We give a complete characterization of both comonotone and not comonotone coherent risk measures in the discrete finite probability space, where each outcome is equally likely. To the best of our knowledge, this is the first work that characterizes \textit{and} distinguishes comonotone and not comonotone coherent risk …
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48 results for “discrete probability measures”
New discrepancy function compares discrete probability measures considering space geometry.
problem Comparing discrete probability measures in a geometrically meaningful way.
method Proposes the Fourier Discrepancy Function, proving convexity, differentiability, and providing gradient formula.
result Proves the Fourier Discrepancy is convex, twice differentiable, and provides an explicit gradient formula.
Algorithm reduces support of discrete measures by integrating against functions.
problem Efficiently reducing the support of discrete measures when .
method Geometric characterization and greedy geometric sampling.
result A new measure with atoms has the same mean as original measure.
Convergence of the financial value of weak information for a sequence of discrete-time marketsmath.PR
Study shows financial value of weak information converges in discrete vs continuous markets.
problem Analyzing financial value of weak information in discrete vs continuous markets.
method Defined minimal probability measure and financial value of weak information, then showed convergence.
result Financial value of weak information converges in discrete vs continuous markets.
Study asset pricing under model uncertainty with discrete time and states.
problem Asset pricing under model uncertainty with discrete time and states.
method Novel definition of arbitrage, investigation of no-arbitrage conditions, expansion to multi-period securities model.
result Necessary and sufficient conditions for no-arbitrage asset pricing under model uncertainty.
The notion of utility maximising entropy (u-entropy) of a probability density, which was introduced and studied by Slomczynski and Zastawniak (Ann. Prob 32 (2004) 2261-2285, arXiv:math.PR/0410115 v1), is extended in two directions. First, the relative u-entropy of two probability measures in arbitrary probability space…
New method optimizes non-linear functionals over probability measures.
problem Optimizing non-linear functionals defined over probability measures.
method N-particle underdamped Langevin algorithm with spacetime discretization.
result Converges globally in total variation distance.
Proves hardness of semi-discrete optimal transport and proposes regularization methods.
problem Computing Wasserstein distance between discrete and non-discrete probability measures.
method Proves hardness, introduces distributionally robust dual optimal transport, regularizes primal objective, uses stochastic gradient descent.
result Regularization schemes and improved convergence guarantees for semi-discrete optimal transport problems.
Wassmap reduces image complexity while preserving key features.
problem Global nonlinear dimensionality reduction in imaging.
method Wassmap uses Wasserstein space and pairwise distances to create isometric embeddings.
result Wassmap can recover parameters of image manifolds like translations and dilations.
A scalable approach to learning from probability measures using quantization.
problem Efficiently comparing and manipulating large sets of probability measures.
method Quantization of probability measures to a fixed support, followed by optimal transport computations.
result Consistency and convergence guarantees for quantized measures in various OT-based tasks.
The paper studies properties of Sliced Wasserstein energy for discrete measures.
problem Optimizing discrete probability measures using Sliced Wasserstein loss.
method Investigates the regularity and optimisation properties of the Sliced Wasserstein energy and its Monte-Carlo approximation.
result Convergence results on the critical points of Monte-Carlo approximations to the Sliced Wasserstein energy.
Paper develops a consistent estimator for discrete mixture models.
problem Estimating mixing probability density in discrete mixture models.
method Develops an consistent estimator under specific conditions.
result Consistent estimation of mixing probability density for every density .
Representations based on random walks can exploit discrete data distributions for clustering and classification. We extend such representations from discrete to continuous distributions. Transition probabilities are now calculated using a diffusion equation with a diffusion coefficient that inversely depends on the dat…
Framework estimates multiple plausible solutions with uncertainty measures.
problem Machine learning models need to propose multiple plausible solutions with meaningful uncertainty.
method Discrete latent variables model one-to-many mappings, allowing effective conditional probability estimation.
result Framework outperforms state-of-the-art in uncertainty estimation and is practical.
Study shows -NN classifier is not universally consistent on but consistent on discrete and specific measure spaces.
problem Consistency of -NN classifier under Wasserstein distance on measure spaces.
method Analysis of -NN classifier properties under Wasserstein distance, use of -finite metric dimension, geodesic structures of Wasserstein spaces.
result Consistency of -NN classifier on specific measure spaces (discrete, Gaussian, wavelet series) but not on .
This paper studies the probability of discrete groups generated by two parabolic elements in Kleinian groups.
problem Estimating the probability of discrete groups generated by two parabolic elements in Kleinian groups.
method Introduced geometrically natural probability measures and used computational investigation into the Riley slice.
result Identified the precise probability that a Fuchsian group generated by two parabolic Möbius transformations is discrete.
