Probabilistic Interpretation Research Articles

CUT-OFF DISTRIBUTIONS AND PROBABILISTIC INTERPRETATIONS OF THE GENERAL FRACTIONAL DERIVATIVES WITH MEMORY EFFECT

The physical/probabilistic meanings of fractional derivatives or integrals are very basic and important problems in fractional calculus. In this paper, we investigate the probabilistic interpretations of the general fractional derivatives with memory effect and their connections with cut-off distributions. Some examples like the cut-off Rayleigh distribution and the Atangana–Gómez fractional derivative are especially discussed. This paper shows that the kernels of the general fractional derivatives with memory effect may be not the same as the density functions of the physical processes described. For example, the kernel of Atangana–Gómez fractional derivative is a Gaussian function but its corresponding probabilistic distribution is a cut-off Rayleigh distribution.

ExSpliNet: An interpretable and expressive spline-based neural network

In this paper we present ExSpliNet, an interpretable and expressive neural network model. The model combines ideas of Kolmogorov neural networks, ensembles of probabilistic trees, and multivariate B-spline representations. We give a probabilistic interpretation of the model and show its universal approximation properties. We also discuss how it can be efficiently encoded by exploiting B-spline properties. Finally, we test the effectiveness of the proposed model on synthetic approximation problems and classical machine learning benchmark datasets.

Alternating Linear Scheme in a Bayesian Framework for Low-Rank Tensor Approximation

Multiway data often naturally occurs in a tensorial format which can be approximately represented by a low-rank tensor decomposition. This is useful because complexity can be significantly reduced and the treatment of large-scale data sets can be facilitated. In this paper, we find a low-rank representation for a given tensor by solving a Bayesian inference problem. This is achieved by dividing the overall inference problem into subproblems where we sequentially infer the posterior distribution of one tensor decomposition component at a time. This leads to a probabilistic interpretation of the well-known iterative algorithm alternating linear scheme (ALS). In this way, the consideration of measurement noise is enabled, as well as the incorporation of application-specific prior knowledge and the uncertainty quantification of the low-rank tensor estimate. To compute the low-rank tensor estimate from the posterior distributions of the tensor decomposition components, we present an algorithm that performs the unscented transform in tensor train format.

Paving property for real stable polynomials and strongly Rayleigh processes

One of the equivalent formulations of the Kadison–Singer problem which was resolved in 2013 by Marcus, Spielman and Srivastava, is the “paving conjecture”. In this paper, we first extend this result to real stable polynomials. We prove that for every multi-affine real stable polynomial satisfying a simple condition, it is possible to partition its set of variables to a small number of subsets such that the “restriction” of the polynomial to each subset has small roots. Then, we derive a probabilistic interpretation of this result. We show that there exists a partition of the underlying space of every strongly Rayleigh process into a small number of sets such that the restriction of the point process to each set has “almost independent” points. This result implies that the dependence structure of strongly Rayleigh processes is constrained—a phenomenon that is to be expected in negatively dependent measures. To prove this result, we introduce and study the notion of kernel polynomial for strongly Rayleigh processes. This notion is a natural generalization of the kernel of determinantal processes. We also derive an entropy bound for strongly Rayleigh processes in terms of the roots of the kernel polynomial which is interesting on its own.

Recognition awareness: adding awareness to pattern recognition using latent cognizance

This study investigates an application of a new probabilistic interpretation of a softmax output to Open-Set Recognition (OSR).Softmax is a mechanism wildly used in classification and object recognition. However, a softmax mechanism forces a model to operate under a closed-set paradigm, i.e., to predict an object class out of a set of pre-defined labels. This characteristic contributes to efficacy in classification, but poses a risk of non-sense prediction in object recognition. Object recognition is often operated under a dynamic and diverse condition. A foreign object—an object of any unprepared class—can be encountered at any time. OSR is intended to address an issue of identifying a foreign object in object recognition.Softmax inference has been re-interpreted with the emphasis of conditioning on the context. This re-interpretation and Bayes theorem have led to an approach to OSR, called Latent Cognizance (LC). LC utilizes what a classifier has learned and provides a simple and fast computation for foreign identification.Our investigation on LC employs various scenarios, using Imagenet 2012 dataset as well as foreign and fooling images. Its potential application to adversarial-image detection is also explored. Our findings support LC hypothesis and show its effectiveness on OSR.

