Unique Probability Research Articles

Adaptive invariant density estimation for ergodic diffusions over anisotropic classes

Consider some multivariate diffusion process $\mathbf{X}=(X_{t})_{t\geq0}$ with unique invariant probability measure and associated invariant density $\rho$, and assume that a continuous record of observations $X^{T}=(X_{t})_{0\leq t\leq T}$ of $\mathbf{X}$ is available. Recent results on functional inequalities for symmetric Markov semigroups are used in the statistical analysis of kernel estimators $\widehat{\rho}_{T}=\widehat{\rho}_{T}(X^{T})$ of $\rho$. For the basic problem of estimation with respect to $\mathrm{sup}$-norm risk under anisotropic Holder smoothness constraints, the proposed approach yields an adaptive estimator which converges at a substantially faster rate than in standard multivariate density estimation from i.i.d. observations.

Functional Trajectories Before and After Major Surgery in Older Adults.

We hypothesized that distinct sets of functional trajectories can be identified in the year before and after major surgery, with unique transition probabilities from pre to postsurgical functional trajectories, and that outcomes would be better among participants undergoing elective versus nonelective surgery. Major surgery is common and can be highly morbid in older persons. The relationship between the course of disability (ie, functional trajectory) before and after surgery in older adults has not been well-studied for most operations. Prospective cohort study of 754 community-living persons 70 years or older. The analytic sample included 250 participants who underwent their first major surgery during the study period. Before surgery, 4 functional trajectories were identified: no disability (n = 60, 24.0%), and mild (n = 84, 33.6%), moderate (n = 73, 29.2%), and severe (n = 33, 13.2%) disability. After surgery, 4 functional trajectories were identified: rapid (n = 39, 15.6%), gradual (n = 76, 30.4%), partial (n = 70, 28.0%), and little (n = 57, 22.8%) improvement. Rapid improvement was seen for n = 31 (51.7%) participants with no disability before surgery, but was uncommon among those with mild disability (n = 8, 9.5%) and was not observed in the moderate and severe trajectory groups. For participants with mild to moderate disability before surgery, gradual improvement (n = 46, 54.8%) and partial improvement (n = 36, 49.3%) were most common. Most participants with severe disability (n = 27, 81.8%) before surgery exhibited little improvement. Outcomes were better for participants undergoing elective versus nonelective surgery. Functional prognosis in the year after major surgery is highly dependent on premorbid function.

Detection and differentiation of application layer DDoS attack from flash events using fuzzy‐GA computation

Distributed Denial-of-Service (DDoS) attacks are serious threats in the data center application, mainly affecting the web server. Even though there are various techniques to detect and mitigate such attacks so far they fail to meet in the case of application layer attack and Flash Events (FE). In the paper, we aim at detecting application layer DDoS attacks and distinguish it from FE. We have considered a DDoS attack model and selected the parameters in the incoming packets that correspond in causing the attack. Based on the attack model we have analysed the statistical parameters of the incoming packets such as inter-arrival time, the probability of uniqueness of an IP address in given time frame and the unavailability of HTTP (Hyper Text Transfer Protocol) GET acknowledgment bit in the header field. These parameters are the input to the Fuzzy classification model. We have used Genetic Algorithm (GA) to provide an optimised value range for the input parameters. The optimised values are now applied to Fuzzy logic to identify whether the web accessing clients shows the behavior of attack, normal or FE. The experimental results show that Fuzzy-GA model provides an accuracy of 98.4% in detecting DDoS attack and 97.3% in detecting FE..

Large deviations for randomly connected neural networks: II. State-dependent interactions

Abstract We continue the analysis of large deviations for randomly connected neural networks used as models of the brain. The originality of the model relies on the fact that the directed impact of one particle onto another depends on the state of both particles, and they have random Gaussian amplitude with mean and variance scaling as the inverse of the network size. Similarly to the spatially extended case (see Cabana and Touboul (2018)), we show that under sufficient regularity assumptions, the empirical measure satisfies a large deviations principle with a good rate function achieving its minimum at a unique probability measure, implying, in particular, its convergence in both averaged and quenched cases, as well as a propagation of a chaos property (in the averaged case only). The class of model we consider notably includes a stochastic version of the Kuramoto model with random connections.

Persistence in Stochastic Lotka-Volterra Food Chains with Intraspecific Competition.

