A new class of nonparametric prior distributions, termed Beta-Binomial stick-breaking process, is proposed. By allowing the underlying length random variables to be dependent through a Beta marginals Markov chain, an appealing discrete random probability measure arises. The chain's dependence parameter controls the ordering of the stick-breaking weights, and thus tunes the model's label-switching ability. Also, by tuning this parameter, the resulting class contains the Dirichlet process and the Geometric process priors as particular cases, which is of interest for fast convergence of MCMC implementations. Some properties of the model are discussed and a density estimation algorithm is proposed and tested with simulated datasets.
This paper proposes the beta binomial autoregressive moving average model (BBARMA) for modeling quantized amplitude data and bounded count data. The BBARMA model estimates the conditional mean of a beta binomial distributed variable observed over the time by a dynamic structure including: (i) autoregressive and moving average terms; (ii) a set of regressors; and (iii) a link function. Besides introducing the new model, we develop parameter estimation, detection tools, an out-of-signal forecasting scheme, and diagnostic measures. In particular, we provide closed-form expressions for the conditi…
Penalized likelihood inference for beta-binomial meta-analysis of proportions of rare events
The Graduate Institute for Advanced Studies, The Graduate University for Advanced Studies（高级研究研究生院，高级研究院）；Department of Interdisciplinary Statistical Mathematics, The Institute of Statistical Mathematics（统计数学系，统计数学研究所）
Modeling sparse data such as microbiome and transcriptomics (RNA-seq) data is very challenging due to the exceeded number of zeros and skewness of the distribution. Many probabilistic models have been used for modeling sparse data, including Poisson, negative binomial, zero-inflated Poisson, and zero-inflated negative binomial models. One way to identify the most appropriate probabilistic models for zero-inflated or hurdle models is based on the p-value of the Kolmogorov-Smirnov (KS) test. The main challenge for identifying the probabilistic model is that the model parameters are typically unk…
In this paper, we consider Bayesian point estimation and predictive density estimation in the binomial case. After presenting preliminary results on these problems, we compare the risk functions of the Bayes estimators based on the truncated and untruncated beta priors and obtain dominance conditions when the probability parameter is less than or equal to a known constant. The case where there are both a lower bound restriction and an upper bound restriction is also treated. Then our problems are shown to be related to similar problems in the Poisson case. Finally, numerical studies are presen…
A class of discrete probability distributions contains distributions with limited support. A typical example is some variant of a Likert scale, with response mapped to either the $\{1, 2, \ldots, 5\}$ or $\{-3, -2, \ldots, 2, 3\}$ set. An interesting subclass of discrete distributions with finite support are distributions limited to two parameters and having no more than one change in probability monotonicity. The main contribution of this paper is to propose a family of distributions fitting the above description, which we call the Generalised Score Distribution (GSD) class. The proposed GSD …
In behavioral and psychiatric research, data consisting of a per-subject proportion of "successes" and "failures" over a finite number of trials often arise. This kind of clustered binary data are usually non-normally distributed, which can cause issues with parameter estimation and predictions if the usual general linear model is applied and sample size is small. Here we studied the performances of some of the available analytic methods applicable to the analysis of proportion data; namely linear regression, Poisson regression, beta-binomial regression and Generalized Linear Mixed Models (GLM…
We show that the stick-breaking construction of the beta process due to Paisley, et al. (2010) can be obtained from the characterization of the beta process as a Poisson process. Specifically, we show that the mean measure of the underlying Poisson process is equal to that of the beta process. We use this underlying representation to derive error bounds on truncated beta processes that are tighter than those in the literature. We also develop a new MCMC inference algorithm for beta processes, based in part on our new Poisson process construction.
