We extend the Fourier cosine method to discrete probability distributions, achieving faster convergence rates.
problem Extending Fourier cosine method to discrete probability distributions.
method Spectral filters and convergence rates analysis.
result Spectral filters achieve one order faster convergence rates than previously recognized.
New formulae identify discrete probability laws without needing normalization constants.
problem Characterizing non-normalized discrete probability distributions.
method Derive explicit formulae for mass functions using Stein's method.
result Developed tools for solving statistical problems without normalization constants.
Sharp bounds for high-probability estimation of discrete distributions.
problem Estimating discrete distributions with high probability under χ2-divergence. method Sharp upper and lower bounds for the classical Laplace estimator, and characterization of minimax high-probability risk for any estimator.
result Sharp bounds for high-probability estimation of discrete distributions can be achieved through a simple smoothing strategy.
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.
Refined analysis of Mitra's algorithm for discrete mixtures.
problem Classifying general discrete mixture distribution models.
method Spectral clustering tailored to bipartite stochastic block models.
result Improved separation conditions for probability distributions.
SDE automatically recovers interpretable discrete distributions.
problem Limited interpretable discrete probability laws.
method Unsupervised framework using symbolic density estimation.
result Accurately recovers interpretable discrete distributions.
In this expository paper we illustrate the generality of game theoretic probability protocols of Shafer and Vovk (2001) in finite-horizon discrete games. By restricting ourselves to finite-horizon discrete games, we can explicitly describe how discrete distributions with finite support and the discrete pricing formulas…
Study error bounds in evaluating distributional computational graphs.
problem Error analysis in evaluating graphs with inputs as probability distributions.
method Establish non-asymptotic error bounds using Wasserstein-1 distance.
result Non-asymptotic error bounds for discretization errors in distributional computational graphs.
In this paper a quantitative analysis of the ruin probability in finite time of discrete risk process with proportional reinsurance and investment of finance surplus is focused on. It is assumed that the total loss on a unit interval has a light-tailed distribution -- exponential distribution and a heavy-tailed distrib…
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.
Probability distributions produced by the cross-entropy loss for ordinal classification problems can possess undesired properties. We propose a straightforward technique to constrain discrete ordinal probability distributions to be unimodal via the use of the Poisson and binomial probability distributions. We evaluate …
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.
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…
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.
This research explores various sampling methods and probability distributions for hard alignment in sequence-to-sequence TTS synthesis.
problem Improving alignment accuracy in sequence-to-sequence text-to-speech synthesis.
method Investigated various sampling methods (greedy, beam, random) and probability distributions (Bernoulli, Concrete) for hard alignment.
result Deterministic search is more preferable than stochastic search for natural alignment transition.
Optimal testing of discrete distributions with high probability, achieving sample complexity bounds.
problem Testing discrete distributions with high probability accuracy.
method Characterizing sample complexity as a function of parameters like δ, providing sample-optimal testers.
result Optimal algorithms for closeness and independence testing, achieving within constant factors of information-theoretic lower bounds.
Exact Bayesian inference for discrete models using probability generating functions.
problem Discrete statistical models with infinite support and continuous priors.
method Probabilistic programming language with automatic differentiation and probability generating functions.
result Genfer tool provides exact solutions for a wide range of inference problems.
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.
A new method for categorical variational inference using discrete normalizing flows.
problem Challenges in optimizing variational approximations for discrete latent variables.
method Differentiable reparameterization using a mixture of discrete normalizing flows.
result Improves optimization of evidence lower bound and reduces sensitivity to hyperparameters.
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…
Flow based models such as Real NVP are an extremely powerful approach to density estimation. However, existing flow based models are restricted to transforming continuous densities over a continuous input space into similarly continuous distributions over continuous latent variables. This makes them poorly suited for m…
New algorithms for efficient return distribution approximation in reinforcement learning.
problem Efficiently approximating unknown return distributions in reinforcement learning.
method Introduced novel distributional dynamic programming algorithms for arbitrary probabilistic reward mechanisms.
result Proved error bounds for the algorithms in Wasserstein and Kolmogorov--Smirnov distances.
We describe discrete restricted Boltzmann machines: probabilistic graphical models with bipartite interactions between visible and hidden discrete variables. Examples are binary restricted Boltzmann machines and discrete naive Bayes models. We detail the inference functions and distributed representations arising in th…
Moate Simulation improves accuracy and speed of financial derivative pricing.
problem Efficiently pricing financial derivatives with high accuracy.
method Discrete time simulation of probability distributions using Moate Simulation.
result Moate Simulation provides highly accurate distributions for financial derivatives pricing.
Unified discrete diffusion for categorical data simplifies training and sampling.
problem Training and sampling in discrete diffusion models for categorical data.
method Mathematical simplifications and elegant unification of discrete-time and continuous-time discrete diffusion.
result Unified Simplified Discrete Denoising Diffusion (USD3) outperforms SOTA baselines.
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.
In quantitative finance, it is often necessary to analyze the distribution of the sum of specific functions of observed values at discrete points of an underlying process. Examples include the probability density function, the hedging error, the Asian option, and statistical hypothesis testing. We propose a method to c…
Paper proposes DAG-DB for learning discrete DAGs via backpropagation.
problem Learning Directed Acyclic Graphs (DAGs) from data.
method DAG-DB uses Discrete Backpropagation with I-MLE and Straight-Through Estimation.
result DAG-DB learns DAGs effectively using probabilistic sampling and backpropagation.
