The article proposes and investigates the performance of two Bayesian nonparametric estimation procedures in the context of benchmark dose estimation in toxicological animal experiments. The methodology is illustrated using several existing animal dose-response data sets and is compared with traditional parametric methods available in standard benchmark dose estimation software (BMDS), as well as with a published model-averaging approach and a frequentist nonparametric approach. These comparisons together with simulation studies suggest that the nonparametric methods provide a lot of flexibility in terms of model fit and can be a very useful tool in benchmark dose estimation studies, especially when standard parametric models fail to fit to the data adequately.
Please cite this article in press as: N. Guha et al., A variational Bayesian approach for inverse problems with skew-t error distributions, J. Comput. Phys. (2015), http://dx.
AbstractIn this work, we develop a novel robust Bayesian approach to inverse problems with data errors following a skew-t distribution. A hierarchical Bayesian model is developed in the inverse problem setup. The Bayesian approach contains a natural mechanism for regularization in the form of a prior distribution, and a LASSO type prior distribution is used to strongly induce sparseness. We propose a variational type algorithm by minimizing the Kullback-Leibler divergence between the true posterior distribution and a separable approximation. The proposed method is illustrated on several two-dimensional linear and nonlinear inverse problems, e.g. Cauchy problem and permeability estimation problem.
In this paper, we develop a Bayesian multiscale approach based on a multiscale finite element method. Because of scale disparity in many multiscale applications, computational models can not resolve all scales. Various subgrid models are proposed to represent un-resolved scales. Here, we consider a probabilistic approach for modeling unresolved scales using the Multiscale Finite Element Method (cf., [1,2]). By representing dominant modes using the Generalized Multiscale Finite Element, we propose a Bayesian framework, which provides multiple inexpensive (computable) solutions for a deterministic problem. These approximate probabilistic solutions may not be very close to the exact solutions and, thus, many realizations are needed. In this way, we obtain a rigorous probabilistic description of approximate solutions. In the paper, we consider parabolic and wave equations in heterogeneous media. In each time interval, the domain is divided into subregions. Using residual information, we design appropriate prior and posterior distributions. The likelihood consists of the residual minimization. To
In this paper, we study porous media flows in heterogeneous stochastic media. We propose an efficient forward simulation technique that is tailored for variational Bayesian inversion. As a starting point, the proposed forward simulation technique decomposes the solution into the sum of separable functions (with respect to randomness and the space), where each term is calculated based on a variational approach. This is similar to Proper Generalized Decomposition (PGD). Next, we apply a multiscale technique to solve for each term (as in [1]) and, further, decompose the random function into 1D fields. As a result, our proposed method provides an approximation hierarchy for the solution as we increase the number of terms in the expansion and, also, increase the spatial resolution of each term. We use the hierarchical solution distributions in a variational Bayesian approximation to perform uncertainty quantification in the inverse problem. We conduct a detailed numerical study to explore the performance of the proposed uncertainty quantification technique and show the theoretical posterior concentration.
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