
On a Rapid Simulation of the Dirichlet Process
We describe a simple and efficient procedure for approximating the Lévy measure of a Gamma(α,1) random variable. We use this approximation to derive a finite sumrepresentation that converges almost surely to Ferguson's representation of the Dirichlet process based on arrivals of a homogeneous Poisson process. We compare the efficiency of our approximation to several other well known approximations of the Dirichlet process and demonstrate a substantial improvement.
07/04/2011 ∙ by Mahmoud Zarepour, et al. ∙ 0 ∙ shareread it

Statistical Reasoning: Choosing and Checking the Ingredients, Inferences Based on a Measure of Statistical Evidence with Some Applications
The features of a logically sound approach to a theory of statistical reasoning are discussed. A particular approach that satisfies these criteria is reviewed. This is seen to involve selection of a model, model checking, elicitation of a prior, checking the prior for bias, checking for priordata conflict and estimation and hypothesis assessment inferences based on a measure of evidence. A longstanding anomalous example is resolved by this approach to inference and an application is made to a practical problem of considerable importance which, among other novel aspects of the analysis, involves the development of a relevant elicitation algorithm.
02/17/2018 ∙ by Luai Al Labadi, et al. ∙ 0 ∙ shareread it

The TwoSample Problem Via Relative Belief Ratio
This paper deals with a new Bayesian approach to the twosample problem. More specifically, let x=(x_1,...,x_n_1) and y=(y_1,...,y_n_2) be two independent samples coming from unknown distributions F and G, respectively. The goal is to test the null hypothesis H_0: F=G against all possible alternatives. First, a Dirichlet process prior for F and G is considered. Then the change of their Cramérvon Mises distance from a priori to a posteriori is compared through the relative belief ratio. Many theoretical properties of the procedure have been developed and several examples have been discussed, in which the proposed approach shows excellent performance.
05/17/2018 ∙ by Luai Al Labadi, et al. ∙ 0 ∙ shareread it

On onesample Bayesian tests for the mean
This paper deals with a new Bayesian approach to the standard onesample z and t tests. More specifically, let x_1,...,x_n be an independent random sample from a normal distribution with mean μ and variance σ^2. The goal is to test the null hypothesis H_0: μ=μ_1 against all possible alternatives. The approach is based on using the wellknown formula of the KullbakLeibler divergence between two normal distributions (sampling and hypothesized distributions selected in an appropriate way). The change of the distance from a priori to a posteriori is compared through the relative belief ratio (a measure of evidence). Eliciting the prior, checking for priordata conflict and bias are also considered. Many theoretical properties of the procedure have been developed. Besides it's simplicity, and unlike the classical approach, the new approach possesses attractive and distinctive features such as giving evidence in favor of the null hypothesis. It also avoids several undesirable paradoxes, such as Lindley's paradox that may be encountered by some existing Bayesian methods. The use of the approach has been illustrated through several examples.
03/03/2019 ∙ by Ibrahim Abdelrazeq, et al. ∙ 0 ∙ shareread it

KullbackLeibler Divergence for Bayesian Nonparametric Model Checking
Bayesian nonparametric statistics is an area of considerable research interest. While recently there has been an extensive concentration in developing Bayesian nonparametric procedures for model checking, the use of the Dirichlet process, in its simplest form, along with the KullbackLeibler divergence is still an open problem. This is mainly attributed to the discreteness property of the Dirichlet process and that the KullbackLeibler divergence between any discrete distribution and any continuous distribution is infinity. The approach proposed in this paper, which is based on incorporating the Dirichlet process, the KullbackLeibler divergence and the relative belief ratio, is considered the first concrete solution to this issue. Applying the approach is simple and does not require obtaining a closed form of the relative belief ratio. A Monte Carlo study and real data examples show that the developed approach exhibits excellent performance.
03/02/2019 ∙ by Luai Al Labadi, et al. ∙ 0 ∙ shareread it

A Bayesian Nonparametric Estimation to Entropy
A Bayesian nonparametric estimator to entropy is proposed. The derivation of the new estimator relies on using the Dirichlet process and adapting the wellknown frequentist estimators of Vasicek (1976) and Ebrahimi, Pflughoeft and Soofi (1994). Several theoretical properties, such as consistency, of the proposed estimator are obtained. The quality of the proposed estimator has been investigated through several examples, in which it exhibits excellent performance.
03/02/2019 ∙ by Luai Al Labadi, et al. ∙ 0 ∙ shareread it

