Pairwise Independent Research Articles

BMT independence

We introduce the notion of BMT independence, enabling the study of arbitrary mixtures of Boolean, monotone, and tensor independence and generalizing the notion of BM independence of J. Wysoczanski. pairwise independence relations are encoded through a directed graph, which in turn determines the way mixed moments must be computed. Corresponding Central and Poisson-Type Limit Theorems are provided along with an explicit construction to realize BMT independent random variables as bounded operators on the tensor product of Hilbert spaces.

Статистичний критерій перевірки незалежності внутрішніх станів та виходів криптопримітиву, що генерує (псевдо)випадкові послідовності

The article is devoted to creation and justification of new statistical criterion of pairwise independence of binary sequences from given set, which are considered as realization of random variables. The corresponding algorithm, which fulfills the checking of pairwise independence, is formulated in details. This algorithm is necessary tool for statistical veri-fication of cryptographic quality of different cryptoprimitives, which functioning is con-nected with random/ pseudorandom sequences generation — such as ran-dom/pseudorandom sequences generators or stream ciphers. Usage of the obtained criteri-on allows independence checking not only for output sequences, but also for its interme-diate state or inputs. Note that such independence is necessary for unpredictability of out-put sequences.

Almost pairwise independence and resilience to deep learning attacks

Almost pairwise independence (API) is a quantitative property of a class of functions that is desirable in many cryptographic applications. This property is satisfied by Learning with errors (LWE)-mappings and by special Substitution-Permutation Networks (SPN). API block ciphers are known to be resilient to differential and linear cryptanalysis attacks. Recently, security of protocols against neural network-based attacks became a major trend in cryptographic studies. Therefore, it is relevant to study the hardness of learning a target function from an API class of functions by gradient-based methods. We propose a theoretical analysis based on the study of the variance of the gradient of a general machine learning objective with respect to a random choice of target function from a class. We prove an upper bound and verify that, indeed, such a variance is extremely small for API classes of functions. This implies the resilience of actual LWE-based primitives against deep learning attacks, and to some extent, the security of SPNs. The hardness of learning reveals itself in the form of the barren plateau phenomenon during the training process, or in other words, in a low information content of the gradient about the target function. Yet, we emphasize that our bounds hold for the case of a regular parameterization of a neural network and the gradient may become informative if a class is mildly pairwise independent and a parameterization is non-regular. We demonstrate our theory in experiments on the learnability of LWE mappings.

On connectivity in random graph models with limited dependencies

AbstractWe consider random graph models in which the events describing the inclusion of potential edges have to be independent of each other if the corresponding edges are non‐adjacent and ask: what is the minimum probability , such that for any distribution (in this model) on graphs with vertices in which each potential edge has a marginal probability of being present at least , a graph drawn from is connected with non‐zero probability? The answer to this question is sensitive to the formalization of the independence condition. We introduce a strict hierarchy of five conditions, which give rise to at least three different functions . For each condition, we provide upper and lower bounds for . For the strongest condition, the coloring model, we show that for . In contrast, for the weakest condition, pairwise independence, we show that lies within of the threshold for completely arbitrary distributions.

A note on the exponentiation approximation of the birthday paradox

This note sheds new light on the exponentiation approximation of the probability that all K individuals have distinct birthdays across N calendar days. The exponentiation approximation imposes a pairwise independence assumption, which does not hold in general. We sidestep this assumption by deriving the conditional probability for each pair of individuals to have distinct birthdays given that previous pairs do. An interesting implication is that the conditional probability decreases in a step-function form—not in a strictly monotonical form—as more pairs are restricted to have distinct birthdays. The source of the step-function structure is identified and illustrated. We also establish the equivalence between the pairwise approach and another common approach based on permutations of all individuals.