This paper is devoted to the analysis of a simple Lotka-Volterra food chain evolving in a stochastic environment. It can be seen as the companion paper of Hening and Nguyen (J Math Biol 76:697-754, 2018b) where we have characterized the persistence and extinction of such a food chain under the assumption that there is no intraspecific competition among predators. In the current paper, we focus on the case when all the species experience intracompetition. The food chain we analyze consists of one prey and [Formula: see text] predators. The jth predator eats the [Formula: see text]st species and is eaten by the [Formula: see text]st predator; this way each species only interacts with at most two other species-the ones that are immediately above or below it in the trophic chain. We show that one can classify, based on the invasion rates of the predators (which we can determine from the interaction coefficients of the system via an algorithm), which species go extinct and which converge to their unique invariant probability measure. We obtain stronger results than in the case with no intraspecific competition because in this setting we can make use of the general results of Hening and Nguyen (Ann Appl Probab 28:1893-1942, 2018a). Unlike most of the results available in the literature, we provide an in-depth analysis for both non-degenerate and degenerate noise. We exhibit our general results by analyzing trophic cascades in a plant-herbivore-predator system and providing persistence/extinction criteria for food chains of length [Formula: see text].

Asymptotic harvesting of populations in random environments.

We consider the harvesting of a population in a stochastic environment whose dynamics in the absence of harvesting is described by a one dimensional diffusion. Using ergodic optimal control, we find the optimal harvesting strategy which maximizes the asymptotic yield of harvested individuals. To our knowledge, ergodic optimal control has not been used before to study harvesting strategies. However, it is a natural framework because the optimal harvesting strategy will never be such that the population is harvested to extinction-instead the harvested population converges to a unique invariant probability measure. When the yield function is the identity, we show that the optimal strategy has a bang-bang property: there exists a threshold [Formula: see text] such that whenever the population is under the threshold the harvesting rate must be zero, whereas when the population is above the threshold the harvesting rate must be at the upper limit. We provide upper and lower bounds on the maximal asymptotic yield, and explore via numerical simulations how the harvesting threshold and the maximal asymptotic yield change with the growth rate, maximal harvesting rate, or the competition rate. We also show that, if the yield function is [Formula: see text] and strictly concave, then the optimal harvesting strategy is continuous, whereas when the yield function is convex the optimal strategy is of bang-bang type. This shows that one cannot always expect bang-bang type optimal controls.

Invariant measure for quantum trajectories

We study a class of Markov chains that model the evolution of a quantum system subject to repeated measurements. Each Markov chain in this class is defined by a measure on the space of matrices, and is then given by a random product of correlated matrices taken from the support of the defining measure. We give natural conditions on this support that imply that the Markov chain admits a unique invariant probability measure. We moreover prove the geometric convergence towards this invariant measure in the Wasserstein metric. Standard techniques from the theory of products of random matrices cannot be applied under our assumptions, and new techniques are developed, such as maximum likelihood-type estimations.

Statistical properties of quadratic polynomials with a neutral fixed point

We describe the statistical properties of the dynamics of the quadratic polynomials P_\alpha(z)=e^{2\pi \alpha \mathbf{i}} z+ z^2 on the complex plane, with \alpha of high type. In particular, we show that these maps are uniquely ergodic on their measure-theoretic attractors, and the unique invariant probability is a physical measure describing the statistical behaviour of typical orbits in the Julia set. This confirms a conjecture of Pérez–Marco on the unique ergodicity of hedgehog dynamics, in this class of maps.

Early adolescent adversity inflates threat estimation in females and promotes alcohol use initiation in both sexes.

Childhood adversity is associated with exaggerated threat processing and earlier alcohol use initiation. Conclusive links remain elusive, as childhood adversity typically co-occurs with detrimental socioeconomic factors, and its impact is likely moderated by biological sex. To unravel the complex relationships among childhood adversity, sex, threat estimation, and alcohol use initiation, we exposed female and male Long-Evans rats to early adolescent adversity (EAA). In adulthood, >50 days following the last adverse experience, threat estimation was assessed using a novel fear discrimination procedure in which cues predict a unique probability of footshock: danger (p = 1.00), uncertainty (p = .25), and safety (p = .00). Alcohol use initiation was assessed using voluntary access to 20% ethanol, >90 days following the last adverse experience. During development, EAA slowed body weight gain in both females and males. In adulthood, EAA selectively inflated female threat estimation, exaggerating fear to uncertainty and safety, but promoted alcohol use initiation across sexes. Meaningful relationships between threat estimation and alcohol use initiation were not observed, underscoring the independent effects of EAA. Results isolate the contribution of EAA to adult threat estimation, alcohol use initiation, and reveal moderation by biological sex. (PsycINFO Database Record

Exponential rate of convergence independent of the dimension in a mean-field system of particles

EXPONENTIAL RATE OF CONVERGENCE INDEPENDENT OF THE DIMENSION IN A MEAN-FIELD SYSTEM OF PARTICLESThis article deals with a mean-field model. We consider a large number of particles interacting through their empirical law. We know that there is a unique invariant probability for this diffusion.We look at functional inequalities. In particular, we briefly show that the diffusion satisfies a Poincaré inequality. Then, we establish a so-called WJ-inequality, which is independent of the number of particles.