Uncertainty-Assisted Image-Processing for Human-Robot Close Collaboration

The safety of human workers has been the main concern in human-robot close collaboration. Along with rapidly developed artificial intelligence techniques, deep learning models using two-dimensional images have become feasible solutions for human motion detection. These models serve as “sensors” in the closed-loop system that involve humans and robots. Most existing methods that detect human motion using images do not consider the uncertainty from the deep learning model itself. The mappings established by deep learning models should not be taken blindly, and thus uncertainty should be a natural part of this type of sensor. In particular, model uncertainty should be explicitly quantified and incorporated into robot motion control to guarantee safety. With this motivation, to rigorously quantify the uncertainty of these “sensors”, this letter proposes a probabilistic interpretation method and automatically provides a framework to benefit from a deep model's uncertainty. Experimental data from human-robot collaboration has been collected and used to validate the proposed method. A training strategy is proposed to efficiently train surrogate models that learn to refine the prediction of the main Bayesian models. The proposed framework is also compared with Ego hands benchmark showing a 4.7% increase in mIoU.

The promises and perils of predictive politics

Rachel Friedman’s Probable Justice and Jeffrey Friedman’s Power without Knowledge explore the promises and pitfalls of the application of predictive tools to the solution of social and political problems. Rachel Friedman argues that a fundamental duality in philosophical interpretations of probability allowed social insurance schemes to successfully accommodate two rival visions of liberal justice over the centuries. But in focusing on ideas around probability, she misses the limitations of the experts who put these ideas into practice and threatened to undermine them in the process. Jeffrey Friedman, by contrast, is centrally concerned with the limitations of experts. While he shows how these undermine one rather narrow conception of technocratic legitimacy, he avoids examining their implications for democratic legitimacy understood more broadly.

A three-way decision method with pre-order relations

Three-way decision (3WD) theory has been widely used in solving multi-attribute decision-making (MADM) problems, however the calculation method for semantic interpretations of loss functions and conditional probabilities has not been reasonably addressed. Thus, by virtue of existing studies on 3WD and MADM, a novel 3WD method to MADM is explored in the current paper, which is titled the 3W-MADM method. First, a pair of novel pre-order relations are designed by virtue of the classic TOPSIS (Technique for Order Preference by Similarity to an Ideal Solution) method, and a new scheme for calculating conditional probabilities is further established. Then, the calculation scheme of loss functions and thresholds is put forward via attribute evaluation values and ideal solutions. In particular, several primary steps and the algorithm of the 3W-MADM method are summed up. Finally, the practicality and validity of the proposed method are demonstrated by a realistic case of lymphography from the UCI database. Moreover, the stability and superiority of the presented method are shown by several experimental analyses.

Finite dimensional systems of free fermions and diffusion processes on spin groups

In this article, we are concerned with “finite dimensional fermions,” by which we mean vectors in a finite dimensional complex space embedded in the exterior algebra over itself. These fermions are spinless but possess the characterizing anticommutativity property. We associate invariant complex vector fields on the Lie group Spin(2n + 1) to the fermionic creation and annihilation operators. These vector fields are elements of the complexification of the regular representation of the Lie algebra so(2n+1). As such, they do not satisfy the canonical anticommutation relations; however, once they have been projected onto an appropriate subspace of L2(Spin(2n + 1)), these relations are satisfied. We define a free time evolution of this system of fermions in terms of a symmetric positive-definite quadratic form in the creation–annihilation operators. The realization of fermionic creation and annihilation operators brought by the (invariant) vector fields allows us to interpret this time evolution in terms of a positive self-adjoint operator that is the sum of a second order operator, which generates a stochastic diffusion process, and a first order complex operator, which strongly commutes with the second order operator. A probabilistic interpretation is given in terms of a Feynman–Kac-like formula with respect to the diffusion process associated with the second order operator.