Tight Probability Bounds with Pairwise Independence

Tight Probability Bounds with Pairwise Independence

Asymptotic behavior of tail distortion risk measure for aggregate weight-adjusted losses

<p style='text-indent:20px;'>Consider a portfolio of <inline-formula><tex-math id="M1">\begin{document}$ n $\end{document}</tex-math></inline-formula> losses accompanied with <inline-formula><tex-math id="M2">\begin{document}$ n $\end{document}</tex-math></inline-formula> stochastic loss adjustment factors. This paper establishes some asymptotic formulas of the tail distortion risk measure for aggregate weight-adjusted heavy-tailed losses under the framework of multivariate regular variation, pairwise quasi-asymptotic independence or arbitrary dependence. As an application, the corresponding results on the asymptotics for the risk concentration based on tail distortion risk measure are also derived. Several examples and simulation studies are provided to better illustrate the obtained results.</p>

Limit Theorems for Deviation Means of Independent and Identically Distributed Random Variables

We derive a strong law of large numbers, a central limit theorem, a law of the iterated logarithm and a large deviation theorem for so-called deviation means of independent and identically distributed random variables. (For the strong law of large numbers, we suppose only pairwise independence instead of (total) independence.) The class of deviation means is a special class of M-estimators or more generally extremum estimators, which are well studied in statistics. The assumptions of our limit theorems for deviation means seem to be new and weaker than the known ones for M-estimators in the literature. In particular, our results on the strong law of large numbers and on the central limit theorem generalize the corresponding ones for quasi-arithmetic means due to de Carvalho (Am Stat 70(3):270–274, 2016) and the ones for Bajraktarević means due to Barczy and Burai (Aequ Math 96(2):279–305, 2022).

Validation of Trauma-Informed Care Instruments: Emergency Department Environment and Transitional Secondary Environment.

Trauma-informed care is a paradigm of care that requires health care practitioners to understand multiple types of traumas and their effects on the trauma survivor and then incorporate that knowledge into practice. However, there are few psychometrically robust instruments to evaluate trauma-informed care, and none have been applied in the trauma patient setting. The purpose of this article is to validate two trauma-informed care instruments in the trauma patient setting. Exploratory factor analysis and simultaneous pairwise marginal independence testing procedures were conducted on the "Emergency Department Environment" and the "Transitional Secondary Environment" instruments from September 2020 to November 2020. Descriptive statistics were reported for the content experts participating in the instrument validation. Exploratory data analysis results for each trauma-informed care statement in the tools indicate multidimensionality of trauma-informed care core values, with statistically significant (p < .05) overlap between two or more trauma-informed care core values. After accounting for item interdependence, the associations between the trauma-informed care core values do not appear to be spurious consequences of response interdependence. The two instruments showed high levels of internal consistency supporting the multidimensional models posited by the trauma-informed care framework. The findings will allow for routine monitoring and early detection of gaps in health care provider behaviors in the individualized care of trauma survivors. This will enable identification of trends in trauma care delivery and inform trauma-informed care education for health care providers, ultimately enhancing the healing of trauma survivors.

Marginal Distance and Hilbert-Schmidt Covariances-Based Independence Tests for Multivariate Functional Data

We study the pairwise and mutual independence testing problem for multivariate functional data. Using a basis representation of functional data, we reduce this problem to testing the independence of multivariate data, which may be high-dimensional. For pairwise independence, we apply tests based on distance and Hilbert-Schmidt covariances as well as their marginal versions, which aggregate these covariances for coordinates of random processes. In the case of mutual independence, we study asymmetric and symmetric aggregating measures of pairwise dependence. A theoretical justification of the test procedures is established. In extensive simulation studies and examples based on a real economic data set, we investigate and compare the performance of the tests in terms of size control and power. An important finding is that tests based on distance and Hilbert-Schmidt covariances are usually more powerful than their marginal versions under linear dependence, while the reverse is true under non-linear dependence.

Metacommunity dynamics and the detection of species associations in co‐occurrence analyses: Why patch disturbance matters

