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.
This paper uses RBM to calculate conditional probabilities for nonlinear system identification.
problem Challenges in obtaining probability distributions for nonlinear system identification.
method Modified RBM to calculate joint, input, and conditional probabilities.
result The method outperforms other black-box models in noisy, complex systems.
Non-trivialization probability of arc system in 3D space
problem Defining and generalizing the knotting probability of an arc diagram in 3D space
method Transforming polygonal arcs in 3D space into unique arc diagrams
result Introducing and generalizing the Non-Trivialization probability (NT probability) for arc systems in 3D space
Identifies conditions for multiple invariant probabilities in Markov kernels.
problem Global irreducibility and recurrence do not guarantee uniqueness of invariant probabilities.
method Uses Jordan decomposition of the difference of two invariant probabilities.
result A Markov kernel has more than one invariant probability if and only if it admits a visible absorbing decomposition.
NT probability measures knotting in 3D arc systems.
problem Measuring knotting in 3D arc systems.
method Transforming polygonal arcs into unique diagrams, generalizing NT probability.
result Properties of NT probability for 3D arc systems are shown.
Estimates class posterior probabilities without using scores from classifiers.
problem Estimating class posterior probabilities for new points in classification tasks.
method Varying prior probabilities to derive the ratio of pdf's at point x, directly determining class posterior probabilities.
result A method to estimate posterior probabilities without relying on classification scores.
Probability calibration trees improve accuracy of probability estimates.
problem Improving accuracy and calibration of probability estimates from classifiers.
method Probability calibration trees modify logistic model trees to learn different models in regions of the input space.
result Probability calibration trees outperform isotonic regression and Platt scaling in terms of root mean squared error.
Interpretable classifier improves accuracy through probability series expansion.
problem Improving classifier accuracy while maintaining interpretability.
method Directly measures class probabilities from training data, refines predictions through series expansion.
result Achieves comparable accuracy to Random Forests on four datasets.
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 …
This work improves deep neural network probability estimation methods.
problem Estimating probabilities from high-dimensional data with inherent uncertainty.
method Investigates and compares methods for probability estimation using deep neural networks, proposing a new method that promotes consistent probabilities.
result The new method outperforms existing approaches on most metrics on simulated and real-world data.
Proposes unimodal probability distributions for better ordinal classification.
problem Undesired properties of cross-entropy loss distributions for ordinal classification.
method Uses Poisson and binomial distributions to constrain discrete ordinal probability distributions to be unimodal.
result Obtains promising results on deep learning ordinal image datasets.
Tutorial on estimating SVM class probabilities.
problem Estimating class probabilities for SVM models.
method Compute implied posterior probabilities via isotonic regression.
result Calibrated implied posterior probabilities for SVMs.
Explicit formula derived for Slepian process boundary non-crossing probabilities.
problem Calculating boundary non-crossing probabilities for Slepian processes.
method Derived explicit formula and approximation formula for general continuous boundaries.
result Easy to implement formulas for boundary non-crossing probabilities.
This paper studies geometrical structure of the manifold of escort probability distributions and shows its new applicability to information science. In order to realize escort probabilities we use a conformal transformation that flattens so-called alpha-geometry of the space of discrete probability distributions, which…
Estimating IPM is as hard as estimating under IPM, both requiring similar optimal rates.
problem Estimating Integral Probability Metrics (IPMs) between probability measures.
method Study of minimax optimal rates for IPM estimation and under IPM estimation based on samples.
result Minimax optimal rates for estimating IPM and estimating under IPM are multiplicatively equivalent.
Defines financial models without probability theory.
problem Establishing martingale theory without probability.
method Introducing supermartingales, martingales, and semimartingales in continuous price paths.
result Probability-free versions of martingale results established.
A new method for adapting to label shifts using class probability matching.
problem Adapting to label shifts where class probabilities differ between source and target domains.
method Class Probability Matching using Kernel Methods (CPMKM) framework.
result CPMKM outperforms existing methods on real datasets.
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.
Study generalizes property elicitation to imprecise probabilities.
problem Minimizing risk over imprecise probability distributions.
method Maximin risk minimization over a set of imprecise probabilities.
result Conditions for elicitability of IP-properties.
Study on identifying probability distributions from random data, showing computable partial learners exist.
problem Identifying probability distributions from random data samples.
method Algorithmic learning theory approach, focusing on computable probability measures and high oracles.
result Characterization of oracles that compute explanatory learners for computable probability measures.
Study on the probability of immunity and its bounds.
problem Estimating the probability of immunity and its bounds.
method Derive necessary and sufficient conditions for non-immunity and ε-bounded immunity; introduce indirect immunity; propose sensitivity analysis.
result Estimate the probability of benefit and produce tighter bounds of the probability of benefit.
Deep learning approximates poker probabilities efficiently.
problem Intractable calculation of poker probabilities using combinatorics.
method Deep learning to approximate probabilities from Monte Carlo simulations.
result Deep learning model provides high-accuracy probabilities efficiently.
We link probability density functions to Fisher information metrics.
problem Constructing probability density functions from Fisher information metrics.
method Utilizing the spatially disjoint product of probability density functions and their Fisher information metric tensors.
result A method for constructing arbitrary Riemannian Fisher information metric tensors.
The paper addresses probability calibration for incomplete sequences.
problem Improving probability estimates from incomplete sequences.
method Adapting traditional calibration techniques to sequences of varying lengths.
result Proposed methods improve probability calibration for modern sequential models.
Functional approach calculates path probabilities in stochastic motion.
problem Calculating path probabilities in stochastic motion.
method Functional technique applied to derive path probability distribution.
result General formula derived for path probability distribution.