Poincaré-type inequalities for stable densities

In a recent paper we introduced the concept of fractional score, a generalization of the linear score function, well-known in theoretical statistics. As the Gaussian density is closely related to the linear score, the fractional score function allows to identify Levy stable laws as the (unique) probability densities for which the score of a random variable X is proportional to $$-X$$ . We use this analogy to extend to stable laws the classical Poincare inequality for Gaussian densities. Application of this inequality allows to obtain bounds on moments of stable laws, and a sharp one-dimensional version of Hardy–Poincare inequality.

Status threat, not economic hardship, explains the 2016 presidential vote

This study evaluates evidence pertaining to popular narratives explaining the American public's support for Donald J. Trump in the 2016 presidential election. First, using unique representative probability samples of the American public, tracking the same individuals from 2012 to 2016, I examine the "left behind" thesis (that is, the theory that those who lost jobs or experienced stagnant wages due to the loss of manufacturing jobs punished the incumbent party for their economic misfortunes). Second, I consider the possibility that status threat felt by the dwindling proportion of traditionally high-status Americans (i.e., whites, Christians, and men) as well as by those who perceive America's global dominance as threatened combined to increase support for the candidate who emphasized reestablishing status hierarchies of the past. Results do not support an interpretation of the election based on pocketbook economic concerns. Instead, the shorter relative distance of people's own views from the Republican candidate on trade and China corresponded to greater mass support for Trump in 2016 relative to Mitt Romney in 2012. Candidate preferences in 2016 reflected increasing anxiety among high-status groups rather than complaints about past treatment among low-status groups. Both growing domestic racial diversity and globalization contributed to a sense that white Americans are under siege by these engines of change.

Scalable and exact sampling method for probabilistic generative graph models

Interest in modeling complex networks has fueled the development of multiple probabilistic generative graph models (PGGMs). PGGMs are statistical methods that model the network distribution and match common characteristics of real world networks. Recently, scalable sampling algorithms for well known PGGMs, made the analysis of large-scale, sparse networks feasible for the first time. However, it has been demonstrated that these scalable sampling algorithms do not sample from the original underlying distribution, and sometimes produce very unlikely graphs. To address this, we extend the algorithm proposed in Moreno et al. (in: IEEE 14th international conference on data mining, pp 440–449, 2014) for a single model and develop a general solution for a broad class of PGGMs. Our approach exploits the fact that PGGMs are typically parameterized by a small set of unique probability values—this enables fast generation via independent sampling of groups of edges with the same probability value. By sampling within groups, we remove bias due to conditional sampling and probability reallocation. We show that our grouped sampling methods are both provably correct and efficient. Our new algorithm reduces time complexity by avoiding the expensive rejection sampling step previously necessary, and we demonstrate its generality, by outlining implementations for six different PGGMs. We conduct theoretical analysis and empirical evaluation to demonstrate the strengths of our algorithms. We conclude by sampling a network with over a billion edges in 95 s on a single processor.

Energy and reserve scheduling under ambiguity on renewable probability distribution

This paper presents a novel methodology to devise a least-cost energy and reserve scheduling under uncertainty in renewable energy sources (RES) and equipment outages. The uncertainty in renewable production is accounted for by exogenously simulated scenarios, as customary in stochastic programming, whereas outages of generators and/or transmission lines are addressed via adjustable robust optimization. The precise characterization of the RES output by means of a unique probability distribution is a challenging task. Hence, we provide a general formulation that allows the consideration of a set of “credible” probability distributions. In this manner, the system operator's ambiguity aversion to uncertainty in renewable production is accounted for. Our proposed methodology determines the least-cost energy and reserve scheduling through a three-level model. Structurally, the upper level defines a least-cost scheduling and, under uncertainty in renewable production, the middle level identifies the worst contingency for the given operating point. The lower level then utilizes the scheduling provided by the upper-level to determine the best redispatch. In order to control the system equilibrium, we adapt risk constraint techniques to handle the system imbalance uncertainty and ensure a reliable operating level. To solve the multi-level problem, we propose an algorithm that combines Benders decomposition and column-and-constraint generation techniques to approximate the risk measure while scheduling power and reserves. The effectiveness of the proposed model and the importance of considering ambiguity are demonstrated through a case study with real data from the Great Britain power system network.

