Nilabja Guha

Nonparametric Bayesian Methods for Benchmark Dose Estimation

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.

A variational Bayesian approach for inverse problems with skew-t error distributions

In 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.

Multilevel approximate Bayesian approaches for flows in highly heterogeneous porous media and their applications

Estimation of quantities related to high-contrast flow problems such as permeability field plays an important role in porous media characterization. A Generalized Multiscale Finite Element Method (GMsFEM) can be used for solving parameter-dependent (or stochastic) flow problems with multiscale nature. A hierarchy of approximations of different resolution can be provided by GMsFEM. Hence, it can be coupled with Multilevel Markov Chain Monte Carlo (MLMCMC) to generate samples in different levels and form the multilevel estimator. KarhunenLoeve Expansion (KLE) is used to parameterize the underlying random field by a function of Gaussian random field. Instead of MCMC, an Approximate Bayesian Computation (ABC) method can be used within the Multilevel Monte Carlo framework. ABC can be incorporated in different levels to reduce the computational cost and to produce an approximate solution by ensembling different levels.

Bayesian and variational Bayesian approaches for flows in heterogeneous random media

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.