Additive representations of non-additive measures and the choquet integral

This paper studies some new properties of set functions (and, in particular, “non-additive probabilities” or “capacities”) and the Choquet integral with respect to such functions, in the case of a finite domain. We use an isomorphism between non-additive measures on the original space (of states of the world) and additive ones on a larger space (of events), and embed the space of real-valued functions on the former in the corresponding space on the latter. This embedding gives rise to the following results:

the Choquet integral with respect to any totally monotone capacity is an average over minima of the integrand;

the Choquet integral with respect to any capacity is the difference between minima of regular integrals over sets of additive measures;

under fairly general conditions one may define a “Radon-Nikodym derivative” of one capacity with respect to another;

the “optimistic” pseudo-Bayesian update of a non-additive measure follows from the Bayesian update of the corresponding additive measure on the larger space.

We also discuss the interpretation of these results and the new light they shed on the theory of expected utility maximization with respect to non-additive measures.     

Duality in non-additive expected utility theory

Some duality problems in expected utility theory, raised by the introduction of non-additive probabilities, are examined. These problems do not arise if the probability measure is symmetric; i.e. has the property of complementary additivity. Additional, mild properties of coherence of conditional probabilities imply full additivity of the unconditional measure.     

A strong law of large numbers for non-additive probabilities

In this paper, with the notion of independence for random variables under upper expectations, we derive a strong law of large numbers for non-additive probabilities. This result is a natural extension of the classical Kolmogorov’s strong law of large numbers to the case where the probability is no longer additive. As an application of our result, we give most frequent interpretation for Bernoulli-type experiments with ambiguity.     

Laws of large numbers of negatively correlated random variables for capacities

Our aim is to present some limit theorems for capacities.We consider a sequence of pairwise negatively correlated random variables.We obtain laws of large numbers for upper probabilities and 2-alternating capacities,using some results in the classical probability theory and a non-additive version of Chebyshev’s inequality and Boral-Contelli lemma for capacities.     

Independence for full conditional probabilities: Structure,factorization, non-uniqueness,and Bayesian networks

This paper examines concepts of independence for full conditional probabilities; that is, for set-functions that encode conditional probabilities as primary objects, and that allow conditioning on events of probability zero. Full conditional probabilities have been used in economics, in philosophy, in statistics, in artificial intelligence. This paper characterizes the structure of full conditional probabilities under various concepts of independence; limitations of existing concepts are examined with respect to the theory of Bayesian networks. The concept of layer independence (factorization across layers) is introduced; this seems to be the first concept of independence for full conditional probabilities that satisfies the graphoid properties of Symmetry, Redundancy, Decomposition, Weak Union, and Contraction. A theory of Bayesian networks is proposed where full conditional probabilities are encoded using infinitesimals, with a brief discussion of hyperreal full conditional probabilities.     

Quasi-Stationarity of Discrete-Time Markov Chains with Drift to Infinity

We consider a discrete-time Markov chain on the non-negative integers with drift to infinity and study the limiting behavior of the state probabilities conditioned on not having left state 0 for the last time. Using a transformation, we obtain a dual Markov chain with an absorbing state such that absorption occurs with probability 1. We prove that the state probabilities of the original chain conditioned on not having left state 0 for the last time are equal to the state probabilities of its dual conditioned on non-absorption. This allows us to establish the simultaneous existence, and then equivalence, of their limiting conditional distributions. Although a limiting conditional distribution for the dual chain is always a quasi-stationary distribution in the usual sense, a similar statement is not possible for the original chain.     

Role of update dynamics in the collective cooperation on the spatial snowdrift games: Beyond unconditional imitation and replicator dynamics

In this paper, we investigate the role of update or imitation rules in the spatial snowdrift game on regular lattices. Three different update rules, including unconditional imitation (UI), replicator dynamics (RD) and the Moran process, are utilized to update the strategies of focal players during the game process in the spatial snowdrift on the lattice. We observe that the aggregate cooperation level between players is largely elevated by using the Moran process in the spatial snowdrift game, when compared to the UI or replicator dynamics. Meanwhile, we carefully explore the dynamical evolution of frequency of cooperators and the cluster formation pattern for these three update rules. Moreover, it is also shown that the evolutionary behavior under the Moran update is independent of and insensitive to the randomly initial configurations of cooperators and defectors. The current results clearly indicate that the introduction of moderate randomness in the strategy update will highly promote the maintenance and persistence of cooperation among selfish individuals, which will be greatly instrumental to deeply understand the evolution of cooperation within many natural, biological and social systems.     

Improved trial methods for a class of generalized Bernoulli problems

The aim of this article is to develop improved trial methods for the solution of a generalized exterior Bernoulli free boundary problem. At the free boundary, we prescribe the Neumann boundary condition and update the free boundary with the help of the remaining Dirichlet boundary condition. Appropriate update rules are obtained by expanding the state's Dirichlet data at the actual boundary via a Taylor expansion of first and second order. The resulting trial methods converge linearly for both cases, although the trial method based on the second order Taylor expansion is much more robust. Nevertheless, via results of shape sensitivity analysis, we are able to modify the update rules such that their convergence is improved. The feasibility of the proposed trial methods and their performance is demonstrated by numerical results.     

Products of Capacities Derived from Additive Measures

One of the unanswered questions in non-additive measure theory is how to define product of non-additive measures. Most of the approaches that have already been presented only work for discrete measures. In this paper a new approach is presented for not necessarily discrete non-additive measures that are in a certain relation with additive measures, usually this means that they are somehow derived from the additive measures.     