A novel approach to computing barycenters on graph-supported probability measures.
problem Computing weighted averages of measures on graphs.
method Dynamic optimal transport formulation on the simplex, gradient descent on the probability simplex.
result Intrinsic gradient descent provides a coherent framework for synthesizing and analyzing measures on graphs.
With model uncertainty characterized by a convex, possibly non-dominated set of probability measures, the agent minimizes the cost of hedging a path dependent contingent claim with given expected success ratio, in a discrete-time, semi-static market of stocks and options. Based on duality results which link quantile he…
Different approaches to defining dynamic market risk measures are available in the literature. Most are focused or derived from probability theory, economic behavior or dynamic programming. Here, we propose an approach to define and implement dynamic market risk measures based on recursion and state economy representat…
The paper develops approximations for Pearson's chi-square statistic and applies them to confidence intervals.
problem Finding confidence intervals for strictly convex functions of discrete distribution weights.
method Non-asymptotic local normal approximation for multinomial probabilities, deriving bounds and coupling inequalities.
result Developed methods to find confidence intervals for negative entropy of discrete distributions.
The new notion of maturity-independent risk measures is introduced and contrasted with the existing risk measurement concepts. It is shown, by means of two examples, one set on a finite probability space and the other in a diffusion framework, that, surprisingly, some of the widely utilized risk measures cannot be used…
A new method for multilevel clustering using Wasserstein means.
problem Simultaneously partitioning data in each group and discovering grouping patterns among groups.
method Joint optimization over spaces of discrete probability measures with Wasserstein distance metrics, including variants that admit fast optimization.
result Consistency properties for estimates of both local and global clusters are established.
Withdrawal Success Optimizationq-fin.PM
Maximizes probability of completing investment schedules with optimal portfolio weights.
problem Optimizing probability of completing investment schedules with optimal portfolio weights.
method Computing maximum probability and optimal portfolio weight functions for various rebalancing schedules.
result Noticeable improvements in probability to complete schedules with optimal portfolio weights.
Study optimal investment strategies under model uncertainty in discrete markets.
problem Maximizing utility in markets with model uncertainty.
method Alternative framework for model uncertainty, using stochastic processes.
result Optimal investment strategies exist under certain conditions.
New method for scalable barycenter computation using Wasserstein gradient flows.
problem Scalability and integration of label information in barycenter computation.
method Gradient flows in Wasserstein space, time discretization, mini-batch optimal transport, modular regularization, task-aware functions, supervised information integration.
result Empirically validated new state-of-the-art barycenter solver with labeled barycenters outperforming unlabeled ones.
Efficient methods estimate concordance probability for big data.
problem Efficiently calculating concordance probability in large datasets.
method Proposes two estimation methods for discrete and continuous settings.
result Estimators are accurate and computationally efficient.
New algorithms minimize MMD to approximate probability measures efficiently.
problem Approximating probability measures by representative point sets.
method Sequential greedy minimization of maximum mean discrepancy (MMD) over candidate sets, with mini-batch variants.
result Consistency of proposed algorithms and mini-batch variants established.
This work finds mixed equilibria in machine learning problems using measures and simultaneous gradient ascent-descent.
problem Finding pure equilibria in machine learning problems is computationally hard.
method Entropic regularization, simultaneous gradient ascent-descent, and particle discretization in the Wasserstein metric.
result Global convergence towards the global equilibrium in mixed equilibria problems.
Study convergence of simulated annealing in continuous and discrete settings.
problem Analyzing convergence rate of simulated annealing methods.
method Apply Eyring-Kramers law to prove polynomial decay of tail probabilities.
result Explicit rate of convergence for continuous and discrete simulated annealing.
Develops non-standard analysis for coherent risk estimation.
problem Estimating coherent risk measures in financial contexts.
method Non-standard analysis, hyperfinite representations, discrete Kusuoka formulae, plug-in asymptotics.
result Uniform almost sure consistency and asymptotic normality of spectral plug-in estimators.
Investigates trading with integer constraints in discrete time.
problem Trading with discrete, integer quantities under integer constraints.
method Establishes a novel theory of integer arbitrage-free pricing and hedging for non-rational price processes.
result The set of prices of a contingent claim is either empty or dense in an interval.
Framework for robust control under model uncertainty, improving financial derivatives hedging.
problem Model uncertainty in financial derivatives hedging.
method Dynamic programming principle for solving one-step optimization problems.
result Robust hedging strategy outperforms model-based strategies during adverse scenarios.
