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We attempt image restoration in the framework of the Baysian inference. Recently, it has been shown that under a certain criterion the MAP (Maximum A Posterior) estimate, which corresponds to the minimization of energy, can be outperformed…

Disordered Systems and Neural Networks · Physics 2009-11-07 Hayaru Shouno , Koji Wada , Masato Okada

For large model spaces, the potential entrapment of Markov chain Monte Carlo (MCMC) based methods with spike-and-slab priors poses significant challenges in posterior computation in regression models. On the other hand, maximum a posteriori…

Methodology · Statistics 2026-02-25 Shamriddha De , Joyee Ghosh

In order to estimate model parameters and circumvent possible difficulties encountered with the likelihood function, we propose to replace the likelihood in the formula of the posterior distribution by a function depending on a contrast.…

Statistics Theory · Mathematics 2008-12-18 S. Soubeyrand , F. Carpentier , N. Desassis , J. Chadœuf

Gaussian multiplicative noise is commonly used as a stochastic regularisation technique in training of deterministic neural networks. A recent paper reinterpreted the technique as a specific algorithm for approximate inference in Bayesian…

Machine Learning · Statistics 2017-11-09 Jiri Hron , Alexander G. de G. Matthews , Zoubin Ghahramani

Sensor noise sources cause differences in the signal recorded across pixels in a single image and across multiple images. This paper presents a Bayesian approach to decomposing and characterizing the sensor noise sources involved in imaging…

We consider the concept of Bayes risk in the context of finite-dimensional ill-posed linear inverse problem with Gaussian prior and noise models. In this note, we rederive the following well-known result: in the present Gaussian linear…

Statistics Theory · Mathematics 2023-12-27 Alen Alexanderian

Bayesian learning provides a unified skeleton to solve the electrophysiological source imaging task. From this perspective, existing source imaging algorithms utilize the Gaussian assumption for the observation noise to build the likelihood…

Machine Learning · Computer Science 2025-08-07 Yuanhao Li , Badong Chen , Zhongxu Hu , Keita Suzuki , Wenjun Bai , Yasuharu Koike , Okito Yamashita

We consider a Bayesian approach to model selection in Gaussian linear regression, where the number of predictors might be much larger than the number of observations. From a frequentist view, the proposed procedure results in the penalized…

Statistics Theory · Mathematics 2010-09-14 Felix Abramovich , Vadim Grinshtein

We present a parametric deterministic formulation of Bayesian inverse problems with input parameter from infinite dimensional, separable Banach spaces. In this formulation, the forward problems are parametric, deterministic elliptic partial…

Analysis of PDEs · Mathematics 2015-05-27 Ch. Schwab , A. M. Stuart

A central theme in classical algorithms for the reconstruction of discontinuous functions from observational data is perimeter regularization via the use of the total variation. On the other hand, sparse or noisy data often demands a…

Optimization and Control · Mathematics 2020-04-13 Oliver R. A. Dunbar , Matthew M. Dunlop , Charles M. Elliott , Viet Ha Hoang , Andrew M. Stuart

Compressed sensing deals with the reconstruction of sparse signals using a small number of linear measurements. One of the main challenges in compressed sensing is to find the support of a sparse signal. In the literature, several bounds on…

Information Theory · Computer Science 2009-11-26 Ali Hormati , Amin Karbasi , Soheil Mohajer , Martin Vetterli

Since the seminal work of Venkatakrishnan et al. in 2013, Plug & Play (PnP) methods have become ubiquitous in Bayesian imaging. These methods derive Minimum Mean Square Error (MMSE) or Maximum A Posteriori (MAP) estimators for inverse…

We propose a machine-learning algorithm for Bayesian inverse problems in the function-space regime based on one-step generative transport. Building on the Mean Flows, we learn a fully conditional amortized sampler with a neural-operator…

Machine Learning · Statistics 2026-03-17 Zilan Cheng , Li-Lian Wang , Zhongjian Wang

This paper considers the objective comparison of stochastic models to solve inverse problems, more specifically image restoration. Most often, model comparison is addressed in a supervised manner, that can be time-consuming and partly…

Computation · Statistics 2020-10-14 Benjamin Harroué , Jean-François Giovannelli , Marcelo Pereyra

Estimating conditional dependence graphs and precision matrices are some of the most common problems in modern statistics and machine learning. When data are fully observed, penalized maximum likelihood-type estimators have become standard…

Machine Learning · Statistics 2019-04-09 Roger Fan , Byoungwook Jang , Yuekai Sun , Shuheng Zhou

We study linear models under heavy-tailed priors from a probabilistic viewpoint. Instead of computing a single sparse most probable (MAP) solution as in standard deterministic approaches, the focus in the Bayesian compressed sensing…

Computer Vision and Pattern Recognition · Computer Science 2014-03-05 George Papandreou , Alan Yuille

Bayesian approach, as a useful tool for quantifying uncertainties, has been widely used for solving inverse problems of partial differential equations (PDEs). One of the key difficulties for employing Bayesian approach for the issue is how…

Numerical Analysis · Mathematics 2026-02-09 Junxiong Jia , Qian Zhao , Zongben Xu , Deyu Meng , Yee Leung

This paper tackles the problem of jointly estimating the noise covariance matrix alongside states (parameters such as poses and points) from measurements corrupted by Gaussian noise and, if available, prior information. In such settings,…

Robotics · Computer Science 2025-08-13 Kasra Khosoussi , Iman Shames

Maximum a posteriori and Bayes estimators are two common methods of point estimation in Bayesian Statistics. It is commonly accepted that maximum a posteriori estimators are a limiting case of Bayes estimators with 0-1 loss. In this paper,…

Statistics Theory · Mathematics 2018-02-23 Robert Bassett , Julio Deride

In high dimension, it is customary to consider Lasso-type estimators to enforce sparsity. For standard Lasso theory to hold, the regularization parameter should be proportional to the noise level, yet the latter is generally unknown in…

Machine Learning · Statistics 2017-10-19 Mathurin Massias , Olivier Fercoq , Alexandre Gramfort , Joseph Salmon