Existence of Probability Measures With Given Marginals

We show that if f is a probability density on Rn wrt Lebesgue measure (or any absolutely continuous measure) and 0 ≤ f ≤ 1, then there is another density g with only the values 0 and 1 and with the same (n−1)-dimensional marginals in any finite number of directions. This sharpens, unifies and extends the results of Lorentz and of Kellerer. Given a pair of independent random variables 0 ≤ X, Y ≤ 1, we further study functions 0 ≤ ϕ ≤ 1 such that Z = ϕ (X,Y) satisfies E(Z|X) = X and E(Z|Y) = Y. If there is a solution then there also is a nondecreasing solution ϕ(x,y). These results are applied to tomography and baseball

Inequalities and Positive-Definite Functions Arising From a Problem in Multidimensional Scaling

We solve the following variational problem: Find the maximum of E ∥ X−Y ∥ subject to E ∥ X ∥2 ≤ 1, where X and Y are i.i.d. random n-vectors, and ∥⋅∥ is the usual Euclidean norm on Rn. This problem arose from an investigation into multidimensional scaling, a data analytic method for visualizing proximity data. We show that the optimal X is unique and is (1) uniform on the surface of the unit sphere, for dimensions n ≥ 3, (2) circularly symmetric with a scaled version of the radial density ρ/(1−ρ2)1/2, 0 ≤ ρ ≤1, for n=2, and (3) uniform on an interval centered at the origin, for n=1 (Plackett\u27s theorem). By proving spherical symmetry of the solution, a reduction to a radial problem is achieved. The solution is then found using the Wiener-Hopf technique for (real) n \u3c 3. The results are reminiscent of classical potential theory, but they cannot be reduced to it. Along the way, we obtain results of independent interest: for any i.i.d. random n-vectors X and Y,E ∥ X−Y ∥ ≤ E ∥ X+Y ∥. Further, the kernel Kp, β(x,y) = ∥ x+y ∥βp− ∥x−y∥βp, x, y∈Rn and ∥ x ∥ p=(∑|xi|p)1/p, is positive-definite, that is, it is the covariance of a random field, Kp,β(x,y) = E [ Z(x)Z(y) ] for some real-valued random process Z(x), for 1 ≤ p ≤ 2 and 0 \u3c β ≤ p ≤ 2 (but not for β \u3ep or p\u3e2 in general). Although this is an easy consequence of known results, it appears to be new in a strict sense. In the radial problem, the average distance D(r1,r2) between two spheres of radii r1 and r2 is used as a kernel. We derive properties of D(r1,r2), including nonnegative definiteness on signed measures of zero integral

Some Problems in Probabilistic Tomography

Given probability distributions F1 , F2 , . . ., Fk on R and distinct directions θ1, . . ., θk, one may ask whether there is a probability measure μ on R2 such that the marginal of μ in direction θj is Fj, j = 1, . . ., k. For example for k = 3 we ask what the marginal of μ at 45° can be if the x and y marginals are each say standard normal? In probabilistic language, if X and Y are each standard normal with an arbitrary joint distribution, what can the distribution of X + Y or X - Y be? This type of question is familiar to probabilists and is also familiar (except perhaps in that μ is positive) to tomographers, but is difficult to answer in special cases. The set of distributions for Z = X - Y is a convex and compact set, C, which contains the single point mass Z ≡ 0 since X ≡ Y, standard normal, is possible. We show that Z can be 3-valued, Z=0, ±a for any a, each with positive probability, but Z cannot have any (genuine) two-point distribution. Using numerical linear programming we present convincing evidence that Z can be uniform on the interval [-ε, ε] for ε small and give estimates for the largest such ε. The set of all extreme points of C seems impossible to determine explicitly. We also consider the more basic question of finding the extreme measures on the unit square with uniform marginals on both coordinates, and show that not every such measure has a support which has only one point on each horizontal or vertical line, which seems surprising

Review of measures of worksite environmental and policy supports for physical activity and healthy eating