A differentiable classification of certain locally free actions of Lie groups

Let L be a Lie group and let M be a compact manifold with dimension dim(L) + 1. Let Φ be a locally free action of L on M having class C r with r ≥ 2. Let R be the radical of L and let χ1, . . ., χ n be the characters of the adjoint action of {itR}. Finally, let Δ be the modular function of R. Under the assumption that none of the identities Δ×|χ i | = |χ j |α hold for any α ∈ [0, 1], one shows that Φ is the restriction to L of a locally free and transitive C r action of a larger Lie group. A second result is the existence of a unique Φ-invariant probability measure on {itM}; that measure is induced by a C r−1 nonsingular volume form. What makes that theorem all the more interesting is that certain of the Lie groups under consideration are not amenable.

Eternal inflation: when probabilities fail

In eternally inflating cosmology, infinitely many pocket universes are seeded. Attempts to show that universes like our observable universe are probable amongst them have failed, since no unique probability measure is recoverable. This lack of definite probabilities is taken to reveal a complete predictive failure. Inductive inference over the pocket universes, it would seem, is impossible. I argue that this conclusion of impossibility mistakes the nature of the problem. It confuses the case in which no inductive inference is possible, with another in which a weaker inductive logic applies. The alternative, applicable inductive logic is determined by background conditions and is the same, non-probabilistic logic as applies to an infinite lottery. This inductive logic does not preclude all predictions, but does affirm that predictions useful to deciding for or against eternal inflation are precluded.

On the uniqueness of global multiple SLEs

This article focuses on the characterization of global multiple Schramm–Loewner evolutions (SLE). The chordal SLE describes the scaling limit of a single interface in various critical lattice models with Dobrushin boundary conditions, and similarly, global multiple SLEs describe scaling limits of collections of interfaces in critical lattice models with alternating boundary conditions. In this article, we give a minimal amount of characterizing properties for the global multiple SLEs: we prove that there exists a unique probability measure on collections of pairwise disjoint continuous simple curves with a certain conditional law property. As a consequence, we obtain the convergence of multiple interfaces in the critical Ising, FK-Ising and percolation models.

A user-friendly condition for exponential ergodicity in randomly switched environments

We consider random switching between finitely many vector fields leaving positively invariant a compact set. Recently, Li, Liu and Cui showed in [12] that if one the vector fields has a globally asymptotically stable (G.A.S.) equilibrium from which one can reach a point satisfying a weak Hormander-bracket condition, then the process converges in total variation to a unique invariant probability measure. In this note, adapting the proof in [12] and using results of [5], the assumption of a G.A.S. equilibrium is weakened to the existence of an accessible point at which a barycentric combination of the vector fields vanishes. Some examples are given which demonstrate the usefulness of this condition.

A Class of Parseval Fusion Frames for Probabilistic Erasures

The fusion frame, as a generated frame, has a wide range of applications in engineering. In the data erasures problem, for any fusion frame, trying to find its optimal dual fusion frame is an important problem. In our paper we plan to discuss the problem of probabilistic erasures by using the probabilistic model. We construct a fusion frame {W, w} such that its dual fusion frame {W, v} is the unique probability optimal dual frames in some cases. Such fusion frame is called probability uniform Parseval fusion frame. We discuss the existence of the probability uniform Parseval fusion frame and the applications of probability uniform Parseval fusion frame. Then we get the relationship of probability uniform Parseval fusion frame and their optimal dual fusion frame. And then we can see the advantage of probability uniform Parseval fusion frame from some properties.

Subjective Expected Utility with Imperfect Perception

In many decisions under uncertainty, there are constraints on both the available information and the feasible actions. The agent can only make certain observations of the state space, and she cannot make them with perfect accuracy — she has imperfect perception. Likewise, she can only perform acts that transform states continuously into outcomes, and perhaps satisfy other regularity conditions. To incorporate such constraints, we modify the Savage decision model by endowing the state space S and outcome space X with topological structures. We axiomatically characterize a Subjective Expected Utility (SEU) representation of conditional preferences, involving a continuous utility function on X (unique up to positive affine transformations), and a unique probability measure on a Boolean algebra B of regular open subsets of S. We also obtain SEU representations involving a Borel measure on the Stone space of B — a “subjective” state space encoding the agent’s imperfect perception.