A Beta-Binomial-Logit model is a Beta-Binomial model with covariate information incorporated via a logistic regression. Posterior propriety of a Bayesian Beta-Binomial-Logit model can be data-dependent for improper hyper-prior distributions. Various researchers in the literature have unknowingly used improper posterior distributions or have given incorrect statements about posterior propriety because checking posterior propriety can be challenging due to the complicated functional form of a Beta-Binomial-Logit model. We derive data-dependent necessary and sufficient conditions for posterior pr…
In this paper, we address the problem of testing goodness-of-fit for discrete distributions, where we focus on the geometric distribution. We define new likelihood-based goodness-of-fit tests using the beta-geometric distribution and the type I discrete Weibull distribution as alternative distributions. The tests are compared in a simulation study, where also the classical goodness-of-fit tests are considered for comparison. Throughout the paper we consider conditional testing given a minimal sufficient statistic under the null hypothesis, which enables the calculation of exact p-values. For t…
We consider a problem of ecological inference, in which individual-level covariates are known, but labeled data is available only at the aggregate level. The intended application is modeling voter preferences in elections. In Rosenman and Viswanathan (2018), we proposed modeling individual voter probabilities via a logistic regression, and posing the problem as a maximum likelihood estimation for the parameter vector beta. The likelihood is a Poisson binomial, the distribution of the sum of independent but not identically distributed Bernoulli variables, though we approximate it with a heteros…
We characterize the combinatorial structure of conditionally-i.i.d. sequences of negative binomial processes with a common beta process base measure. In Bayesian nonparametric applications, such processes have served as models for latent multisets of features underlying data. Analogously, random subsets arise from conditionally-i.i.d. sequences of Bernoulli processes with a common beta process base measure, in which case the combinatorial structure is described by the Indian buffet process. Our results give a count analogue of the Indian buffet process, which we call a negative binomial Indian…
Center for Advanced Mathematics in Energy Research Applications（能源研究应用高级数学中心）；Synchrotron Infrared Structural Biology Program（同步辐射红外结构生物学计划）；Molecular Biophysics and Integrated Bioimaging Division（分子生物物理与整合生物成像部）
The beta distribution is a basic distribution serving several purposes. It is used to model data, and also, as a more flexible version of the uniform distribution, it serves as a prior distribution for a binomial probability. The bivariate beta distribution plays a similar role for two probabilities that have a bivariate binomial distribution. We provide a new multivariate distribution with beta marginal distributions, positive probability over the unit square, and correlations over the full range. We discuss its extension to three or more dimensions.
The binomial and Poisson distributions have interesting relationships with the beta and gamma distributions, respectively, which involve their cumulative distribution functions and the use of conjugate priors in Bayesian statistics. We briefly discuss these relationships and some properties resulting from them which play an important role in the construction of exact nested two-sided confidence intervals and the computation of two-tailed P-values. The purpose of this article is to show that such relationships also exist between the hypergeometric distribution and a special case of the Polya (o…
Discrete random probability measures are a key ingredient of Bayesian nonparametric inferential procedures. A sample generates ties with positive probability and a fundamental object of both theoretical and applied interest is the corresponding random number of distinct values. The growth rate can be determined from the rate of decay of the small frequencies implying that, when the decreasingly ordered frequencies admit a tractable form, the asymptotics of the number of distinct values can be conveniently assessed. We focus on the geometric stick-breaking process and we investigate the effect …
We derive a construction of the beta process that allows for the atoms with significant measure to be drawn first. Our representation is based on an extension of the Sethuraman (1994) construction of the Dirichlet process, and therefore we refer to it as a stick-breaking construction. Our first proof uses a limiting case argument of finite arrays. To this end, we present a finite sieve approximation to the beta process that parallels that of Ishwaran & Zarepour (2002) and prove its convergence to the beta process. We give a second proof of the construction using Poisson process machinery. We u…
In a review on the contribution of J.B.S. Haldane to the development of the Bayes factor hypothesis test (arXiv:1511.08180), Etz and Wagenmakers focus on Haldane's proposition of a mixture prior in a genetic example (Haldane 1932, A note on inverse probability. Mathematical Proceedings of the Cambridge Philosophical Society, 28, 55-61.). As Haldane never followed up on these ideas, it is difficult to gauge his motivation and intentions. I argue that contrary to Haldane's stated intention of replacing flat priors with more reasonable assumptions, he actually chose in this example an unreasonabl…
New formulas for the moments about zero of the Non-central Chi-Squared and the Non-central Beta distributions are achieved by means of novel approaches. The mixture representation of the former model and a new expansion of the ascending factorial of a binomial are the main ingredients of the first approach, whereas the second one hinges on an interesting relationship of conditional independence and a simple conditional density of the latter model. Then, a simulation study is carried out in order to pursue a twofold purpose: providing numerical validations of the derived moment formulas on one …
University of Wisconsin-Madison（威斯康星大学麦迪逊分校）；Duke University（杜克大学）
The non-asymptotic tail bounds of random variables play crucial roles in probability, statistics, and machine learning. Despite much success in developing upper bounds on tail probability in literature, the lower bounds on tail probabilities are relatively fewer. In this paper, we introduce systematic and user-friendly schemes for developing non-asymptotic lower bounds of tail probabilities. In addition, we develop sharp lower tail bounds for the sum of independent sub-Gaussian and sub-exponential random variables, which match the classic Hoeffding-type and Bernstein-type concentration inequal…