A Bayesian Nonparametric Test for Assessing Multivariate Normality
In this paper, a novel Bayesian nonparametric test for assessing multivariate normal models is presented. While there are extensive frequentist and graphical methods for testing multivariate normality, it is challenging to find Bayesian counterparts. The proposed approach is based on the use of the Dirichlet process and Mahalanobis distance. More precisely, the Mahalanobis distance is employed as a good technique to transform the mvariate problem into a univariate problem. Then the Dirichlet process is used as a prior on the distribution of the Mahalanobis distance. The concentration of the distribution of the distance between the posterior process and the chisquare distribution with m degrees of freedom is compared to the concentration of the distribution of the distance between the prior process and the chisquare distribution with m degrees of freedom via a relative belief ratio. The distance between the Dirichlet process and the chisquare distribution is established based on the AndersonDarling distance. Key theoretical results of the approach are derived. The procedure is illustrated through several examples, in which the proposed approach shows excellent performance.
04/04/2019 ∙ by Luai Al Labadi, et al. ∙ 0 ∙ shareread it

Using prior expansions for priordata conflict checking
Any Bayesian analysis involves combining information represented through different model components, and when different sources of information are in conflict it is important to detect this. Here we consider checking for priordata conflict in Bayesian models by expanding the prior used for the analysis into a larger family of priors, and considering a marginal likelihood score statistic for the expansion parameter. Consideration of different expansions can be informative about the nature of any conflict, and extensions to hierarchically specified priors and connections with other approaches to priordata conflict checking are discussed. Implementation in complex situations is illustrated with two applications. The first concerns testing for the appropriateness of a LASSO penalty in shrinkage estimation of coefficients in linear regression. Our method is compared with a recent suggestion in the literature designed to be powerful against alternatives in the exponential power family, and we use this family as the prior expansion for constructing our check. A second application concerns a problem in quantum state estimation, where a multinomial model is considered with physical constraints on the model parameters. In this example, the usefulness of different prior expansions is demonstrated for obtaining checks which are sensitive to different aspects of the prior.
02/27/2019 ∙ by David J. Nott, et al. ∙ 0 ∙ shareread it

Male Underperformance in Undergraduate Engineering Mathematical Courses: Causes and Solution Strategy
The performance of students in Mathematics and its allied fields has been a topic of great interest for mathematical educationalists worldwide. In this paper, we study the student performance in one of the most math heavy fieldsEngineering. An analysis of the performance of students using a sample from calculus courses across all fields within Engineering at the University of Sharjah, UAE reveled a trend of girls performing better than the boys. To be able to apply corrective strategies to handle this issue, a survey was carried out focusing on micro issues which we as educationalist could deal with at the college level and the university level. The results of the survey pinpointed out clear reasons for this grade disparity among the genders, thus allowing us to propose some immediate and practical solutions to deal with this scenario.
07/01/2019 ∙ by Luai Al Labadi, et al. ∙ 0 ∙ shareread it

A Bayesian Semiparametric Gaussian Copula Approach to a Multivariate Normality Test
In this paper, a Bayesian semiparametric copula approach is used to model the underlying multivariate distribution F_true. First, the Dirichlet process is constructed on the unknown marginal distributions of F_true. Then a Gaussian copula model is utilized to capture the dependence structure of F_true. As a result, a Bayesian multivariate normality test is developed by combining the relative belief ratio and the Energy distance. Several interesting theoretical results of the approach are derived. Finally, through several simulated examples and a real data set, the proposed approach reveals excellent performance.
07/03/2019 ∙ by Luai Al Labadi, et al. ∙ 0 ∙ shareread it

Measuring Bayesian Robustness Using Rényi's Divergence and Relationship with PriorData Conflict
This paper deals with measuring the Bayesian robustness of classes of contaminated priors. Two different classes of priors in the neighbourhood of the elicited prior are considered. The first one is the wellknown ϵcontaminated class, while the second one is the geometric mixing class. The proposed measure of robustness is based on computing the curvature of Rényi's divergence between posterior distributions. The relationship between robustness and prior data conflict has been studied. Through two examples, a strong connection between robustness and priordata conflict has been found.
05/15/2019 ∙ by Luai Al Labadi, et al. ∙ 0 ∙ shareread it
Luai Al Labadi
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