Mixture Of Beta Densities Research Articles

Semi-Parametric Estimation Using Bernstein Polynomial and a Finite Gaussian Mixture Model.

The central focus of this paper is upon the alleviation of the boundary problem when the probability density function has a bounded support. Mixtures of beta densities have led to different methods of density estimation for data assumed to have compact support. Among these methods, we mention Bernstein polynomials which leads to an improvement of edge properties for the density function estimator. In this paper, we set forward a shrinkage method using the Bernstein polynomial and a finite Gaussian mixture model to construct a semi-parametric density estimator, which improves the approximation at the edges. Some asymptotic properties of the proposed approach are investigated, such as its probability convergence and its asymptotic normality. In order to evaluate the performance of the proposed estimator, a simulation study and some real data sets were carried out.

A Bayesian hierarchical model for estimating and partitioning Bernstein polynomial density functions

A Bayesian hierarchical model for simultaneously estimating and partitioning probability density functions is presented. Individual density functions are flexibly modeled using Bernstein densities, which are mixtures of beta densities whose parameters depend only on the number of mixture components. A prior distribution is placed on the number of mixture components, and the mixture weights are expressed as increments of a distribution function G. A Dirichlet process prior is placed on G and the parameters of the Dirichlet process, the baseline distribution and the precision parameter, are treated as random. A mixture of a product of beta densities is used to partition subjects into groups, with subjects in the same group sharing information via a common baseline distribution. Inference is carried out using Markov chain Monte Carlo. A computing algorithm based on the constructive definition of the Dirichlet process is offered, for both a fixed number of groups and an unknown number of groups. When the number of groups is unknown, a birth–death algorithm is used to make inference regarding the number of groups. The model is demonstrated using radiologist-specific distributions of percent mammographic density.

Semiparametric Bayesian Testing Procedure for Noninferiority Trials with Binary Endpoints

A semiparametric testing approach based on the Bayes factor is developed for non-inferiority trials with binary endpoints. The proposed method is shown to work for a broad class of hypotheses by accommodating a variety of dissimilarity measures between two binomial parameters. Two of the unique features of the proposed testing procedure include: (i) construction of a flexible class of conjugate priors using a mixture of beta densities to maintain approximate equality of prior probabilities of the competing hypotheses; and (ii) automatic determination of the cutoff value of the Bayes factor to facilitate the decision making process. In contrast to the use of Jeffreys's rule of thumb, two forms of total weighted average error criteria are used to determine the cutoff value. Through several simulation studies it is demonstrated that the proposed Bayesian procedure has competitive frequentist properties of controlling type I error as compared to the default frequentist test and meanwhile the proposed criterion improves the statistical power, especially in small samples. The method is further illustrated using the data from a streptococcal pharyngitis clinical trial.

Experimental Validation of a Probabilistic Framework for Microarray Data Analysis

With the primary objective of developing fundamental probability models that can be used for drawing rigorous statistical inference from microarray data, we have presented in a previous publication, theoretical results for characterizing the entire microarray data set as an ensemble. Specifically, we established, from first principles, that under reasonable assumptions, the distribution of microarray intensities follows the gamma model, and consequently that the underlying theoretical distribution for the entire set of fractional intensities is a mixture of beta densities. This probabilistic framework was then used to develop a rigorous statistical inference methodology whose outcome, for each gene, is an ordered triplet: a raw computed fractional (or relative) change in expression level; an associated probability that this number indicates lower, higher, or no differential expression; and a measure of confidence associated with the stated result. In this paper we validate the probabilistic framework and associated statistical inference methodology through confirmatory experimental studies of gene expression in Saccharomyces cerevisiae using Affymetrix Genechips®. The array data were analyzed using the probabilistic framework, and 9 genes-with indeterminate expression status according to the standard 2-fold change criteria, but for which our probabilistic method indicated high expression status probabilities-were selected for higher precision characterization. In particular, for genes CGR1, GOS1, ICS2, PCL5 and PLB1, the high probabilities of being differentially expressed (up or down) were found to be in excellent agreement with the expression status determined by the independent, high precision confirmatory experiments. These confirmatory experiments, using the high precision, medium throughput polonies technique, confirmed that the probabilistic framework performs quite well in correctly identifying the expression status of genes in general, but especially differentially expressed genes that would otherwise not have been identifiable using the standard 2-fold change criteria.

A bias-reduced approach to density estimation using Bernstein polynomials

Mixtures of Beta densities have led to different methods of density estimation for univariate data assumed to have compact support. One such method relies on Bernstein polynomials and leads to good approximation properties for the resulting estimator of the underlying density f. In particular, if f is twice continuously differentiable, this estimator can be shown to attain the optimal nonparametric convergence rate of n −4/5 in terms of mean integrated squared error (MISE). However, this rate cannot be improved upon directly when relying on the usual Bernstein polynomials, no matter what other assumptions are made on the smoothness of f. In this note, we show how a simple method of bias reduction can lead to a Bernstein-based estimator that does achieve a higher rate of convergence. Precisely, we exhibit a bias-corrected estimator that achieves the optimal nonparametric MISE rate of n −8/9 when the underlying density f is four times continuously differentiable on its support.

A probabilistic framework for microarray data analysis: Fundamental probability models and statistical inference

Gene expression studies generate large quantities of data with the defining characteristic that the number of genes (whose expression profiles are to be determined) exceed the number of available replicates by several orders of magnitude. Standard spot-by-spot analysis still seeks to extract useful information for each gene on the basis of the number of available replicates, and thus plays to the weakness of microarrays. On the other hand, because of the data volume, treating the entire data set as an ensemble, and developing theoretical distributions for these ensembles provides a framework that plays instead to the strength of microarrays. We present theoretical results that under reasonable assumptions, the distribution of microarray intensities follows the Gamma model, with the biological interpretations of the model parameters emerging naturally. We subsequently establish that for each microarray data set, the fractional intensities can be represented as a mixture of Beta densities, and develop a procedure for using these results to draw statistical inference regarding differential gene expression. We illustrate the results with experimental data from gene expression studies on Deinococcus radiodurans following DNA damage using cDNA microarrays.

Rates of convergence for the posterior distributions of mixtures of Betas and adaptive nonparametric estimation of the density

In this paper, we investigate the asymptotic properties of nonparametric Bayesian mixtures of Betas for estimating a smooth density on $[0,1]$. We consider a parametrization of Beta distributions in terms of mean and scale parameters and construct a mixture of these Betas in the mean parameter, while putting a prior on this scaling parameter. We prove that such Bayesian nonparametric models have good frequentist asymptotic properties. We determine the posterior rate of concentration around the true density and prove that it is the minimax rate of concentration when the true density belongs to a H\"{o}lder class with regularity $\beta$, for all positive $\beta$, leading to a minimax adaptive estimating procedure of the density. We also believe that the approximating results obtained on these mixtures of Beta densities can be of interest in a frequentist framework.

A mixture model for the PDF of a diffusing scalar in a turbulent flow

This paper offers physical and statistical arguments that lead to the representation of the probability density function of a scalar diffusing in a turbulent flow as a mixture of two probability density functions. Particular attention is given to the high-concentration tails which are consistently observed to be described accurately by a generalized Pareto distribution. A 4-parameter construct, using a mixture of beta densities, is shown to provide an adequate representation of measured concentration distributions of wind-tunnel, grid turbulence, plume data. By using either measured or theoretically predicted moments to conduct statistical inference, it is expected that the proposed model will be flexible enough to apply to a wide variety of experimental data sets, including those obtained from large-scale environmental flows.