Set Of Probability Distributions Research Articles

On an elementary characterization of the increasing convex ordering, by an application

In this short note, we present a simple characterization of the increasing convex ordering on the set of probability distributions on ℝ. We show its usefulness by providing a very short proof of a comparison result for M/GI/1 queues due to Daley and Rolski, and obtained by completely different means.

On an elementary characterization of the increasing convex ordering, by an application

In this short note, we present a simple characterization of the increasing convex ordering on the set of probability distributions on ℝ. We show its usefulness by providing a very short proof of a comparison result for M/GI/1 queues due to Daley and Rolski, and obtained by completely different means.

Memorial: Nikolai Nikolaevitch Chentsov

Memorial: Nikolai Nikolaevitch Chentsov

Efficient inference in Bayes networks as a combinatorial optimization problem

A number of exact algorithms have been developed in recent years to perform probabilistic inference in Bayesian belief networks. The techniques used in these algorithms are closely related to network structures, and some of them are not easy to understand and implement. We consider the problem from the combinatorial optimization point of view and state that efficient probabilistic inference in a belief network is a problem of finding an optimal factoring given a set of probability distributions. From this viewpoint, previously developed algorithms can be seen as alternative factoring strategies. In this paper, we define a combinatorial optimization problem, the optimal factoring problem, and discuss application of this problem in belief networks. We show that optimal factoring provides insight into the key elements of efficient probabilistic inference, and demonstrate simple, easily implemented algorithms with excellent performance.

Instruction window size trade-offs and characterization of program parallelism

Detecting independent operations is a prime objective for computers that are capable of issuing and executing multiple operations simultaneously. The number of instructions that are simultaneously examined for detecting those that are independent is the scope of concurrency detection. The authors present an analytical model for predicting the performance impact of varying the scope of concurrency detection as a function of available resources, such as number of pipelines in a superscalar architecture. The model developed can show where a performance bottleneck might be: insufficient resources to exploit discovered parallelism, insufficient instruction stream parallelism, or insufficient scope of concurrency detection. The cost associated with speculative execution is examined via a set of probability distributions that characterize the inherent parallelism in the instruction stream. These results were derived using traces from a Multiflow TRACE SCHEDULING compacting FORTRAN 77 and C compilers. The experiments provide misprediction delay estimates for 11 common application-level benchmarks under scope constraints, assuming speculative, out-of-order execution and run time scheduling. The throughput prediction of the analytical model is shown to be close to the measured static throughput of the compiler output. >

Non-homogeneous de Finetti-type theorems

De Finetti-type theorems in a more general sense state that a certain convex set of probability distributions on an infinite product space, defined by symmetry conditions, is a simplex, and give a description, of its extreme points. In the classical examples these are homogeneous product measures. Motivated by work of Diaconiset al.(4) we describe a general method to obtain certain nonhomogeneous product measures as extreme solutions to the defining symmetry equations, and give examples.

Foundations for a robust theory of decision making: the simple case

We provide axiomatic foundations to model imprecision in a subject’s preferences and beliefs through a set of probability distributions and a set of utility functions, in the simple case.

Stochastic dominance under Bayesian learning

The concept of first-order stochastic dominance defined on distributions is inadequate in models with learning. We extend this concept to the space of distributions on distributions. We discuss conditions under which for all common observations one person's beliefs (over a set of probability distributions) dominate another person's beliefs by first-order stochastic dominance. We obtain sufficient conditions for this partial order and show that the sufficient conditions are necessary, provided that the underlying distributions satisfy an additional assumption. These conditions can be verified without taking any observations. Applications are discussed.

