Logical and algorithmic properties of stable conditional independence

The logical and algorithmic properties of stable conditional independence (CI) as an alternative structural representation of conditional independence information are investigated. We utilize recent results concerning a complete axiomatization of stable conditional independence relative to discrete probability measures to derive perfect model properties of stable conditional independence structures. We show that stable CI can be interpreted as a generalization of Markov networks and establish a connection between sets of stable CI statements and propositional formulas in conjunctive normal form. Consequently, we derive that the implication problem for stable CI is coNP-complete. Finally, we show that Boolean satisfiability (SAT) solvers can be employed to efficiently decide the implication problem and to compute concise, non-redundant representations of stable CI, even for instances involving hundreds of random variables.     

Conditional independence,conditional mixing and conditional association

Some properties of conditionally independent random variables are studied. Conditional versions of generalized Borel-Cantelli lemma, generalized Kolmogorov’s inequality and generalized Hájek-Rényi inequality are proved. As applications, a conditional version of the strong law of large numbers for conditionally independent random variables and a conditional version of the Kolmogorov’s strong law of large numbers for conditionally independent random variables with identical conditional distributions are obtained. The notions of conditional strong mixing and conditional association for a sequence of random variables are introduced. Some covariance inequalities and a central limit theorem for such sequences are mentioned.     

Conditional independence structure and its closure: Inferential rules and algorithms

In this paper, we deal with conditional independence models closed with respect to graphoid properties. Such models come from different uncertainty measures, in particular in a probabilistic setting. We study some inferential rules and describe methods and algorithms to compute efficiently the closure of a set of conditional independence statements.     

Independence concepts in evidence theory

We study three conditions of independence within evidence theory framework. The first condition refers to the selection of pairs of focal sets. The remaining two ones are related to the choice of a pair of elements, once a pair of focal sets has been selected. These three concepts allow us to formalize the ideas of lack of interaction among variables and among their (imprecise) observations. We illustrate the difference between both types of independence with simple examples about drawing balls from urns. We show that there are no implication relationships between both of them. We also study the relationships between the concepts of “independence in the selection” and “random set independence”, showing that they cannot be simultaneously satisfied, except in some very particular cases.     

Computing p-values in conditional independence models for a contingency table

We present a Markov chain Monte Carlo (MCMC) method for generating Markov chains using Markov bases for conditional independence models for a four-way contingency table. We then describe a Markov basis characterized by Markov properties associated with a given conditional independence model and show how to use the Markov basis to generate random tables of a Markov chain. The estimates of exact p-values can be obtained from random tables generated by the MCMC method. Numerical experiments examine the performance of the proposed MCMC method in comparison with the χ ² approximation using large sparse contingency tables.     

Induced and higher-dimensional stable independence

We provide several crucial technical extensions of the theory of stable independence notions in accessible categories. In particular, we describe circumstances under which a stable independence notion can be transferred from a subcategory to a category as a whole, and examine a number of applications to categories of groups and modules, extending results of [16]. We prove, too, that under the hypotheses of [11], a stable independence notion immediately yields higher-dimensional independence as in [26].     

The reduction and fusion of fuzzy covering systems based on the evidence theory

This paper studies reduction of a fuzzy covering and fusion of multi-fuzzy covering systems based on the evidence theory and rough set theory. A novel pair of belief and plausibility functions is defined by employing a method of non-classical probability model and the approximation operators of a fuzzy covering. Then we study the reduction of a fuzzy covering based on the functions we presented. In the case of multiple information sources, we present a method of information fusion for multi-fuzzy covering systems, by which objects can be well classified in a fuzzy covering decision system. Finally, by using the method of maximum flow, we discuss under what conditions, fuzzy covering approximation operators can be induced by a fuzzy belief structure.     

On tests for selection of variables and independence under multivariate regression models

The authors consider various procedures for testing the hypotheses of independence of two sets of variables and certain regression coefficients are zero under multivariate regression model. Various properties of these procedures and the asymptotic distributions associated with these procedures are also considered.     

基于条件事件代数系统的条件证据组合与条件信任组合

讨论基于条件事件代数系统上的条件证据与条件信任问题,给出了条件证据的组合方法,由此可作出在条件下对目标支持的判断,我们还给出了综合各条件证据的无条件目标语据组合,它可看作是从全局角度对目标支持的判断,最后我们给出了基于多值映射的条件信任的组合。     

On Conditional Independence,Mixing, and Association

Abstract

This is a follow-up to a recent article by Prakasa Rao [15 Prakasa Rao , B.L.S. 2008 . Conditional independence, conditional mixing and association . Annals of the Institute of Statistical Mathematics AISM , doi: 10.1007/S10463-007-0152-2 . [Google Scholar]] on conditional independence, conditional mixing and conditional association. The purpose of this article is to derive rigorously some results following from conditioning. To this end, a brief review is presented of the concepts of conditional independence of events, classes of events, and random variables, followed by a conditional version of a factorization theorem, as well as a first installment of some basic results. Next, the concepts of conditional covariance and variance are introduced, and a second installment of basic results follows. Furthermore, a certain representation of the covariance is established in detail, followed by a conditional version of it, as well as a generalization. The concept of the conditional characteristic function is also recalled, and a certain inequality is established. Finally, the concept of conditional positive (negative) quadrant dependence, as well as that of conditional positive (negative) association are introduced. The article concludes with the derivation of the conditional versions of some known results, regarding positive (negative) association. This is done anticipating that conditional association (and also conditional mixing) will prove to be of significant applicability.     

