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.     

The emergence of some of the nonlogical paradoxes of the theory of sets, 1903–1908

The goal of this paper is to discuss why, how, and when nonlogical set-theoretic paradoxes were discovered. No one factor dominated the whole outlook; instead, it is argued that these paradoxes were not a simple and direct consequence of the ones discovered by Bertrand Russell and published for the first time in his The Principles of Mathematics (1903).     

The Goodman–Nguyen relation within imprecise probability theory

The Goodman–Nguyen relation is a partial order generalising the implication (inclusion) relation to conditional events. As such, with precise probabilities it both induces an agreeing probability ordering and is a key tool in a certain common extension problem. Most previous work involving this relation is concerned with either conditional event algebras or precise probabilities. We investigate here its role within imprecise probability theory, first in the framework of conditional events and then proposing a generalisation of the Goodman–Nguyen relation to conditional gambles. It turns out that this relation induces an agreeing ordering on coherent or C-convex conditional imprecise previsions. In a standard inferential problem with conditional events, it lets us determine the natural extension, as well as an upper extension. With conditional gambles, it is useful in deriving a number of inferential inequalities.     

Some Paradoxes in Special Relativity and the Resolutions

The special theory of relativity is the foundation of modern physics, but its unusual postulate of invariant vacuum speed of light results in a number of plausible paradoxes. This situation leads to radical criticisms and suspicions against the theory of relativity. In this paper, from the perspective that the relativity is nothing but a geometry, we give a uniform resolution to some famous and typical paradoxes such as the ladder paradox, the Ehrenfest’s rotational disc paradox. The discussion shows that all the paradoxes are caused by misinterpretation of concepts. We misused the global simultaneity and the principle of relativity. As a geometry of Minkowski space-time, special relativity can never result in a logical contradiction.     

Smooth sample average approximation of stationary points in nonsmooth stochastic optimization and applications

Inspired by a recent work by Alexander et al. (J Bank Finance 30:583–605, 2006) which proposes a smoothing method to deal with nonsmoothness in a conditional value-at-risk problem, we consider a smoothing scheme for a general class of nonsmooth stochastic problems. Assuming that a smoothed problem is solved by a sample average approximation method, we investigate the convergence of stationary points of the smoothed sample average approximation problem as sample size increases and show that w.p.1 accumulation points of the stationary points of the approximation problem are weak stationary points of their counterparts of the true problem. Moreover, under some metric regularity conditions, we obtain an error bound on approximate stationary points. The convergence result is applied to a conditional value-at-risk problem and an inventory control problem.      

Nonparametric Estimation of Extreme Conditional Quantiles with Functional Covariate

Estimation of the extreme conditional quantiles with functional covariate is an important problem in quantile regression. The existing methods, however, are only applicable for heavy-tailed distributions with a positive conditional tail index. In this paper, we propose a new framework for estimating the extreme conditional quantiles with functional covariate that combines the nonparametric modeling techniques and extreme value theory systematically. Our proposed method is widely applicable, no matter whether the conditional distribution of a response variable Y given a vector of functional covariates X is short, light or heavy-tailed. It thus enriches the existing literature.     

On a generalization of the Dvoretzky–Wald–Wolfowitz theorem with an application to a robust optimization problem

A generalization of the Dvoretzky–Wald–Wolfowitz theorem to the case of conditional expectations is provided assuming that the σ-field on the state space has no conditional atoms. Neither continuity nor compactness assumptions are made on the model. An application to a robust (maxmin) optimization problem is given.     

Singularities in the dynamics of systems with non-ideal constraints

The fundamental problem of dynamics involving determining the generalised accelerations and reactions of constraints as a function of the applied forces is considered for mechanical systems with k ⩾ 1 non-ideal geometrical constraints. A relation is established between this problem and the analysis of the singularities of piecewise-smooth mappings of a space R^k into itself. For Coulomb-type friction, a criterion for there to be no paradoxes is obtained and it is shown that when k = 1 possible singularities are convolutions, while when k = 2 they are subdivided into a fold, a casp and a double fold. The well-known Painlevé-Klein example is considered in detail for cases of bilateral and unilateral contacts; a complete list of possible paradoxical situations is presented for the first time.     

Conditional Stability Estimates for Ill-Posed PDE Problems by Using Interpolation

The focus of this article is on conditional stability estimates for ill-posed inverse problems in partial differential equations. Conditional stability estimates have been obtained in related literature by a couple different methods. In this article, we propose a method called interpolation method, which is based on interpolation in variable Hilbert scales. We provide the theoretical background of this method and show that optimal conditional stability estimates are obtained. The capabilities of our method are illustrated by a comprehensive collection of different inverse and ill-posed PDE problems containing elliptic and parabolic problems, one source problem and the problem of analytic continuation.     

求解一种带有微分方程约束的条件变分问题的多目标最优化方法

Aiming at the conditional variational problem with the bi-objective functionals and the differential equational constraint in the optimal design of the electrostatic lenses, the conditional variational problem is transformed into multi-objective optimization problem by means of the spline function and the integral transformation. For solving the transformed problem, the analytic representation formula of the optimal solution about the original problem is obtained with regard to the voltage distribution and the electron trajectory. It will provide a new effective method for the design of the electrostatic lenses.     

