribution to 1/(1 + g), which encompasses as special …
The computation of two Bayesian predictive distributions which are discrete mixtures of incomplete beta functions is considered. The number of iterations can easily become large for these distributions and thus, the accuracy of the result can be questionable. Therefore, existing algorithms for that class of mixtures are improved by introducing round-off error calculation into the stopping rule. A further simple modification is proposed to deal with possible underflows that may prevent recurrence to work properly.
Shannon and Renyi entropies are quantitative measures of uncertainty in a data set. They are developed by Renyi in the context of entropy theory. These measures have been studied in the case of the multivariate t-distributions. We extend these tools to the class of multivariate skew t-distributions and then to more families of finite mixture of multivariate skew t distributions. In particular, by using generalized Holder inequality and some properties of multinomial theorem, we find upper and lower bounds of entropies for these families. An approximate value of these entropies can be calculate…
Harvard University（哈佛大学）；Universita di Pavia（帕维亚大学）；University of Kent（肯特大学）；MD Anderson Cancer Center（MD安德森癌症中心）
Many popular Bayesian nonparametric priors can be characterized in terms of exchangeable species sampling sequences. However, in some applications, exchangeability may not be appropriate. We introduce a {novel and probabilistically coherent family of non-exchangeable species sampling sequences characterized by a tractable predictive probability function with weights driven by a sequence of independent Beta random variables. We compare their theoretical clustering properties with those of the Dirichlet Process and the two parameters Poisson-Dirichlet process. The proposed construction provides …
Thanks to the reparameterization trick, deep latent Gaussian models have shown tremendous success recently in learning latent representations. The ability to couple them however with nonparamet-ric priors such as the Dirichlet Process (DP) hasn't seen similar success due to its non parameteriz-able nature. In this paper, we present an alternative treatment of the variational posterior of the Dirichlet Process Deep Latent Gaussian Mixture Model (DP-DLGMM), where we show that the prior cluster parameters and the variational posteriors of the beta distributions and cluster hidden variables can be…
This paper studies sparse density estimation via $\ell_1$ penalization (SPADES). We focus on estimation in high-dimensional mixture models and nonparametric adaptive density estimation. We show, respectively, that SPADES can recover, with high probability, the unknown components of a mixture of probability densities and that it yields minimax adaptive density estimates. These results are based on a general sparsity oracle inequality that the SPADES estimates satisfy. We offer a data driven method for the choice of the tuning parameter used in the construction of SPADES. The method uses the gen…
We present an alternative approach to the Bayesian nonparametric analysis of conditional species richness under two-parameter Poisson Dirichlet priors. We rely on a known characterization by deletion of classes property and on results for Beta-Binomial distributions. Besides leading to simplified and much more direct proofs, our proposal provides a new scale mixture representation of the conditional asymptotic law.
Grouped data in form of income shares have been conventionally used to estimate income inequality due to the lack of availability of individual records. Most prior research on economic inequality relies on lower bounds of inequality measures in order to avoid the need to impose a parametric functional form to describe the income distribution. These estimates neglect income differences within shares, introducing, therefore, a potential source of measurement error. The aim of this paper is to explore a nuanced alternative to estimate income inequality, which leads to a reliable representation of…
