Pair Of Probability Distributions Research Articles

On Stability Estimate of Optimal Transport Maps Using m-th Polynomial Convexity

In 1781, Gaspard Monge first proposed the practical problem of relocating building materials while minimizing workers’ effort. Mathematically, the problem can be reiterated as finding a mapping T0 that transforms a random variable (X) following probability measure (μ) into a random variable (Y) following probability measure (ν), with minimal cost. Afterward, it has been widely studied and applied in statistics, machine learning, and economics, which concern the study of “distance” between usually a pair of probability distributions. The focus of this paper is centered on investigating and generalizing stability estimates for optimal transport plans, particularly through the lens of strong polynomial convexity. Building on previous research using plug-in estimators to strengthen the convergence rate of discrete or semi-discrete estimators for optimal transport plans, this paper introduces a novel stability estimate leveraging L-Lipschitz continuity and a paradigmatic methodology based on polynomial convexity, the understanding of which remains limited.

On the OWA Aggregation with Probabilistic Inputs

We discuss the use of the ordered weighted averaging (OWA) operator in multi-criteria decision problems as a means of aggregating the individual criteria satisfactions. We emphasize the need for ordering the arguments, the criteria satisfactions, when using the OWA operator. We consider the situation where the criteria satisfactions have some uncertainty, are finite probability distributions and note the requirement of having to order probability distributions. We introduce the idea of using pairwise stochastic dominance to provide the necessary ordering relationship over the probability distributions. We note that while this approach is appropriate, it is often not possible, since the presence of a stochastic dominance relationship between all pairs of probability distributions is not always the case, the relationship is not complete. To circumvent this problem we introduce an approach called the probabilistic exceedance method (PEM), which allows us to provide a surrogate for the OWA aggregation of probability distributions that doesn't require a linear ordering over the probability distributions. We look at this in both the cases in which the criteria have equal and unequal importances.

A model-learner pattern for bayesian reasoning

A Bayesian model is based on a pair of probability distributions, known as the prior and sampling distributions. A wide range of fundamental machine learning tasks, including regression, classification, clustering, and many others, can all be seen as Bayesian models. We propose a new probabilistic programming abstraction, a typed Bayesian model, which is based on a pair of probabilistic expressions for the prior and sampling distributions. A sampler for a model is an algorithm to compute synthetic data from its sampling distribution, while a learner for a model is an algorithm for probabilistic inference on the model. Models, samplers, and learners form a generic programming pattern for model-based inference. They support the uniform expression of common tasks including model testing, and generic compositions such as mixture models, evidence-based model averaging, and mixtures of experts. A formal semantics supports reasoning about model equivalence and implementation correctness. By developing a series of examples and three learner implementations based on exact inference, factor graphs, and Markov chain Monte Carlo, we demonstrate the broad applicability of this new programming pattern.

Wasserstein distances in the analysis of time series and dynamical systems

A new approach based on Wasserstein distances, which are numerical costs of an optimal transportation problem, allows us to analyze nonlinear phenomena in a robust manner. The long-term behavior is reconstructed from time series, resulting in a probability distribution over phase space. Each pair of probability distributions is then assigned a numerical distance that quantifies the differences in their dynamical properties. From the totality of all these distances a low-dimensional representation in a Euclidean space is derived, in which the time series can be classified and statistically analyzed. This representation shows the functional relationships between the dynamical systems under study. It allows us to assess synchronization properties and also offers a new way of numerical bifurcation analysis. The statistical techniques for this distance-based analysis of dynamical systems are presented, filling a gap in the literature, and their application is discussed in a few examples of datasets arising in physiology and neuroscience, and in the well-known Hénon system.

Proof of the Mixing Theorem for Statistical Systems in Classical Physics

In a general measure-theoretic context, it is proved that the action of a stochastic map on a pair of probability distributions can be characterized by the criterion of decreasing mixing distance.

Foundations of Risk Measurement. I. Risk As Probable Loss

This paper seeks to get behind specific contextual referents of risky situations to consider characteristics of risk that apply to many situations. It is guided by previous theoretical and empirical research in perceived risk, and focuses on the joint effects on risk of loss probability and the distribution of losses. The approach taken follows modern axiomatic theory by proposing conditions on a relation “is at least as risky as” between pairs of probability distributions over an outcome variable. Several sets of axioms for risk that characterize different forms for risk measurement are presented.

On axiomatic characterization of some non-additive measures of information

An axiomatic characterization of non-additive measures of information associated with a pair of probability distributions having the same number of elements has been given. This quantity under additional suitable postulates leads to the non-additive Entropy, Directed-Divergence and Inaccuracy of one or more parameters.

On a characterization of unified measure in information theory

A characterization of the unified measure associated with a pair of probability distributions, containing Kullaback's relative information and Kerridge's inaccuracy as particular cases, has been made in this paper through a set of axioms.

On axiomatic characterization of information-theoretic measures

An axiomatic characterization of an information-theoretic quantity associated with a pair of probability distributions having the same number of elements has been given. This quantity, under additional suitable conditions, leads to Kullback's information and Kerridge's inaccuracy concepts. By modifying one of the axioms, the two-parameter generalization of these is obtained.