Estimating a model through the conditional MLE

The estimation problem of a model through the conditional maximum likelihood estimator (MLE) is explored. The estimated model is compared using the two dual Kullback-Leibler losses with that through the unconditional MLE. The former is found to be superior to the latter under familiar models. This result is applicable to the model selection problem. These suggest a novel extensive use of the conditional likelihood, since the traditional use of the conditional likelihood was restricted only on inference for the structural parameter.     

Estimating functions for conditional inference: Many nuisance parameter case

When there is a complete sufficient statistic for the nuisance parameter which depends on the parameter of interest then there are locally optimal unbiased estimating functions, but generally there is no globally optimal estimating function. We consider conditioning on the minimal sufficient statistic for the nuisance parameter and find the conditional linear optimal unbiased estimating function. Since the nuisance parameter is totally eliminated in the conditional model there is no intrinsic problem in setting up conditional tests of significance and confidence intervals. A compromise between conditional and unconditional optimum estimating functions is suggested. The techniques are illustrated on three examples including the well known common means problem. The proposed hypothesis testing and confidence interval procedures work reasonably well for the examples considered.     

Paul Painlevé and his contribution to science

The life and career of the great French mathematician and politician Paul Painlevé is described. His contribution to the analytical theory of nonlinear differential equations was significant. The paper outlines the achievements of Paul Painlevé and his students in the investigation of an interesting class of nonlinear second-order equations and new equations defining a completely new class of special functions, now called the Painlevé transcendents. The contribution of Paul Painlevé to the study of algebraic nonintegrability of the N-body problem, his remarkable observations in mechanics, in particular, paradoxes arising in the dynamics of systems with friction, his attempt to create the axiomatics of mechanics and his contribution to gravitation theory are discussed.     

Solution of equations in Boolean algebra

The explicit form of solutions of Boolean equations with one unknown is obtained. The effectiveness of the method is demonstrated for a number of equations whose solution previously has been found only in “tabular” form. The proposed approach leads to a method for solving systems of equations in Boolean set algebra. We use it to analyze the famous paradoxes of set theory, such as the barber paradox and the liar paradox, as well as Russell's and Cantor's paradoxes. Translated from Nelineinaya Dinamika i Upravlenie, pp. 119–132, 1999.     

Optimal Control and Filtering of the Reproduction Law of a Branching Process

A finite horizon control problem for the reproduction law of a branching process is studied. Some examples with complete information are tackled via the Hamilton–Jacobi–Bellman equation. A partially observable control of the cardinality of the population using the information given by the splitting process is formulated. Though there is correlation between the state and the observations and the observation process has unbounded intensity, a Girsanov-type change of probability measure can be set and the filtering equation for the unnormalized conditional distribution (the Zakai equation) can be derived. Strong uniqueness for the Zakai equation and, as a consequence, also for the Kushner–Stratonovich equation is obtained. A separated control problem is introduced, in which the dynamics are represented by the splitting process and the unnormalized conditional distribution. By the strong uniqueness for the Zakai equation, equivalence between the partially observable control problem and the separated one is proved.     

Density Deconvolution of Different Conditional Distributions

Recently, a new technique to circumvent the ill-posedness of the deconvolution problem has been suggested. This technique is based on what is known as multi-channel convolution system. In this paper, we modify and develop this technique in order to adapt it for statistical use. We then apply it to the problem of estimation of deconvolution density in the case of different conditional densities. This method enables us to combine equations efficiently for any set of conditional densities and to construct estimators in cases where the characteristic functions of the conditional distributions vanish at some points, as it happens in the case of uniform and triangular distributions.     

An adaptive multiphase approach for large unconditional and conditional p-median problems

A multiphase approach that incorporates demand points aggregation, Variable Neighbourhood Search (VNS) and an exact method is proposed for the solution of large-scale unconditional and conditional p-median problems. The method consists of four phases. In the first phase several aggregated problems are solved with a “Local Search with Shaking” procedure to generate promising facility sites which are then used to solve a reduced problem in Phase 2 using VNS or an exact method. The new solution is then fed into an iterative learning process which tackles the aggregated problem (Phase 3). Phase 4 is a post optimisation phase applied to the original (disaggregated) problem. For the p-median problem, the method is tested on three types of datasets which consist of up to 89,600 demand points. The first two datasets are the BIRCH and the TSP datasets whereas the third is our newly geometrically constructed dataset that has guaranteed optimal solutions. The computational experiments show that the proposed approach produces very competitive results. The proposed approach is also adapted to cater for the conditional p-median problem with interesting results.     

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 information criteria for selecting variables in linear mixed models

In this paper, we consider the problem of selecting the variables of the fixed effects in the linear mixed models where the random effects are present and the observation vectors have been obtained from many clusters. As the variable selection procedure, here we use the Akaike Information Criterion, AIC. In the context of the mixed linear models, two kinds of AIC have been proposed: marginal AIC and conditional AIC. In this paper, we derive three versions of conditional AIC depending upon different estimators of the regression coefficients and the random effects. Through the simulation studies, it is shown that the proposed conditional AIC’s are superior to the marginal and conditional AIC’s proposed in the literature in the sense of selecting the true model. Finally, the results are extended to the case when the random effects in all the clusters are of the same dimension but have a common unknown covariance matrix.     

Asymptotic Expansion in Nonlinear Filtering with Homogenization

The problem of nonlinear filtering is studied for a class of diffusions whose statistics depend periodically on the state and a small parameter ε . Our purpose here is to show that, under some assumptions, the conditional density of the filtering problem admits an asymptotic expansion (see [2]). Accepted 9 September 1999