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We consider the accuracy of an approximate posterior distribution in nonparametric regression problems by combining posterior distributions computed on subsets of the data defined by the locations of the independent variables. We show that…
We present some properties of measures (q-Gaussian) that orthogonalize the set of q-Hermite polynomials. We also present an algorithm for simulating i.i.d. sequences of random variables having q-Gaussian distribution.
We recently proposed a general algorithm for approximating nonstandard Bayesian posterior distributions by minimization of their Kullback-Leibler divergence with respect to a more convenient approximating distribution. In this note we offer…
In computational inverse problems, it is common that a detailed and accurate forward model is approximated by a computationally less challenging substitute. The model reduction may be necessary to meet constraints in computing time when…
Exponential random graph models are an important tool in the statistical analysis of data. However, Bayesian parameter estimation for these models is extremely challenging, since evaluation of the posterior distribution typically involves…
This paper addresses the issue of inversion in cases where (1) the observation system is modeled by a linear transformation and additive noise, (2) the problem is ill-posed and regularization is introduced in a Bayesian framework by an a…
Probabilistic machine learning models are distinguished by their ability to integrate prior knowledge of noise statistics, smoothness parameters, and training data uncertainty. A common approach involves modeling data with Gaussian…
We propose to use L\'evy {\alpha}-stable distributions for constructing priors for Bayesian inverse problems. The construction is based on Markov fields with stable-distributed increments. Special cases include the Cauchy and Gaussian…
In Bayesian inverse problems, the posterior distribution is used to quantify uncertainty about the reconstructed solution. In practice, Markov chain Monte Carlo algorithms often are used to draw samples from the posterior distribution.…
This paper considers properties of an optimization based sampler for targeting the posterior distribution when the likelihood is intractable and auxiliary statistics are used to summarize information in the data. Our reverse sampler…
The Bayesian solution to a statistical inverse problem can be summarised by a mode of the posterior distribution, i.e. a MAP estimator. The MAP estimator essentially coincides with the (regularised) variational solution to the inverse…
This paper tackles efficient methods for Bayesian inverse problems with priors based on Whittle--Mat\'ern Gaussian random fields. The Whittle--Mat\'ern prior is characterized by a mean function and a covariance operator that is taken as a…
In this paper we study algorithms to find a Gaussian approximation to a target measure defined on a Hilbert space of functions; the target measure itself is defined via its density with respect to a reference Gaussian measure. We employ the…
We propose a dimension reduction technique for Bayesian inverse problems with nonlinear forward operators, non-Gaussian priors, and non-Gaussian observation noise. The likelihood function is approximated by a ridge function, i.e., a map…
We present a new primitive for quantum algorithms that implements a discrete Hermite transform efficiently, in time that depends logarithmically in both the dimension and the inverse of the allowable error. This transform, which maps basis…
Many inverse problems focus on recovering a quantity of interest that is a priori known to exhibit either discontinuous or smooth behavior. Within the Bayesian approach to inverse problems, such structural information can be encoded using…
In Bayesian theory, calculating a posterior probability distribution is highly important but usually difficult. Therefore, some methods have been put forward to deal with such problem, among which, the most popular one is the asymptotic…
Accelerated algorithms for maximum likelihood image reconstruction are essential for emerging applications such as 3D tomography, dynamic tomographic imaging, and other high dimensional inverse problems. In this paper, we introduce and…
Solving ill-posed inverse problems by Bayesian inference has recently attracted considerable attention. Compared to deterministic approaches, the probabilistic representation of the solution by the posterior distribution can be exploited to…
We propose a general framework using spike-and-slab prior distributions to aid with the development of high-dimensional Bayesian inference. Our framework allows inference with a general quasi-likelihood function. We show that highly…