A new framework for probabilistic learning using Maximum Probability Theorem.
problem Challenges in defining and quantifying model probabilities in probabilistic learning.
method Introduces a new probabilistic framework based on Maximum Probability Theorem, defining models as events with quantified probability measures.
result The probability of a model is invariant to reparameterization and depends solely on the likelihood function.
Investigates statistical properties of perturb-softmax and perturb-argmax distributions.
problem Underexplored statistical properties of Gumbel-Softmax and Gumbel-Argmax distributions.
method Investigates convexity and differentiability to determine completeness and minimality of these distributions.
result Identifies parameters that admit complete and minimal representation of probability distributions.
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.
Maps sets to probability distributions to minimize information loss.
problem Learning to map sets to probability distributions to preserve information.
method Relates set operations to probability distribution interpolations and demonstrates a preliminary solution.
result Experimental results show the effectiveness of the set embedding approach.
This paper analyzes the probability flow in the stock market using the Black-Scholes model.
problem The non-conservation of probability in the stock market.
method Expressed the Black-Scholes equation in Hamiltonian form and analyzed the flow of probability.
result Conditions under which probability might be conserved in the market, challenging the non-Hermitian nature of the Black-Scholes Hamiltonian.
Bayesian method for high-dimensional categorical data analysis.
problem Difficulties in probability modeling for high-dimensional data.
method Bayesian learning of clique tree structure.
result Optimal clique tree structure for probability modeling.
Paper constructs unfaithful probability distributions in binary causal graphs.
problem Unfaithful probability distributions in binary causal graphs.
method Constructs unfaithful probability distributions in binary causal graphs.
result Examples of unfaithful probability distributions in binary causal graphs.
Study improves estimation of rare language model outputs.
problem Estimating probabilities of rare outputs in language models.
method Importance sampling vs. activation extrapolation for low probability estimation.
result Importance sampling outperforms activation extrapolation.
The study evaluates Bregman divergences for learning crowd probabilities.
problem Learning crowd probabilities from global perspectives.
method Adapting machine learning models to target probability distributions using Bregman divergences.
result Special attention is needed when constructing objective functions for neural network optimization.
Quantum probability metrics improve distribution comparison in high dimensions.
problem Challenges in comparing probability distributions, especially in high-dimensional and non-compact domains.
method Quantum probability metrics (QPMs) derived from quantum state spaces, overcoming limitations of MMD.
result QPMs offer enhanced sensitivity to subtle distributional differences in high dimensions and improve performance in generative modeling.
AI can learn true probabilities if data and assumptions align.
problem Understanding when AI models can accurately represent true objective probabilities.
method Proved conditions under which AI can learn true probabilities.
result Conditions for learning true probabilities are identified.
A method for diffusion on probability simplex for generative models.
problem Tension between continuous and discrete data in diffusion models.
method Proposes using softmax function applied to Ornstein-Uhlenbeck Process on probability simplex.
result Method extends to bounded image generation.
The study examines Fisher-Riemann geodesics for nonparametric probability densities.
problem Understanding nonparametric probability densities using Fisher-Riemann geometry.
method Obtaining Fisher-Riemann geodesics as a limit of parametric cases with increasing parameters.
result The weak limit approach for nonparametric probability densities.
Bayesian approach approximates probability functions of Gaussian mixtures.
problem Approximating probability functions of non-spherical Gaussian mixtures.
method Bayesian decomposition, spherical radial decomposition, random sampling.
result Established differentiability and integral representation of gradient for probability functions.
Conditional probabilities modeled using Riemann-Theta Boltzmann Machines.
problem Modeling conditional probabilities in Boltzmann machines.
method Deriving conditional density functions from Riemann-Theta Boltzmann machines.
result Conditional densities can be directly inferred from Riemann-Theta Boltzmann machines.
Geometric calculus on probability simplex using Wasserstein metric.
problem Calculus on the probability simplex with Wasserstein metric.
method Embedding probability simplex in positive measure space with nonlinear metric tensor, deriving Christoffel symbols, connections, curvature tensors, and operators.
result Established geometric computations on probability manifold and density space, connecting Fisher-Rao and optimal transport metrics.
TensorFlow Probability introduces JointDistributions for probabilistic programming.
problem Specifying models in probabilistic programming languages.
method Declarative representations of directed graphical models.
result JointDistributions for TensorFlow Probability.
We describe a Groebner basis of relations among conditional probabilities in a discrete probability space, with any set of conditioned-upon events. They may be specialized to the partially-observed random variable case, the purely conditional case, and other special cases. We also investigate the connection to generali…
Paper simplifies calculating causation probabilities and ranks root causes.
problem Computational challenges in assessing causal relationships.
method Algorithmic simplifications and novel methodological framework for Root Cause Analysis.
result Significantly reduces computational complexity for calculating causation probabilities.
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.
Paper sets minimax bounds for Wasserstein distribution estimation.
problem Estimating a probability distribution using Wasserstein distance.
method Uses metric properties and weak moment assumptions.
result Upper and lower bounds on statistical minimax rates.
The law of total probability may be deployed in binary classification exercises to estimate the unconditional class probabilities if the class proportions in the training set are not representative of the population class proportions. We argue that this is not a conceptually sound approach and suggest an alternative ba…
This study redefines probability for finite outcomes using axioms and examples.
problem Defining probability for finite outcomes and preserving information.
method Developed three axioms for relative probability functions and provided examples and a system for their composition.
result Proved the topological closure of the relative probability space, preserving information under limits.