INTRODUCTION: Obesity prevention strategies are needed that target multiple settings, including the worksite. The objective of this study was to assess the state of science concerning available measures of worksite environmental and policy supports for physical activity (PA) and healthy eating (HE). METHODS: We searched multiple databases for instruments used to assess worksite environments and policies. Two commonly cited instruments developed by state public health departments were also included. Studies that were published from 1991 through 2013 in peer-reviewed publications and gray literature that discussed the development or use of these instruments were analyzed. Instrument administration mode and measurement properties were documented. Items were classified by general health topic, 5 domains of general worksite strategy, and 19 subdomains of worksite strategy specific to PA or HE. Characteristics of worksite measures were described including measurement properties, length, and administration mode, as well as frequencies of items by domain and subdomain. RESULTS: Seventeen instruments met inclusion criteria (9 employee surveys, 5 manager surveys, 1 observational assessment, and 2 studies that used multiple administration modes). Fourteen instruments included reliability testing. More items were related to PA than HE. Most instruments (n = 10) lacked items in the internal social environment domain. The most common PA subdomains were exercise facilities and lockers/showers; the most common HE subdomain was healthy options/vending. CONCLUSION: This review highlights gaps in measurement of the worksite social environment. The findings provide a useful resource for researchers and practitioners and should inform future instrument development

The long-term effects of invasive signal crayfish (Pacifastacus leniusculus) on instream macroinvertebrate communities

Non-native species represent a significant threat to indigenous biodiversity and ecosystem functioning worldwide. It is widely acknowledged that invasive crayfish species may be instrumental in modifying benthic invertebrate community structure, but there is limited knowledge regarding the temporal and spatial extent of these effects within lotic ecosystems. This study investigates the long term changes to benthic macroinvertebrate community composition following the invasion of signal crayfish, Pacifastacus leniusculus, into English rivers. Data from long-term monitoring sites on 7 rivers invaded by crayfish and 7 rivers where signal crayfish were absent throughout the record (control sites) were used to examine how invertebrate community composition and populations of individual taxa changed as a result of invasion. Following the detection of non-native crayfish, significant shifts in invertebrate community composition were observed at invaded sites compared to control sites. This pattern was strongest during autumn months but was also evident during spring surveys. The observed shifts in community composition following invasion were associated with reductions in the occurrence of ubiquitous Hirudinea species (Glossiphonia complanata and Erpobdella octoculata), Gastropoda (Radix spp.), Ephemeroptera (Caenis spp.), and Trichoptera (Hydropsyche spp.); although variations in specific taxa affected were evident between regions and seasons. Changes in community structure were persistent over time with no evidence of recovery, suggesting that crayfish invasions represent significant perturbations leading to permanent changes in benthic communities. The results provide fundamental knowledge regarding non-native crayfish invasions of lotic ecosystems required for the development of future management strategies

The Hilbertian Tensor Norm and Entangled Two-Prover Games

We study tensor norms over Banach spaces and their relations to quantum information theory, in particular their connection with two-prover games. We consider a version of the Hilbertian tensor norm $\gamma_2$ and its dual $\gamma_2^*$ that allow us to consider games with arbitrary output alphabet sizes. We establish direct-product theorems and prove a generalized Grothendieck inequality for these tensor norms. Furthermore, we investigate the connection between the Hilbertian tensor norm and the set of quantum probability distributions, and show two applications to quantum information theory: firstly, we give an alternative proof of the perfect parallel repetition theorem for entangled XOR games; and secondly, we prove a new upper bound on the ratio between the entangled and the classical value of two-prover games.Comment: 33 pages, some of the results have been obtained independently in arXiv:1007.3043v2, v2: an error in Theorem 4 has been corrected; Section 6 rewritten, v3: completely rewritten in order to improve readability; title changed; references added; published versio

Attacker Control and Impact for Confidentiality and Integrity

Language-based information flow methods offer a principled way to enforce strong security properties, but enforcing noninterference is too inflexible for realistic applications. Security-typed languages have therefore introduced declassification mechanisms for relaxing confidentiality policies, and endorsement mechanisms for relaxing integrity policies. However, a continuing challenge has been to define what security is guaranteed when such mechanisms are used. This paper presents a new semantic framework for expressing security policies for declassification and endorsement in a language-based setting. The key insight is that security can be characterized in terms of the influence that declassification and endorsement allow to the attacker. The new framework introduces two notions of security to describe the influence of the attacker. Attacker control defines what the attacker is able to learn from observable effects of this code; attacker impact captures the attacker's influence on trusted locations. This approach yields novel security conditions for checked endorsements and robust integrity. The framework is flexible enough to recover and to improve on the previously introduced notions of robustness and qualified robustness. Further, the new security conditions can be soundly enforced by a security type system. The applicability and enforcement of the new policies is illustrated through various examples, including data sanitization and authentication