Abstract Many statistical methods attempt to detect species associations—and so infer interspecific interactions—from species co‐occurrence patterns. Habitat heterogeneity and out‐of‐equilibrium colonization histories are well recognized as potentially causing species associations, even when interactions are absent. The potential for patch disturbance, a classical component of metacommunity dynamics, to also drive spurious species associations has however been overlooked. Using a new general metacommunity model, we derive mathematical predictions regarding how patch disturbance would affect the patterns of species associations detected in ‘null’ co‐occurrence matrices. We also conduct numerical simulations to test our predictions and to compare the performance of several widespread statistical methods, including direct tests of pairwise independence, matrix permutation approaches and joint species distribution modelling. We show how classical metacommunity dynamics can produce statistical associations, both positive and negative, even when species do not interact, when there is no habitat heterogeneity, and at equilibrium. This occurs as soon as there is some rate of patch disturbance (i.e. simultaneous extinction of several species in a patch) and/or a finite life span of patches, a common feature of a broad range of plant, animal or microbial systems. Patch disturbance can compromise species co‐occurrence analyses and cause the artefactual detection of species associations if not taken into account. Including patch age (i.e. the time since the last patch disturbance event) as a covariate in a joint species distribution model can resolve the artefact. However, this requires additional data that often are not available in practice. We argue that the consequences of patch disturbance should not be underestimated when analysing species distribution patterns in metacommunity‐like systems. Read the free Plain Language Summary for this article on the Journal blog.

Pairwise Independence May Not Imply Independence: New Illustrations and a Generalization

A number of standard textbooks that are followed in a junior/senior level course or in a first-year graduate level course in mathematical statistics and probability, routinely include one single basic illustration, obviously in its variant forms, to highlight an important point: pairwise independence may not imply (mutual) independence. We earnestly believe that beginning students appreciate more examples to clarify these key issues. Hence, we hope that our new sets of nontrivial illustrations from Section 2 will help our audience. Next, in Section 3, we extend the notion to q-wise independence with a large set of illustrations using both discrete and continuous random variables showing that q-wise independence may not imply (mutual) independence. We reasonably assure that this discourse is immediately accessible to juniors/seniors and first-year graduate students.

Weak independence of events and the converse of the Borel–Cantelli Lemma

The converse of the Borel–Cantelli Lemma states that if {Ai}i=1∞ is a sequence of independent events such that ∑P(Ai)=∞, then almost surely infinitely many of these events will occur. Erdős and Rényi proved that it is sufficient to weaken the condition of independence to pairwise independence. Later, several other weakenings of the condition appeared in the literature. The aim of this paper is to provide a collection of conditions, all of which imply that almost surely infinitely many of the events occur, and determine the complete implicational relationship between them.Many of these results are known, or follow from known results, however, they are not widely known among non-specialists. Yet, the results can be extremely useful for areas outside of probability theory, as evidenced by the original motivation of this paper emerging from infinite combinatorics. Our proofs are aimed to be accessible to a general mathematical audience.

Comment on “Strong Quantum Darwinism and Strong Independence Are Equivalent to Spectrum Broadcast Structure”

In a recent Letter [Phys. Rev. Lett. 122, 010403 (2019)], an equivalence is proposed between the so-called Spectrum Broadcast Structure for a system-multienvironment quantum state, and the conjunction of two information-theory notions: (a) Strong Quantum Darwinism; and (b) Strong Independence. Here, we show that the mathematical formulation of condition (b) by the authors (namely, the pairwise independence of the fragments of the environment, conditioned on the system), is necessary but not sufficient to ensure the equivalence. We propose a simple counter-example, together with a strengthened formulation of condition (b), ensuring the equivalence proposed by the authors.

A counterexample to the existence of a general central limit theorem for pairwise independent identically distributed random variables

The classical Central Limit Theorem (CLT) is one of the most fundamental results in statistics. It states that the standardized sample mean of a sequence of n mutually independent and identically distributed random variables with finite second moment converges in distribution to a standard Gaussian as n goes to infinity. In particular, pairwise independence of the sequence is generally not sufficient for the theorem to hold. We construct explicitly such a sequence of pairwise independent random variables having a common but arbitrary marginal distribution F (satisfying very mild conditions) and for which no CLT holds. We obtain, in closed form, the asymptotic distribution of the sample mean of our sequence, and find it is asymmetrical for any F. This is illustrated through several theoretical examples for various choices of F. Associated R codes are provided in a supplementary appendix online.