Cyclic symmetry on cyclic and noncyclic sets of probability distributions and the entropies of degree α

Abstract Total symmetry is an important algebraic property of entropy. It can be weakened in the strict sense in several ways. The concept of cyclic symmetry, introduced by Nath and Kaur (1981), is a strictly weaker form of it. They formulated it on Γn, the set of all n-component probability distributions with nonnegative elements, and studied it only in connection with the Shannon entropy (1948). In this paper, the authors have studied the concept of cyclic symmetry for entropies of degree α (α>0) by formulating it on Γn and its cyclic and noncyclic subsets. Four equivalence theorems, based on these ideas, have been established.

The infinitely-many-neutral-alleles diffusion model by ages

We discuss two formulations of the infinitely-many-neutral-alleles diffusion model that can be used to study the ages of alleles. The first one, which was introduced elsewhere, assumes values in the set of probability distributions on the set of alleles, and the ages of the alleles can be inferred from its sample paths. We illustrate this approach by proving a result of Watterson and Guess regarding the probability that the most frequent allele is oldest. The second diffusion model, which is new, assumes values in the set of probability distributions on the set of pairs (x, a), where x is an allele and a is its age. We illustrate this second approach by proving an extension of the Ewens sampling formula to age-ordered samples due to Donnelly and Tavaré.

The infinitely-many-neutral-alleles diffusion model by ages

We discuss two formulations of the infinitely-many-neutral-alleles diffusion model that can be used to study the ages of alleles. The first one, which was introduced elsewhere, assumes values in the set of probability distributions on the set of alleles, and the ages of the alleles can be inferred from its sample paths. We illustrate this approach by proving a result of Watterson and Guess regarding the probability that the most frequent allele is oldest. The second diffusion model, which is new, assumes values in the set of probability distributions on the set of pairs (x, a), where x is an allele and a is its age. We illustrate this second approach by proving an extension of the Ewens sampling formula to age-ordered samples due to Donnelly and Tavaré.

Risk aversion in the theory of expected utility with rank dependent probabilities

Expected utility with rank dependent probability theory is a model of decision-making under risk where the preference relations on the set of probability distributions is represented by the mathematical expectation of a utility function with respect to a transformation of the probability distributions on the set of outcomes. This paper defines, based on Gâteaux differentiability, measures of risk aversion for such preferences which characterize the relation “more risk averse” and applies these measures to the analysis of unconditional and conditional portfolio choice problems.

Uniformity and distance—a vivid example from statistics

In this paper we consider a uniformity on the set of probability distributions which is based on the vivid notion of the risk set of a testing problem.

Should Scoring Rules be “Effective”?

A scoring rule is a reward function for eliciting or evaluating forecasts expressed as discrete or continuous probability distributions. A rule is strictly proper if it encourages the forecaster to state his true subjective probabilities, and effective if it is associated with a metric on the set of probability distributions. Recently, the property of effectiveness (which is stronger than strict properness) has been proposed as a desideratum for scoring rules for continuous forecasts, for reasons of “monotonicity” in keeping the forecaster close to his true probabilities, since in practice the forecast must be chosen from a low-dimensional set of “admissible” distributions. It is shown in this paper that what effectiveness implies, beyond strict properness, is not a monotonicity property but a transitivity property, which is difficult to justify behaviorally. The logarithmic scoring rule is shown to violate the transitivity property, and hence is not effective. The L1 and L∞ metrics are shown to allow no effective scoring rules. Some potential difficulties in interpreting admissible forecasts are also discussed.

Stochastic Analysis of Groundwater Flow and Contaminant Transport in a fractured Rock System

ABSTRACTIt has been possible to develop a stochastic model for groundwater flow and mass transport in a fractured rock system. A large number of statistically independent realizations of a fracture network are generated from a set of probability distributions for parameters defining the fracture geometry. By solving for hydraulic head at the fracture intersections and using data on apertures and porosities, seepage velocities may be calculated and the transport equation solved for each trial of a Monte Carlo simulation. Model output consists of distributions of moving particles and various mass exit times. Applications illustrate the types of model results and the skewed character of particle distributions.

