We give an overview of two approaches to probability theory where lower and upper probabilities, rather than probabilities, are used: Walley's behavioural theory of imprecise probabilities, and Shafer and Vovk's game-theoretic account of probability. We show that the two theories are more closely related than would be …
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48 results for “Probability Theory”
Without probability theory, we define classes of supermartingales, martingales, and semimartingales in idealized financial markets with continuous price paths. This allows us to establish probability-free versions of a number of standard results in martingale theory, including the Dubins-Schwarz theorem, the Girsanov t…
New concept of attitude towards probability introduced in risk sharing problems.
problem Risk sharing problems and attitudes towards probability.
method Generalized definition of probability premium, local approximation, rank-dependent utility model, dual theory.
result Attitude towards probability can be first-order or second-order, depending on the model.
New theory challenges traditional machine learning assumptions.
problem Traditional machine learning theories are critiqued.
method A new theory is proposed and discussed.
result Learning true probabilities is not equivalent to other learning goals.
New theory uses probability sets for data variability, improving machine learning.
problem Variability in data distribution causes learning issues.
method Uses convex sets of probabilities (credal sets) to model data variability.
result Derives bounds for risk of models learned from multiple training sets.
Survey on learning Boolean functions in computational theory.
problem Learning Boolean function classes in computational theory.
method Overview of known results in PAC and related models.
result Discussion of various learning results for Boolean functions.
Study classifies submanifolds in probability simplex.
problem Classifying submanifolds in the probability simplex.
method Complete classification through geometric analysis.
result Doubly totally-umbilical submanifolds identified and classified.
Categorical d-separation criterion simplifies probability graph analysis.
problem Detecting causal relationships in probability distributions.
method Introducing categorical definitions for causal models and d-separation.
result Abstract version of d-separation criterion applies to various probability theories.
Quantum probability theory reveals hidden structure in joint probability distributions.
problem Understanding hidden structure in joint probability distributions.
method Modeling joint probability distributions as density operators and applying partial trace.
result Decoding extra information in reduced density operators that captures subsystem interactions.
Paper explores how risk-averse individuals' willingness to pay for insurance varies with risk probability.
problem Understanding how risk-averse individuals' willingness to pay for insurance varies with risk probability.
method Analyzes willingness to pay (WTP) for partial risk reduction within the dual theory of decision.
result In dual theory, reducing the probability of risk and providing insurance can be complementary if the surplus increases with risk reduction.
Reply to Tetlock et al. on tail risk and probability gap.
problem Expert judgment fails to account for tail risk.
method Comparison of forecasting tournaments and extreme value theory.
result Greater gap between tail expectation and probability properties.
Proposes IPT for modeling complex joint distributions.
problem Lack of closed-form solutions for complex continuous or mixed distributions.
method Observer-centered framework with three independence axioms; derivation of closed-form solutions.
result Closed-form solutions for complex joint distributions under IPT.
In this paper, we develop a general theory of truncated inverse binomial sampling. In this theory, the fixed-size sampling and inverse binomial sampling are accommodated as special cases. In particular, the classical Chernoff-Hoeffding bound is an immediate consequence of the theory. Moreover, we propose a rigorous and…
New Schwarz Lemma for Bergman metrics in bounded domains.
problem Finding bounds for Bergman metrics in bounded domains.
method Using Cauchy-Schwarz inequality from probability theory.
result Established a new Schwarz Lemma for Bergman metrics.
Paper uses stats to predict treatment choice based on illness probability.
problem Improving treatment decision-making in personalized medicine.
method Statistical decision theory with maximum regret evaluation.
result Estimates illness probability for better treatment choice.
New theory extends rank-dependent utility for risk and ambiguity.
problem Modeling decision-making under risk and ambiguity.
method Axiomatizes a new preference relation with ambiguity index, probability weighting, and utility function.
result Extends rank-dependent utility to risk and ambiguity, reducing to existing models under specific conditions.