Comprehensive compartmental model and calibration algorithm for the study of clinical implications of the population-level spread of COVID-19: a study protocol

IntroductionThe complex dynamics of the coronavirus disease 2019 (COVID-19) pandemic has made obtaining reliable long-term forecasts of the disease progression difficult. Simple mechanistic models with deterministic parameters are useful for...

Preventing active re-identification attacks on social graphs via sybil subgraph obfuscation

Active re-identification attacks constitute a serious threat to privacy-preserving social graph publication, because of the ability of active adversaries to leverage fake accounts, a.k.a. sybil nodes, to enforce structural patterns that can be used to re-identify their victims on anonymised graphs. Several formal privacy properties have been enunciated with the purpose of characterising the resistance of a graph against active attacks. However, anonymisation methods devised on the basis of these properties have so far been able to address only restricted special cases, where the adversaries are assumed to leverage a very small number of sybil nodes. In this paper, we present a new probabilistic interpretation of active re-identification attacks on social graphs. Unlike the aforementioned privacy properties, which model the protection from active adversaries as the task of making victim nodes indistinguishable in terms of their fingerprints with respect to all potential attackers, our new formulation introduces a more complete view, where the attack is countered by jointly preventing the attacker from retrieving the set of sybil nodes, and from using these sybil nodes for re-identifying the victims. Under the new formulation, we show that k-symmetry, a privacy property introduced in the context of passive attacks, provides a sufficient condition for the protection against active re-identification attacks leveraging an arbitrary number of sybil nodes. Moreover, we show that the algorithm K-Match, originally devised for efficiently enforcing the related notion of k-automorphism, also guarantees k-symmetry. Empirical results on real-life and synthetic graphs demonstrate that our formulation allows, for the first time, to publish anonymised social graphs (with formal privacy guarantees) that effectively resist the strongest active re-identification attack reported in the literature, even when it leverages a large number of sybil nodes.

Application of Monte-Carlo Simulation Towards a Better Understanding of Bayes’ Theorem in Engineering Education

Bayes' Theorem (BT) is treated in probability theory and statistics. The BT shows how to change the probabilities a priori in view of new evidence, to obtain probabilities a posteriori. With the Bayesian interpretation of probability, the BT is expressed as the probability of an event (or the degree of belief in the occurrence of an event) should be changed, after considering evidence about the occurrence of that event. Bayesian inference is fundamental to Bayesian statistics. An example of practical application of this theorem in Health Systems is to consider the existence of false positives and false negatives in diagnoses. At the Academy, the theme of BT is exposed almost exclusively in its analytical form. With this article, the authors intend to contribute to clarify the logic behind this theorem, and get students better understanding of its important fields of application, using three methods: the classic analytical (Bayesian inference), the frequentist (frequency inference) and the numerical simulation of Monte-Carlo. Thus, it intends to explain BT on a practical and friendly way that provides understanding to students avoiding memorizing the formulas. We provide a spreadsheet that is accessible to any professor. Moreover, we highlight the methodology could be extended to other topics.

First global QCD analysis of the TMD g1T from semi-inclusive DIS data

The worm-gear transverse momentum dependent (TMD) function $g_{1T}$ is one of the eight leading-twist TMDs and has the probabilistic interpretation of finding a longitudinally polarized quark inside a transversely polarized hadron. In this work, we present the first simultaneous extraction of $g_{1T}$ from all the available experimental measurements. The study involves the analysis of COMPASS, HERMES, and Jefferson Lab data on semi-inclusive deep-inelastic scattering using Monte Carlo techniques. We also compare $g_{1T}$ obtained from this experimental data with different theoretical approaches, such as the large-$N_c$ approximation, the Wandzura-Wilczek-type approximation, and lattice QCD.