A Kantorovich Theorem for the Structured PSB Update in Hilbert Space

The convergence behavior of quasi-Newton methods has been well investigated for many update rules. One exception that has to be examined is the PSB update in Hilbert space. Analogous to the SR1 update, the PSB update takes advantage of the symmetry property of the operator, but it does not require the positive definiteness of the operator to work with. These properties are of great practical importance, for example, to solve minimization problems where the starting operator is not positive definite, which is necessary for other updates to ensure local convergence. In this paper, a Kantorovich theorem is presented for a structured PSB update in Hilbert space, where the structure is exploited in the sense of Dennis and Walker. Finally, numerical implications are illustrated by various results on an optimal shape design problem.     

Remarks on the role of conditional probability in data exploration

A careful discussion of the concept of conditional event leads to a sensible use of frequency data as conditional probabilities: as a by-product, the well-known ‘paradoxes’ arising from the so-called confounding effect are avoided.     

Bayesian networks with a logistic regression model for the conditional probabilities

Logistic regression techniques can be used to restrict the conditional probabilities of a Bayesian network for discrete variables. More specifically, each variable of the network can be modeled through a logistic regression model, in which the parents of the variable define the covariates. When all main effects and interactions between the parent variables are incorporated as covariates, the conditional probabilities are estimated without restrictions, as in a traditional Bayesian network. By incorporating interaction terms up to a specific order only, the number of parameters can be drastically reduced. Furthermore, ordered logistic regression can be used when the categories of a variable are ordered, resulting in even more parsimonious models. Parameters are estimated by a modified junction tree algorithm. The approach is illustrated with the Alarm network.     

Detecting Jacobian sparsity patterns by Bayesian probing

In this paper we describe an automatic procedure for successively reducing the set of possible nonzeros in a Jacobian matrix until eventually the exact sparsity pattern is obtained. The dependence information needed in this probing process consist of “Boolean” Jacobian-vector products and possibly also vector-Jacobian products, which can be evaluated exactly by automatic differentiation or approximated by divided differences. The latter approach yields correct sparsity patterns, provided there is no exact cancellation at the current argument.?Starting from a user specified, or by default initialized, probability distribution the procedure suggests a sequence of probing vectors. The resulting information is then used to update the probabilities that certain elements are nonzero according to Bayes’ law. The proposed probing procedure is found to require only O(logn) probing vectors on randomly generated matrices of dimension n, with a fixed number of nonzeros per row or column. This result has been proven for (block-) banded matrices, and for general sparsity pattern finite termination of the probing procedure can be guaranteed. Received: April 29, 2000 / Accepted: September 2001?Published online April 12, 2002     

多目标0-1规划问题的元胞狼群优化算法研究

针对多目标0-1规划问题,首先基于元胞自动机原理和人工狼群智能算法,提出一种元胞狼群优化算法,该算法将元胞机的演化规则与嚎叫信息素更新规则、人工狼群更新规则进行组合,采用元胞及其邻居来增强搜索过程的多样性和分布性,使人工头狼在元胞空间搜索的过程中,增强了人工狼群算法的全局搜索能力,并获得更多的全局非劣解;其次结合多目标0-1规划模型对元胞狼群算法进行了详细的数学描述,定义了人工狼群搜索空间、移动算子、元胞演化规则和非劣解集更新规则,并给出了元胞狼群算法的具体实现步骤;最后通过MATLAB软件对3个典型的多目标0-1规划问题算例进行解算,并将解算结果与其它人工智能算法的结果进行比较,结果表明:元胞狼群算法在多目标0-1规划问题求解方面可获得更多的非劣解集和更优的非劣解,并具有较快的收敛速度和较好的全局寻优能力。     

竞争型二元离散风险模型

本文将竞争型二元经典风险模型离散化,得到了竞争二元离散风险模型,并给出了三种破产概率表达式.     

利用有限域上的向量空间构作带仲裁的认证码

This paper is devoted to constructing an authentication code with arbitration using subspaces of vector spaces over finite fields.Moreover,if we choose the encoding rules of the transmitter and the decoding rules of the receiver according to a uniform probability distribution,then some parameters and the probabilities of successful attacks are computed.     

The use of Markov operators to constructing generalised probabilities

A new approach to constructing generalised probabilities is proposed. It is based on the models using lower and upper previsions, or equivalently, convex sets of probability measures. Our approach uses sets of Markov operators in the role of rules preserving desirability of gambles. The main motivation being the operators of conditional expectations which are usually assumed to reduce riskiness of gambles. Imprecise probability models are then obtained in the ways to be consistent with those desirability preserving rules. The consistency criteria are based on the existing interpretations of models using imprecise probabilities. The classical models based on lower and upper previsions are shown to be a special class of the generalised models. Further, we generalise some standard extension procedures, including the marginal extension and independent products, which can be defined independently of the existing procedures known for standard models.     

Optimal discriminant functions for normal populations

A class of discriminant rules which includes Fisher’s linear discriminant function and the likelihood ratio criterion is defined. Using asymptotic expansions of the distributions of the discriminant functions in this class, we derive a formula for cut-off points which satisfy some conditions on misclassification probabilities, and derive the optimal rules for some criteria. Some numerical experiments are carried out to examine the performance of the optimal rules for finite numbers of samples.     

Saddlepoint approximation at the edges of a conditional sample space

Saddlepoint methods present a convenient way to approximate probabilities associated with canonical sufficient statistic vectors in generalized linear models. Implementing saddlepoint approximations requires calculating maximum likelihood estimators for the associated parameters. When the sufficient statistic vector lies at the edge of the sample space, maximum likelihood estimators may not exist. This paper describes how to modify saddlepoint approximation to work in these cases.     

利用有限域上奇异伪辛几何构作具有仲裁的认证码

利用有限域上奇异伪辛几何构作了一个新的具有仲裁的认证码,计算了这个码的参数,当发方编码规则和收方解码规则按等概率分布选取时,计算出各种攻击成功的概率.