We explain the main concepts of Prospect Theory and Cumulative Prospect Theory within the framework of rational dynamic asset pricing theory. We derive option pricing formulas when asset returns are altered with a generalized Prospect Theory value function or a modified Prelec weighting probability function and introdu…
New theory improves diffusion model convergence for generating data.
problem Improving convergence of diffusion models for data generation.
method Developed a non-asymptotic convergence theory for probability flow ODEs.
result Proves iterations suffice for approximating target distributions.
New theory controls compression change probability without prior knowledge.
problem Controlling compression change probability without prior knowledge.
method New theory on compression function and statistical risk.
result Cardinality of compressed set is a consistent estimator of probability of change of compression.
Inferential Moments of Uncertain Multivariable Systemsphysics.data-an
The article introduces inferential moments for analyzing uncertain multivariable systems.
problem Handling inference tasks in uncertain multivariable systems.
method Bayesian inference and quantification of inferential moments.
result Quantification of inferential moments and their connection to mutual information.
The Wasserstein metric is an important measure of distance between probability distributions, with applications in machine learning, statistics, probability theory, and data analysis. This paper provides upper and lower bounds on statistical minimax rates for the problem of estimating a probability distribution under W…
Investigates financial and economic systems using statistical mechanics and information theory.
problem Complexity, asymmetry, stochasticity, and non-linearity in financial and economic systems.
method Model-based and empirical analyses using statistical mechanics and information theory.
result Derives probability distribution functions for better understanding of financial and economic dynamics.
Estimates growth of reciprocal classes in Hecke groups.
problem Estimating the growth of reciprocal conjugacy classes in Hecke groups.
method Using free product structure and word lengths of reciprocal elements, with tools from basic probability theory.
result Estimates the asymptotic growth of reciprocal conjugacy classes in Hecke groups.
Unification of field theory and maximum entropy methods for learning probability densitiesphysics.data-an
The need to estimate smooth probability distributions (a.k.a. probability densities) from finite sampled data is ubiquitous in science. Many approaches to this problem have been described, but none is yet regarded as providing a definitive solution. Maximum entropy estimation and Bayesian field theory are two such appr…
This review explores entropy applications in data analysis and machine learning.
problem Characterizing probability mass distributions in data analysis and machine learning.
method Review of various entropy types and their applications.
result Entropy's versatility in data analysis and machine learning.
Bayesian probability theory is one of the most successful frameworks to model reasoning under uncertainty. Its defining property is the interpretation of probabilities as degrees of belief in propositions about the state of the world relative to an inquiring subject. This essay examines the notion of subjectivity by dr…
Paper axiomatizes interventional probability distributions.
problem Causal inference and intervention.
method Axiomatization of interventional families.
result Markovian property of intervened distributions.
The paper explores geometry of probability measures and barycenter maps.
problem Understanding the space of probability measures and their barycenter.
method Information geometry, Fisher metric, dualistic structures, divergences, geodesics.
result Recent developments in the geometry of probability measures and barycenter.
New neural network approach mitigates vanishing/exploding gradients.
problem Vanishing and exploding gradients in neural networks.
method Gaussian-Poincaré normalized functions and orthogonal weight matrices.
result High-dimensional probability theory shows gradients disappear with high probability in wide neural networks.
New game approximates mean curvature flow evolution.
problem Approximating geometric mean curvature flow evolution.
method Two-player zero-sum game with probabilistic elements.
result Value function approximates mean curvature flow.
Study on error probabilities of machine learning classification techniques using large deviations theory.
problem Performance analysis of machine learning binary classification techniques.
method Large deviations theory applied to Data-Driven Decision Function (D3F) for error probability analysis.
result Classification error probabilities vanish exponentially, with an asymptotic formula providing precise error rate estimates.
The paper improves the probability flow ODE sampler for faster sampling of natural images.
problem Improving the convergence rate of the probability flow ODE sampler.
method Adapting the probability flow ODE sampler to exploit intrinsic low-dimensional structures in natural image data.
result Achieves a dimension-free convergence rate of in total variation distance, improving upon existing results.