Optimal estimation in random coefficient regression models

In linear regression models with random coefficients, the score function usually involves unknown nuisance parameters in the form of weights. Conditioning with respect to the sufficient statistics for the nuisance parameter, when the parameter of interest is held fixed, eliminates the nuisance parameters and is expected to give reasonably good estimating functions. The present paper adopts this approach to the problem of estimation of average slope in random coefficient regression models. Four sampling situations are discussed. Some asymptotic results are also obtained for a model where neither the regressors nor the random regression coefficients replicate. Simulation studies for normal as well as non-normal models show that the performance of the suggested estimating functions is quite satisfactory.     

Acyclic directed graphs representing independence models

In this paper we study the problem of representing probabilistic independence models, in particular those closed under graphoid properties. We focus on acyclic directed graph (DAG): a new algorithm to build a DAG, given an ordering among random variables, is described and peculiarities and advantages of this approach are discussed. Moreover, we provide a necessary and sufficient condition for the existence of a perfect map representing an independence model and we describe an algorithm based on this characterization.     

Building a binary outranking relation in uncertain, imprecise and multi-experts contexts: The application of evidence theory

We consider multicriteria decision problems where the actions are evaluated on a set of ordinal criteria. The evaluation of each alternative with respect to each criterion may be uncertain and/or imprecise and is provided by one or several experts. We model this evaluation as a basic belief assignment (BBA). In order to compare the different pairs of alternatives according to each criterion, the concept of first belief dominance is proposed. Additionally, criteria weights are also expressed by means of a BBA. A model inspired by ELECTRE I is developed and illustrated by a pedagogical example.     

All for one and one for all: Theoretical models,sociological theory,and mathematical sociology

I make the argument that mathematical sociology uniquely contributes to sociological theory through the theoretical models it develops to bridge the gap between the ideas of sociological theorists and data relevant to their empirical evaluation. My work on intergroup association and social integration, the fruit of a long-time collaboration with Fararo and more recently with Karpiński, is used throughout to illustrate my points.     

A test of independence in some copula models

The purpose of this paper is to provide limit laws for semiparametric estimators of copulas. Some statistical tests of independence are introduced as a consequence of this methodology. We are primarily concernedwith the case where the parameter lies on the boundary of the admissible domain.      

Computational aspects of minimizing conditional value-at-risk

We consider optimization problems for minimizing conditional value-at-risk (CVaR) from a computational point of view, with an emphasis on financial applications. As a general solution approach, we suggest to reformulate these CVaR optimization problems as two-stage recourse problems of stochastic programming. Specializing the L-shaped method leads to a new algorithm for minimizing conditional value-at-risk. We implemented the algorithm as the solver CVaRMin. For illustrating the performance of this algorithm, we present some comparative computational results with two kinds of test problems. Firstly, we consider portfolio optimization problems with 5 random variables. Such problems involving conditional value at risk play an important role in financial risk management. Therefore, besides testing the performance of the proposed algorithm, we also present computational results of interest in finance. Secondly, with the explicit aim of testing algorithm performance, we also present comparative computational results with randomly generated test problems involving 50 random variables. In all our tests, the experimental solver, based on the new approach, outperformed by at least one order of magnitude all general-purpose solvers, with an accuracy of solution being in the same range as that with the LP solvers. János Mayer: Financial support by the national center of competence in research "Financial Valuation and Risk Management" is gratefully acknowledged. The national centers in research are managed by the Swiss National Science Foundation on behalf of the federal authorities.     

Hierarchical linear regression models for conditional quantiles

The quantile regression has several useful features and therefore is gradually developing into a comprehensive approach to the statistical analysis of linear and nonlinear response models, but it cannot deal effectively with the data with a hierarchical structure. In practice, the existence of such data hierarchies is neither accidental nor ignorable, it is a common phenomenon. To ignore this hierarchical data structure risks overlooking the importance of group effects, and may also render many of the traditional statistical analysis techniques used for studying data relationships invalid. On the other hand, the hierarchical models take a hierarchical data structure into account and have also many applications in statistics, ranging from overdispersion to constructing min-max estimators. However, the hierarchical models are virtually the mean regression, therefore, they cannot be used to characterize the entire conditional distribution of a dependent variable given high-dimensional covariates. Furthermore, the estimated coefficient vector (marginal effects) is sensitive to an outlier observation on the dependent variable. In this article, a new approach, which is based on the Gauss-Seidel iteration and taking a full advantage of the quantile regression and hierarchical models, is developed. On the theoretical front, we also consider the asymptotic properties of the new method, obtaining the simple conditions for an n1/2-convergence and an asymptotic normality. We also illustrate the use of the technique with the real educational data which is hierarchical and how the results can be explained.     

A ranking model in uncertain, imprecise and multi-experts contexts: The application of evidence theory

We consider ranking problems where the actions are evaluated on a set of ordinal criteria. The evaluation of each alternative with respect to each criterion may be imperfect and is provided by one or several experts. We model each imperfect evaluation as a basic belief assignment (BBA). In order to rank the BBAs characterizing the performances of the actions according to each criterion, a new concept called RBBD and based on the comparison of these BBAs to ideal or nadir BBAs is proposed. This is performed using belief distances that measure the dissimilarity of each BBA to the ideal or nadir BBAs. A model inspired by Xu et al.’s method is also proposed and illustrated by a pedagogical example.