Power Normalizations in Fine-Grained Image, Few-Shot Image and Graph Classification.

Power Normalizations (PN) are useful non-linear operators which tackle feature imbalances in classification problems. We study PNs in the deep learning setup via a novel PN layer pooling feature maps. Our layer combines the feature vectors and their respective spatial locations in the feature maps produced by the last convolutional layer of CNN into a positive definite matrix with second-order statistics to which PN operators are applied, forming so-called Second-order Pooling (SOP). As the main goal of this paper is to study Power Normalizations, we investigate the role and meaning of MaxExp and Gamma, two popular PN functions. To this end, we provide probabilistic interpretations of such element-wise operators and discover surrogates with well-behaved derivatives for end-to-end training. Furthermore, we look at the spectral applicability of MaxExp and Gamma by studying Spectral Power Normalizations (SPN). We show that SPN on the autocorrelation/covariance matrix and the Heat Diffusion Process (HDP) on a graph Laplacian matrix are closely related, thus sharing their properties. Such a finding leads us to the culmination of our work, a fast spectral MaxExp which is a variant of HDP for covariances/autocorrelation matrices. We evaluate our ideas on fine-grained recognition, scene recognition, and material classification, as well as in few-shot learning and graph classification.

ДИФФЕРЕНЦИАЛЬНОЕ МОДЕЛИРОВАНИЕ ПРОЦЕССОВ РАСПРОСТРАНЕНИЯ И ЛИКВИДАЦИИ ИНФЕКЦИОННЫХ ЗАБОЛЕВАНИЙ ЛЕСНЫХ НАСАЖДЕНИЙ

The destabilization of the sustainable state of forests and the associated deterioration of the sanitary climate are mainly associated with their diseases. One of the main tasks of forest pathology monitoring is the detection of foci of disease of stands and the fight against their spread. In this paper, we propose a mathematical description of the processes of the spread and elimination of infectious diseases of forest plantations using the deterministic SIR (Susceptible, Infected, Recovered) model, the main provisions of which are basic in the interpretation of various kinds of infectious diseases, often taking the form of epidemics. Assuming the constancy of the total number of healthy infected subjects during the spread of the disease and the direct proportional dependence of the infection rate on the number of infected, the results obtained indicate that: a) if at the initial moment of time the number of infected individuals does not exceed a certain deterministic value, then plants do not become infected over time; b) in cases if this number is greater than a fixed value, then the part of individuals susceptible to the disease will become infected. It is shown that despite the abstract nature and the impossibility of probabilistic interpretation of the detection of foci of the disease, the proposed differential model can be effective in solving the problems of the spread, localization and prediction of tree diseases and is used, practically, for any closed biological and ecological system

A Bayesian machine learning approach for spatio-temporal prediction of COVID-19 cases.

Modeling the spread of infectious diseases in space and time needs to take care of complex dependencies and uncertainties. Machine learning methods, and neural networks, in particular, are useful in modeling this sort of complex problems, although they generally lack of probabilistic interpretations. We propose a neural network method embedded in a Bayesian framework for modeling and predicting the number of cases of infectious diseases in areal units. A key feature is that our combined model considers the impact of human movement on the spread of the infectious disease, as an additional random factor to the also considered spatial neighborhood and temporal correlation components. Our model is evaluated over a COVID-19 dataset for 245 health zones of Castilla-Leon (Spain). The results show that a Bayesian model informed by a neural network method is generally able to predict the number of cases of COVID-19 in both space and time, with the human mobility factor having a strong influence on the model, together with the number of infections and deaths in nearby areas.

