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We introduce and study a class of probabilistic generative models, where the latent object is a finite-dimensional diffusion process on a finite time interval and the observed variable is drawn conditionally on the terminal point of the…

Probability · Mathematics 2019-06-03 Belinda Tzen , Maxim Raginsky

We introduce a nonparametric approach for estimating drift and diffusion functions in systems of stochastic differential equations from observations of the state vector. Gaussian processes are used as flexible models for these functions and…

Data Analysis, Statistics and Probability · Physics 2018-08-15 Philipp Batz , Andreas Ruttor , Manfred Opper

We consider a nonparametric Bayesian approach to estimate the diffusion coefficient of a stochastic differential equation given discrete time observations over a fixed time interval. As a prior on the diffusion coefficient, we employ a…

Statistics Theory · Mathematics 2020-07-22 Shota Gugushvili , Frank van der Meulen , Moritz Schauer , Peter Spreij

We study nonparametric estimation of the diffusion coefficient from discrete data, when the observations are blurred by additional noise. Such issues have been developed over the last 10 years in several application fields and in particular…

Statistics Theory · Mathematics 2011-12-30 Marc Hoffmann , Axel Munk , Johannes Schmidt-Hieber

Sampling from the posterior is a key technical problem in Bayesian statistics. Rigorous guarantees are difficult to obtain for Markov Chain Monte Carlo algorithms of common use. In this paper, we study an alternative class of algorithms…

Statistics Theory · Mathematics 2024-08-26 Andrea Montanari , Yuchen Wu

We consider the problem of approximating the product of $n$ expectations with respect to a common probability distribution $\mu$. Such products routinely arise in statistics as values of the likelihood in latent variable models. Motivated…

Computation · Statistics 2017-09-05 Anthony Lee , Simone Tiberi , Giacomo Zanella

Given a statistical model, we propose a novel estimation method that yields randomised estimators for the unknown distribution of an observed random variable. We establish non-asymptotic bounds for the performance of these estimators and…

Statistics Theory · Mathematics 2026-05-06 Yannick Baraud

In this paper, we present a theoretical and computational workflow for the non-parametric Bayesian inference of drift and diffusion functions of autonomous diffusion processes. We base the inference on the partial differential equations…

Computational Engineering, Finance, and Science · Computer Science 2024-11-05 Maximilian Kruse , Sebastian Krumscheid

While efficient distribution learning is no doubt behind the groundbreaking success of diffusion modeling, its theoretical guarantees are quite limited. In this paper, we provide the first rigorous analysis on approximation and…

Machine Learning · Statistics 2023-03-06 Kazusato Oko , Shunta Akiyama , Taiji Suzuki

Consider a process, stochastic or deterministic, obtained by using a numerical integration scheme, or from Monte-Carlo methods involving an approximation to an integral, or a Newton-Raphson iteration to approximate the root of an equation.…

Computational Finance · Quantitative Finance 2010-06-17 Don McLeish

Standard variational lower bounds used to train latent variable models produce biased estimates of most quantities of interest. We introduce an unbiased estimator of the log marginal likelihood and its gradients for latent variable models…

Machine Learning · Computer Science 2020-07-14 Yucen Luo , Alex Beatson , Mohammad Norouzi , Jun Zhu , David Duvenaud , Ryan P. Adams , Ricky T. Q. Chen

In this paper we consider parameter estimation for discretely observed diffusion processes. In particular, we focus on data that are observed at low frequency and methodology that can estimate parameters with uncertainty quantification.…

Computation · Statistics 2026-05-01 Jingning Yao , Ajay Jasra , Sheng Jiang

In this paper we consider the estimation of unknown parameters in Bayesian inverse problems. In most cases of practical interest, there are several barriers to performing such estimation, This includes a numerical approximation of a…

Methodology · Statistics 2025-02-07 Neil K. Chada , Ajay Jasra , Mohamed Maama , Raul Tempone

Discretizations of Langevin diffusions provide a powerful method for sampling and Bayesian inference. However, such discretizations require evaluation of the gradient of the potential function. In several real-world scenarios, obtaining…

Statistics Theory · Mathematics 2021-01-19 Abhishek Roy , Lingqing Shen , Krishnakumar Balasubramanian , Saeed Ghadimi

Diffusion models, which convert noise into new data instances by learning to reverse a diffusion process, have become a cornerstone in contemporary generative modeling. In this work, we develop non-asymptotic convergence theory for a…

Machine Learning · Computer Science 2024-08-06 Gen Li , Yuting Wei , Yuejie Chi , Yuxin Chen

The purpose of the present work is to construct estimators for the random effects in a fractional diffusion model using a hybrid estimation method where we combine parametric and nonparametric thechniques. We precisely consider $n$…

Statistics Theory · Mathematics 2025-06-13 Nesrine Chebli , Hamdi Fathallah , Yousri Slaoui

This paper investigates a financial market where returns depend on an unobservable Gaussian drift process. While the observation of returns yields information about the underlying drift, we also incorporate discrete-time expert opinions as…

Portfolio Management · Quantitative Finance 2021-11-04 Jörn Sass , Dorothee Westphal , Ralf Wunderlich

Machine learning models deployed on medical imaging tasks must be equipped with out-of-distribution detection capabilities in order to avoid erroneous predictions. It is unsure whether out-of-distribution detection models reliant on deep…

Computer Vision and Pattern Recognition · Computer Science 2022-06-28 Sebastian G. Popescu , David J. Sharp , James H. Cole , Konstantinos Kamnitsas , Ben Glocker

We present a Bayesian non-parametric way of inferring stochastic differential equations for both regression tasks and continuous-time dynamical modelling. The work has high emphasis on the stochastic part of the differential equation, also…

Machine Learning · Statistics 2020-06-29 Martin Jørgensen , Marc Peter Deisenroth , Hugh Salimbeni

We consider a reflected Ornstein-Uhlenbeck process $X$ driven by a fractional Brownian motion with Hurst parameter $H\in (0, \frac12) \cup (\frac12, 1)$. Our goal is to estimate an unknown drift parameter $\alpha\in (-\infty,\infty)$ on the…

Statistics Theory · Mathematics 2015-03-24 Chihoon Lee , Jian Song