SFM matches flows on statistical manifolds for better discrete generation.
problem Discrete generation on statistical manifolds with strong prior assumptions.
method Statistical Flow Matching (SFM) on manifold of categorical distributions using Fisher information metric.
result SFM achieves higher sampling quality and likelihood than other models.
We apply multiple testing procedures to the validation of estimated default probabilities in credit rating systems. The goal is to identify rating classes for which the probability of default is estimated inaccurately, while still maintaining a predefined level of committing type I errors as measured by the familywise …
A Universal Approximation Theorem of Deep Neural Networks for Expressing Probability Distributionscs.LG
Deep neural networks can approximate any target probability distribution given certain conditions.
problem Approximating complex probability distributions with deep neural networks.
method Proving the existence of a deep neural network mapping that approximates a target distribution under various integral probability metrics.
result Upper bounds on the size of the neural network in terms of dimension and approximation error for different metrics.
Develops a measure-theoretic framework for complex co-occurrence data.
problem Modeling and interpreting complex co-occurrences in high-dimensional data.
method Introduces measure-theoretic probability and conditional probability, investigates E-integrals.
result Establishes a rigorous measure-theoretic foundation for co-occurrence modeling.
New Fourier metrics equivalent to Wasserstein distances in image processing.
problem Equivalence of Fourier-based and Wasserstein metrics in imaging problems.
method Extensions of Fourier-based metrics to handle different centers of mass and discrete measures, showing equivalence to Wasserstein distances.
result New Fourier metrics are equivalent to Wasserstein distances with explicit constants, improving runtime in image processing.
Foster and Hart proposed an operational measure of riskiness for discrete random variables. We show that their defining equation has no solution for many common continuous distributions including many uniform distributions, e.g. We show how to extend consistently the definition of riskiness to continuous random variabl…
This paper analyzes the bias of inexact MCMC methods in high dimensions.
problem Understanding the bias of inexact MCMC methods in high-dimensional spaces.
method Establishing bounds on Wasserstein distances between inexact MCMC methods and target distributions.
result The asymptotic bias of ULA and uHMC depends on key quantities related to the target distribution or the stationary probability measure of the scheme.
Paper proposes an efficient AL-GP method for CDF/CCDF estimation in UQ.
problem Estimating full probability distribution in forward UQ analysis.
method Active learning-based Gaussian process (AL-GP) metamodelling method.
result Efficient estimation of CDF/CCDF without explicit discretization.
Develops a robust framework for pricing and hedging in discrete-time markets.
problem Pricing and hedging of derivative securities in markets with dynamically and statically traded assets.
method Abstract Fundamental Theorem of Asset Pricing and Pricing--Hedging Duality, minimal measurability assumptions, scenario-based approach.
result Includes model-independent results and extends classical probabilistic approaches.
The problem is sequence prediction in the following setting. A sequence x1,..., xn,... of discrete-valued observations is generated according to some unknown probabilistic law (measure) mu. After observing each outcome, it is required to give the conditional probabilities of the next observation. The measure mu belongs…
Generative Assignment Flows for Representing and Learning Joint Distributions of Discrete Datastat.ML
Generative model for joint discrete distributions using randomized assignment flows.
problem Efficiently representing and sampling from complex joint distributions of discrete variables.
method Randomized assignment flows on the statistical submanifold of factorizing distributions.
result Our model can efficiently represent and sample from any target distribution and assess likelihood of unseen data points.
We interpret policy optimization as Wasserstein gradient flows and develop efficient algorithms.
problem Unclear mathematical principle of policy optimization in reinforcement learning.
method Interpreting policy optimization as Wasserstein gradient flows, developing efficient algorithms to solve the corresponding discrete gradient flows.
result Policy optimization becomes a convex problem in terms of distribution optimization under specified circumstances.
In a model independent discrete time financial market, we discuss the richness of the family of martingale measures in relation to different notions of Arbitrage, generated by a class of significant sets, which we call Arbitrage de la classe . The choice of reflects into the int…
Survey on calibration in machine learning, viewing it as indistinguishability.
problem Evaluating continuous probability predictions in discrete outcome settings.
method Defining and measuring calibration error through indistinguishability.
result Calibration measures quantify distinguishability between hypothesized and real-world outcomes.
We investigate existence and uniqueness of p-means and the median of a probability measure on a Finsler manifold, in relation with the convexity of the support of the measure. We prove that the p-mean is the limit point of a continuous time gradient flow. Under some additional condition which is always satisfied for la…
Novel algorithm solves optimal transport using evolving probability distributions and convolution.
problem Sample-based optimal transport problem.
method Adversarial formulation with convolution of adaptive kernel and evolving measure.
result Algorithm robust to dimensionality and produces complex maps.