A conditional, a fuzzy and a probabilistic interpretation of self-organizing maps

Abstract In this paper we establish a link between fuzzy and preferential semantics for description logics and self-organizing maps (SOMs), which have been proposed as possible candidates to explain the psychological mechanisms underlying category generalization. In particular, we show that the input/output behavior of a SOM after training can be described by a fuzzy description logic interpretation as well as by a preferential interpretation, based on a concept-wise multipreference semantics, which takes into account preferences with respect to different concepts and has been recently proposed for ranked and for weighted defeasible description logics. Properties of the network can be proven by model checking on the fuzzy or on the preferential interpretation. Starting from the fuzzy interpretation, we also provide a probabilistic account for this neural network model.

Born machine model based on matrix product state quantum circuit

Born machine model based on the probability interpretation of the wave function combining quantum information theory with machine learning method provides a new tool to study the generative models. The Born machine model with a general parameterized quantum circuit generally requires the same number of qubits as the sample feature size of the dataset to be processed, while each sample usually contains thousands of features in actual dataset. A novel Born machine model with a matrix product state quantum circuit is proposed, which requires less qubits than that with a general parameterized quantum circuit, so it can make better use of scarce qubit resources in near-term quantum devices. And the presented Born machine model is trained with the maximal mean discrepancy loss function. The learning process of the proposed Born machine model is numerically simulated on the Bars-and-Stripes dataset. The simulation results verify the feasibility of the Born machine model with the matrix product state quantum circuit.

Bayesian Interpretation of Essential Tremor Plus.

Essential tremor (ET) plus is a new tremor classification that was introduced in 2018 by a task force of the International Parkinson and Movement Disorder Society. Patients with ET plus meet the criteria for ET but have one or more additional systemic or neurologic signs of uncertain significance or relevance to tremor (“soft signs”). Soft signs are not sufficient to diagnose another tremor syndrome or movement disorder, and soft signs in ET plus are known to have poor interrater reliability and low diagnostic sensitivity and specificity. Therefore, the clinical significance of ET plus must be interpreted probabilistically when judging whether a patient is more likely to have ET or a combined tremor syndrome, such as dystonic tremor. Such a probabilistic interpretation is possible with Bayesian analysis. This review presents a Bayesian analysis of ET plus in patients suspected of having ET versus a dystonic tremor syndrome, which is the most common differential diagnosis in patients referred for ET. Bayesian analysis of soft signs provides an estimate of the probability that a patient with possible ET is more likely to have an alternative diagnosis. ET plus is a distinct tremor classification and should not be viewed as a subtype of ET. ET plus covers a more-comprehensive phenotyping of people with possible ET, and the clinical interpretation of ET plus is enhanced with Bayesian analysis of associated soft signs.

Remarkable Findings in Fundamental Theory of Quantum Mechanics—Matter Wave and Discrete Time in Physics

The relation of matter wave, which is well-known as a hypothesis proposed by de Broglie in 1923, gave basis for establishing the quantum mechanics. After that, experimental results revealed that a micro particle has a wave nature. However, the theoretical validity of the relation itself has never been revealed since his proposal. Theoretical basis that a micro particle has a wave nature has been thus disregarded in the unsolved state. The diffusion equation having been accepted as Fick’s second law was derived from the theory of Markov process in mathematics. It was then revealed that the diffusivity D depends on an angular momentum of a micro particle in a local space. The fact being unable to discriminate between micro particles in a local space resulted in having to accept the existence of minimum time t0 (>0) in the quantum mechanics. Based on t0 and D obtained here, the theoretical validity of relation of matter wave was confirmed. Denying the density theorem in mathematics for time in physics indicates that the probabilistic interpretation is essentially indispensable for understanding the quantum mechanics. The logical necessity of quantum theory itself is thus understandable through introducing t0 into the Newton mechanics. It is remarkable that the value of t0 between 1.14×10-17 s ≤ t0 ≤1.76×10-14 s obtained here is extremely larger than that of the well-known Planck time tp=5.396×10-44 s.