A New Stochastic Time Optimal Control Problem

Consider a system described by the stochastic differential equation $dx = f(t,x,u)dt + \sigma (t,x)dw$ where w is a Wiener process and u lies in a compact set. If solutions are defined in the sense of Girsanov rather than of Itô, then it is natural to work not with the state at time t but rather with the distribution of $x(t)$. It is shown by an extension of the standard argument that if the state reaches a target set, i.e. a specified set of probability distributions, in finite time then there is a first such time. Next it is shown that if the system is linear in u and if $ - 1 \leqq u \leqq 1$, then the attainable distributions arising from bang-bang controls are weakly dense in the set of attainable distributions. Finally some geometric properties of the bang-bang attainable densities are discussed; for example, the exposed points of the attainable densities are the bang-bang attainable densities.

Axioms for expected utility inn-person games

The von Neumann and Morgenstern utility axioms apply to an individual's preferences on a set of probability distributions that is closed under convex linear combinations. However, in a rectangular game situation, the probability distributions on outcomes that correspond to choices of mixed strategies for the players is not generally closed under convex combinations. Axioms are specified for an individual's preference relation on the space of mixed strategies for then players. These axioms are necessary and sufficient for a multilinear expected utility model that reduces to the von Neumann-Morgenstern form. The utility function is unique up to a positive affine transformation.

A Bayesian Analysis of Some Nonparametric Problems

A Bayesian Analysis of Some Nonparametric Problems

Semiorders and risky choices

If a preference relation on a set is a semiorder then the indifference relation is not necessarily transitive. However, if a preference relation on a set of probability distributions is a semiorder and satisfies a simple sure-thing axiom, then indifference is transitive. Modifications of preference theory under risk that preserve intransitive indifference are suggested.

Some Methodological Problems in the Use of the Neyman Type A and the Negative Binomial Probability Distributions for the Analysis of Spatial Point Patterns

PROBABILITY THEORY provides us with a powerful abstract logical system which we may use as a model for describing and analysing geographical distributions. The abstract formulation given to the mathematical theory of probability by A. Kolmogorov (I933) and subsequent axiomatic treatments (such asJ. M. McCord and R. M. Moroney, I964) have served to increase the power and rigour of the mathematical theory. Nevertheless, the initial development of the theory relied very heavily upon the logical problems inherent in the analysis of games of chance. Mathematicians still rely upon simple 'games' or 'urn models', with rules and conditions fully specified, to give an intuitive meaning to the probability distributions which they derive. W. Feller's (I950) treatment of probability theory is an excellent example of the use of such simple games and urn models to facilitate understanding. But the range of problems which has been effectively tackled using the mathematical theory of probability is very much more extensive than would be implied by the relatively simple coin-tossing or card-dealing experiments used to help identify probability distributions. The mathematical theory of probability is, therefore, a neat demonstration of the point made by C. H. Coombs, H. Raiffa and R. M. Thrall (1954) that the mathematician, primarily concerned with the properties of an abstract theoretical system, requires some kind of simple concrete model to help interpret that abstract system, whereas the social scientist, primarily concerned with empirical problems, looks to the abstract mathematical system to provide a model into which he can 'map' difficult empirical problems so as to facilitate analysis. The basic methodological difficulty which faces the geographer is to establish criteria for judging the validity of this 'mapping' process. Only when such criteria are firmly established may we judge whether the use of a particular mathematical model (be it a system of differential equations, a set of probability distributions, a system of simultaneous equations, a recursive cause-and-effect model, and so on) is appropriate to the particular empirical problem under investigation and, further, judge whether the inferences established by way of the mathematical model are truly representative of the phenomena being observed. The general methodological problem of applying a priori analytical knowledge such as that contained in pure mathematical systems, to a posteriori synthetic empirical problems has exercised both philosophers and empirically minded scientists (S. K6rner, I960;J. S. Coleman, 1964). It is the intention of this paper to examine a particular case of this general methodological problem, namely, the analysis of spatial point patterns by quadrat sampling procedures.