ASSESSING LOCAL PRIOR INFLUENCE IN BAYESIAN ANALYSIS

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MULTIVARIATE EXTREME VALUE DISTRIBUTION AND ITS FISHER INFORMATION MATRIX

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Functional estimation incorporating prior correlation information

This paper deals with the problem of estimating functional data from a functional noise model, i.e., on the basis of the observations of a discrete-time stochastic process in additive white noise which can be correlated with the process. Assuming prior information on the correlation functions involved and using principal component analysis for stochastic processes, a general suboptimum estimation procedure is derived. The proposed solution is valid for smoothing, filtering and prediction problems, can be applied to estimate any operation of the process, such as derivatives, and constitutes a computationally efficient algorithm.     

Statistical inference with partial prior information based on a Gauss-type inequality

Potter and Anderson [1] have developed a Bayesian decision procedure requiring the specification of a class of prior distributions restricted to have a minimal probability content for a given subset of the parameter space. They do not, however, provide a method for the selection of that subset. We show how a generalization of Gauss' inequality can be used to determine the relevant parameter subset.     

Wigner-Yanase skew information vs. quantum Fisher information

Among concepts describing the information contents of quantum mechanical density operators, both the Wigner-Yanase skew information and the quantum Fisher information defined via symmetric logarithmic derivatives are natural generalizations of the classical Fisher information. We will establish a relationship between these two fundamental quantities and show that they are comparable.

     

Direct approach to quantum extensions of Fisher information

By manipulating classical Fisher information and employing various derivatives of density operators, and using entirely intuitive and direct methods, we introduce two families of quantum extensions of Fisher information that include those defined via the symmetric logarithmic derivative, via the right logarithmic derivative, via the Bogoliubov-Kubo-Mori derivative, as well as via the derivative in terms of commutators, as special cases. Some fundamental properties of these quantum extensions of Fisher information are investigated, a multi-parameter quantum Cramér-Rao inequality is established, and applications to characterizing quantum uncertainty are illustrated.      

The fuzzy decision problem: An approach to the point estimation problem with fuzzy information

This paper is devoted to the extension of the Bayesian method for the point estimation, when the available information is ‘vague’.In the nonfuzzy case, the parametric estimation can be approached as a particularization in the statistical decision problem. This motivates us to accomplish the mentioned extension by looking at the parametric estimation in the fuzzy case as a special situation in the fuzzy decision problem (defined by Tanaka, Okuda and Asia).In this way, concepts in the fuzzy decision problem are first ‘expressed’ in the estimation terminology. Then, on the basis of these concepts, we shall introduce some notions and state some interesting results. Finally, several illustrative examples will be exposed.     

Fisher information under restriction of Shannon information in multi-terminal situations

Fisher information generally decreases by summarizing observed data into encoded messages. The present paper studies the amount of Fisher information included in independently summarized messages from correlated information sources; that is, the amount of Fisher information when sequences x ^N and y ^N of N independent observations of random variables x and y are encoded (summarized) independently of each other into meassages m _X and m _Y . The problem is to obtain the maximal amount of Fisher information when the size of the summarized data or Shannon message information is limited. The problem is solved in the case of completely compressed symmetric data summarization. An achievable bound is given in the general case. Information geometry, which is a powerful new differential geometrical method applicable to statistics and systems theory, is applied to this problem, proving its usefulness in information theory as well.The present work is supported in part by Grant-in-Aid for Scientific Research #61030014, Ministry of Education, Science and Culture of Japan.     

Entropy differential metric, distance and divergence measures in probability spaces: A unified approach

The paper is devoted to metrization of probability spaces through the introduction of a quadratic differential metric in the parameter space of the probability distributions. For this purpose, a φ-entropy functional is defined on the probability space and its Hessian along a direction of the tangent space of the parameter space is taken as the metric. The distance between two probability distributions is computed as the geodesic distance induced by the metric. The paper also deals with three measures of divergence between probability distributions and their interrelationships.     

How to make sense of the common prior assumption under incomplete information

Recent contributions have questioned the meaningfulness of the Common Prior Assumption (CPA) in situations of incomplete information. We characterize the CPA in terms of the primitives (individuals' belief hierarchies) without reference to an ex ante stage. The key is to rule out “agreeing to disagree” about any aspect of beliefs. Our results also yield a generalization of single-person Bayesian updating to situations without perfect recall. The entire analysis is carried out locally at the “true state”, using beliefs only, rather than beliefs-plus-knowledge. We discuss the role of truth assumptions on beliefs for a satisfactory notion of the CPA, and point out an important conceptual discontinuity between the case of two and many individuals.     

几类随机线性规划模型及其先验分布的估计

本文研究了随机线发性规划问题的几类基本模型结构．分析了随机线性规划问题对先验分布的依赖关系，给出了解除这种依赖关系的充要条件．最后用“原点距”方法与“信息熵”方法分别对先验分布作出了估计．     

On fixed-width confidence intervals associated with maximum likelihood estimation

A sequential procedure is proposed to determine the sample size for a fixed-width confidence interval for an unknown parameter with its maximum likelihood estimator as the center of the interval. It is established that the sequential procedure is asymptotically consistent and efficient.     

Affine Fisher information inequalities

We show that the Lp Petty projection inequality is a special case of the integral affine (p,λ)-Fisher information inequality for 1?p<n.     

Metric adjusted skew information and uncertainty relation

We show that an uncertainty relation for Wigner-Yanase-Dyson skew information proved by Yanagi (2010) [10] can hold for an arbitrary quantum Fisher information under some conditions. This is a refinement of the result of Gibilisco and Isola (2011) [4].     

Global statistical information in exponential experiments and selection of exponential models

The concept of global statistical information in the classical statistical experiment with independent exponentially distributed samples is investigated. Explicit formulas are evaluated for common exponential families. It is shown that the generalized likelihood ratio test procedure of model selection can be replaced by a generalized information procedure. Simulations in a classical regression model are used to compare this procedure with that based on the Akaike criterion.     

多源验前信息下先验分布的稳健融合方法

根据多种先验分布与似然函数尾部特性的比较,给出了多源验前信息下先验分布的稳健融合方法.讨论了由该方法得到的融合先验分布的后验稳健性问题.最后的数值例子表明,由该方法得到的融合先验分布具有较好的稳健性,进一步验证了该方法的有效性.     

On a refinement of Heisenberg uncertainty relation by means of quantum Fisher information

We show that an uncertainty relation proved by Luo and Yanagi and related to Wigner-Yanase-Dyson information cannot hold for an arbitrary quantum Fisher information. We show that, by changing a costant, a similar trace inequality holds in full generality.     

模糊信息差异及唯一性定理证明

定义了模糊相对熵,基于模糊熵和距离定义了模糊信息差异、拟模糊信息差异,并讨论了模糊信息差异唯一性定理以及模糊信息差异的性质.