Modelling association between two Multiple Response Categorical Variables (MRCV) using generalized log-linear models: the case of urban agriculture in Mutare City

Purpose: This research aimed to apply log-linear modelling to model association between multiple response categorical variables (MRCV) on urban agriculture and enhance data analysis of the paper by Basera, Chakaipa, &amp; Dube (2020) impetus of urban agriculture on open spaces of Mutare City. Research methodology: The research data was obtained from households and farmers in Mutare City - urban and peri-urban (inclusive of plots in Weirmouth Park and Fern Valley area in December 2020. A total of one hundred and fifteen (115) household farmers were surveyed. Results: Simultaneous Pairwise Marginal Independence (SPMI) tests revealed the presence of associations. Log-linear tests revealed a perfect fit based on small standardized Pearson residuals and a strong positive association based on observed and model-predicted odds ratios on-field agricultural activities and use of herbicides. Log-linear and further application of heterogeneity tests revealed partial and near no perfect fit in other pairs of MRCVs with a strong negative association between municipality vacant places and field agricultural activities. Limitations: The research could not carry out log-linear model associations of three or more MRCVs because files exceeded 2GB in memory on both MI.test () function for SPMI tests and genloglin regressions. Contribution: The study contributes to urban agriculture planning especially in enactment of urban agriculture laws, agriculture one stop shop business centers housing farm input supply shops, farm produce shops, and determining fit support that can be rendered to urban farmers. Keywords: Multiple Response Categorical Variables (MRCV), Association, Urban agriculture

High-dimensional consistent independence testing with maxima of rank correlations

Testing mutual independence for high-dimensional observations is a fundamental statistical challenge. Popular tests based on linear and simple rank correlations are known to be incapable of detecting nonlinear, nonmonotone relationships, calling for methods that can account for such dependences. To address this challenge, we propose a family of tests that are constructed using maxima of pairwise rank correlations that permit consistent assessment of pairwise independence. Built upon a newly developed Cramér-type moderate deviation theorem for degenerate U-statistics, our results cover a variety of rank correlations including Hoeffding’s $D$, Blum–Kiefer–Rosenblatt’s $R$ and Bergsma–Dassios–Yanagimoto’s $\tau^{*}$. The proposed tests are distribution-free in the class of multivariate distributions with continuous margins, implementable without the need for permutation, and are shown to be rate-optimal against sparse alternatives under the Gaussian copula model. As a by-product of the study, we reveal an identity between the aforementioned three rank correlation statistics, and hence make a step towards proving a conjecture of Bergsma and Dassios.

The one-way Fubini property and conditional independence: An equivalence result

A process defined by a continuum of random variables with non-degenerate idiosyncratic risk is not jointly measurable with respect to the usual product σ-algebra. We show that the process is jointly measurable in a one-way Fubini extension of the product space if and only if there is a countably generated σ-algebra given which the random variables are essentially pairwise conditionally independent, while their conditional distributions also satisfy a suitable joint measurability condition. Applications include: (i) new characterizations of essential pairwise independence and essential pairwise exchangeability; (ii) when a one-way Fubini extension exists, the need for the sample space to be saturated if there is an essentially random regular conditional distribution with respect to the usual product σ-algebra.

Approximation theorems for spaces of localities

The classical Artin–Whaples approximation theorem allows to simultaneously approximate finitely many different elements of a field with respect to finitely many pairwise inequivalent absolute values. Several variants and generalizations exist, for example for finitely many (Krull) valuations, where one usually requires that these are independent, i.e. induce different topologies on the field. Ribenboim proved a generalization for finitely many valuations where the condition of independence is relaxed for a natural compatibility condition, and Ershov proved a statement about simultaneously approximating finitely many different elements with respect to finitely many possibly infinite sets of pairwise independent valuations. We prove approximation theorems for infinite sets of valuations and orderings without requiring pairwise independence.

Topology-Based Machine Learning Strategy for Cluster Structure Prediction.

In cluster physics, the determination of the ground-state structure of medium-sized and large-sized clusters is a challenge due to the number of local minimal values on the potential energy surface growing exponentially with cluster size. Although machine learning approaches have had much success in materials sciences, their applications in clusters are often hindered by the geometric complexity clusters. Persistent homology provides a new topological strategy to simplify geometric complexity while retaining important chemical and physical information without having to "downgrade" the original data. We further propose persistent pairwise independence (PPI) to enhance the predictive power of persistent homology. We construct topology-based machine learning models to reveal hidden structure-energy relationships in lithium (Li) clusters. We integrate the topology-based machine learning models, a particle swarm optimization algorithm, and density functional theory calculations to accelerate the search of the globally stable structure of clusters.