The discrete sum of geometric Brownian motions plays an important role in modeling stochastic annuities in insurance. It also plays a pivotal role in the pricing of Asian options in mathematical finance. In this paper, we study the probability distributions of the infinite sum of geometric Brownian motions, the sum of …
This work bounds the generalization error of private algorithms for discrete data.
problem Bounding the generalization error of private algorithms for discrete data.
method Information-theoretic approach using relative entropy and the method of types.
result Explicit upper bounds on the generalization error of stable private algorithms for discrete data.
This paper proposes an active learning-based Gaussian process (AL-GP) metamodelling method to estimate the cumulative as well as complementary cumulative distribution function (CDF/CCDF) for forward uncertainty quantification (UQ) problems. Within the field of UQ, previous studies focused on developing AL-GP approaches…
Combines expert knowledge and data for efficient probability distribution inference.
problem Inferring discrete probability distributions using limited data and expert knowledge.
method A novel estimator that weights expert knowledge and empirical data.
result The proposed estimator is always more efficient than either expert or data alone.
We propose a correlated stochastic process of which the novel non-Gaussian probability mass function is constructed by exactly solving moment generating function. The calculation of cumulants and auto-correlation shows that the process is convergent and scale invariant in the large but finite number limit. We demonstra…
Novel approach for estimating joint probability densities using tensor decompositions and dictionaries.
problem Estimating joint probability densities of mixed discrete and continuous variables.
method Low-rank tensor decomposition combined with dictionary learning.
result Better classification and lower error rates compared to existing methods.
We propose a method to classify the causal relationship between two discrete variables given only the joint distribution of the variables, acknowledging that the method is subject to an inherent baseline error. We assume that the causal system is acyclicity, but we do allow for hidden common causes. Our algorithm presu…
New bounds for generative models under weaker assumptions.
problem Establishing convergence guarantees for generative models under weak assumptions.
method Non-asymptotic 2-Wasserstein distance bounds for probability flow ODEs under weak log-concavity and Lipschitz continuity.
result Concrete convergence rates for generative models, including non-log-concave distributions.
The reparameterization trick enables optimizing large scale stochastic computation graphs via gradient descent. The essence of the trick is to refactor each stochastic node into a differentiable function of its parameters and a random variable with fixed distribution. After refactoring, the gradients of the loss propag…
Optimizes risk-neutral probabilities for derivative pricing.
problem Deriving bounds on derivative values under multiple risk-neutral scenarios.
method Convex optimization over the set of risk-neutral probability distributions.
result Tractable finite-dimensional optimization problems for pricing.
Paper studies matching of samples from two distributions with a Gibbs probability weight.
problem Matching two independent i.i.d. samples from two distributions with a weighted cost.
method Uses chaos decomposition of polynomial functions of empirical distributions to derive asymptotics.
result Convergence of resulting random joint distribution to Schrödinger problem solution as N→∞.
A new algorithm for minimizing functions on Wasserstein space.
problem Discretization of continuous Wasserstein gradient flows in machine learning.
method Forward-Backward discretization scheme for minimizing functions with smooth and nonsmooth components.
result The FB scheme converges similarly to proximal gradient algorithms in Euclidean spaces.
Estimates high-dimensional distributions using tree tensor networks.
problem Estimating high-dimensional probability distributions from i.i.d. samples.
method Tree-based tensor formats, empirical risk minimization, L2 contrast, orthogonal bases.
result Effective approximation of classical probabilistic models like Gaussian and graphical models.
We present a deep learning model, DE-LSTM, for the simulation of a stochastic process with an underlying nonlinear dynamics. The deep learning model aims to approximate the probability density function of a stochastic process via numerical discretization and the underlying nonlinear dynamics is modeled by the Long Shor…
This paper shows that one cannot learn the probability of rare events without imposing further structural assumptions. The event of interest is that of obtaining an outcome outside the coverage of an i.i.d. sample from a discrete distribution. The probability of this event is referred to as the "missing mass". The impo…
We present an approach to deep estimation of discrete conditional probability distributions. Such models have several applications, including generative modeling of audio, image, and video data. Our approach combines two main techniques: dyadic partitioning and graph-based smoothing of the discrete space. By recursivel…
Estimates mode of discrete distributions with fewer samples.
problem Identifying the mode of a discrete distribution with high probability.
method Generalizes PPR martingale confidence sequences to handle multiple modes.
result PPR-1v1 stopping rule is asymptotically optimal and significantly more efficient.
In this paper, we propose the discrete time Compound Beta-Binomial Risk Model with by-claims, delayed by-claims and randomized dividends. We then analyze the Gerber-Shiu function for the cases where the dividend threshold d=0 and d>0 under the assumption that the constant discount rate ν∈(0,1). More specifical…
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.
This paper finds a unique partition of a sample space for estimating continuous distributions.
problem Estimating continuous probability distributions from finite samples.
method Equal-probability partition of the sample space using order statistics.
result The partition yields an entropy of log2(N+1) bits, providing a discrete entropy estimate.