We generalize the notion of monetary value measures developed with category theory in [Adachi, 2014] by extending their base category from the category \c{hi} to the category of probability spaces Prob introduced in [Adachi and Ryu, 2016].
Explains how geometry and statistics intertwine, focusing on information geometry.
problem Understanding the interplay between geometry and statistics.
method Introduces differential topology, geometry, probability, and (pre-)Frobenius manifolds.
result Discovers connections between geometry and statistics, particularly in information geometry.
Study on random matrices in deep neural networks with IID entries.
problem Distribution of singular values in product of random matrices for deep neural networks.
method Random matrix theory with a streamlined approach for non-Gaussian data.
result Generalization of macroscopic universality property to non-Gaussian data.
Paper analyzes error exponent in agnostic PAC learning.
problem Analyzing performance of agnostic PAC learning.
method Using error exponent from Information Theory to analyze PAC learning.
result Improved distribution-dependent error exponent for agnostic learning.
Paper develops neural network for distribution regression.
problem Regression with probability measures.
method Develops a novel fully connected neural network (FNN) for distribution inputs.
result Almost optimal learning rates for distribution regression derived.
This work introduces a new metric for comparing imprecise probability models.
problem Quantifying differences between imprecise probability models.
method Integral imprecise probability metric framework based on Choquet integral.
result IIPM enables comparison across different imprecise probability models and quantifies epistemic uncertainty.
We connect Causal inference and low-rank recovery via RDT and free probability theory.
problem Determining the applicability of causal inference via low-rank recovery.
method Random Duality Theory, free probability theory, and mathematical rigor.
result Exact closed-form worst case phase transitions for causal inference.
In this text, we establish the risk model based on AR(1) series and propose the basic model which has a dependent structure under intensity of claim number. Considering some properties of the risk model, we take advantage of newton iteration method to figure out the adjustment coefficient and estimate the exponential u…
Improved bounds for discrete probability distribution estimation under the ℓ∞ norm.
problem Estimating discrete probability distributions under the ℓ∞ norm with improved bounds.
method Minimax bounds in expectation and high-probability tail bounds.
result Resolved open questions posed in Kontorovich and Painsky (JMLR, 2025), including a fully empirical tightest risk bound and identifying the worst-case extremal distribution.
This note explores the mathematical theory to solve modern gamblers ruin problems. We establish a ruin framework and solve for the probability of bankruptcy. We also show how this relates to the expected time to bankruptcy and review the risk neutral probabilities associated an adjustment to asymmetrical views.
Financial Probabilities from Fisher Informationcond-mat.stat-mech
We present a novel synthesis of Fisher information and asset pricing theory that yields a practical method for reconstructing the probability density implicit in security prices. The Fisher information approach to these inverse problems transforms the search for a probability density into the solution of a differential…
Quantum decision making by social agentsphysics.soc-ph
The influence of additional information on the decision making of agents, who are interacting members of a society, is analyzed within the mathematical framework based on the use of quantum probabilities. The introduction of social interactions, which influence the decisions of individual agents, leads to a generalizat…
This paper applies quantum probability theory to model asset returns, avoiding assumptions about quantum effects.
problem Modeling asset returns with classical probability theory.
method Derives a Schrödinger-like trading equation using quantum probability, linking it to traders' decisions and market behaviors.
result Quantum probability can describe multimodal distributions of asset returns without assuming quantum effects.
New tools connect CP to GF inference for better probabilistic prediction.
problem Lack of versatility in conformal prediction for quantifying evidence.
method Imprecise probability theory and generalized fiducial inference.
result Establishes a formal connection between CP and GF inference.
We study the problem of identifying a probability distribution for some given randomly sampled data in the limit, in the context of algorithmic learning theory as proposed recently by Vinanyi and Chater. We show that there exists a computable partial learner for the computable probability measures, while by Bienvenu, M…
Optimizes portfolios with utility theory, diversification, and leverage.
problem Finding optimal portfolio allocation strategies.
method Utility theory, exponential and logarithmic utilities, compound probability distributions, maximum expected utility, generalized mean-variance.
result Enhanced portfolio allocation strategies with natural explanations.