Surrogate modeling based on Gaussian processes (GPs) has received increasing attention in the analysis of complex problems in science and engineering. Despite extensive studies on GP modeling, the developments for functional inputs are scarce. Motivated by an inverse scattering problem in which functional inputs representing the support and material properties of the scatterer are involved in the partial differential equations, a new class of kernel functions for functional inputs is introduced for GPs. Based on the proposed GP models, the asymptotic convergence properties of the resulting mean squared prediction errors are derived and the finite sample performance is demonstrated by numerical examples. In the application to inverse scattering, a surrogate model is constructed with functional inputs, which is crucial to recover the reflective index of an inhomogeneous isotropic scattering region of interest for a given far-field pattern.
In this paper we present Poisson mixture approaches for origin-destination (OD) modeling in transportation analysis. We introduce covariate-based models which incorporate different transport modeling phases and also allow for direct probabilistic inference on link traffic based on Bayesian predictions. Emphasis is placed on the Poisson-inverse Gaussian as an alternative to the commonly-used Poisson-gamma and Poisson-lognormal models. We present a first full Bayesian formulation and demonstrate that the Poisson-inverse Gaussian is particularly suited for OD analysis due to desirable marginal an…
Multivariate compositional count data arise in many applications including ecology, microbiology, genetics, and paleoclimate. A frequent question in the analysis of multivariate compositional count data is what values of a covariate(s) give rise to the observed composition. Learning the relationship between covariates and the compositional count allows for inverse prediction of unobserved covariates given compositional count observations. Gaussian processes provide a flexible framework for modeling functional responses with respect to a covariate without assuming a functional form. Many scient…
We consider the P(M,lambda,tau) maintenance policy of a dam using the total discounted and long-run average costs, when the input process is inverse Gaussian.
In this paper, we present some asymptotic properties of the normalized inverse-Gaussian process. In particular, when the concentration parameter is large, we establish an analogue of the empirical functional central limit theorem, the strong law of large numbers and the Glivenko-Cantelli theorem for the normalized inverse-Gaussian process and its corresponding quantile process. We also derive a finite sum-representation that converges almost surely to the Ferguson and Klass representation of the normalized inverse-Gaussian process. This almost sure approximation can be used to simulate efficie…
In this work we consider Bayesian inference problems with intractable likelihood functions. We present a method to compute an approximate of the posterior with a limited number of model simulations. The method features an inverse Gaussian Process regression (IGPR), i.e., one from the output of a simulation model to the input of it. Within the method, we provide an adaptive algorithm with a tempering procedure to construct the approximations of the marginal posterior distributions. With examples we demonstrate that IGPR has a competitive performance compared to some commonly used algorithms, es…
In this paper we consider Bayesian estimation for the parameters of inverse Gaussian distribution. Our emphasis is on Markov Chain Monte Carlo methods. We provide complete implementation of the Gibbs sampler algorithm. Assuming an informative prior, Bayes estimates are computed using the output of the Gibbs sampler and also from Lindley's approximation method. Maximum Likelihood and Uniformly Minimum Variance Unbiased estimates are obtained as well. We also compute Highest Posterior Density credible intervals, exact confidence intervals as well as "percentile" and "percentile-t" bootstrap appr…
In this paper we propose a generalized Gaussian process concurrent regression model for functional data where the functional response variable has a binomial, Poisson or other non-Gaussian distribution from an exponential family while the covariates are mixed functional and scalar variables. The proposed model offers a nonparametric generalized concurrent regression method for functional data with multi-dimensional covariates, and provides a natural framework on modeling common mean structure and covariance structure simultaneously for repeatedly observed functional data. The mean structure pr…
In this paper, we develop simple, yet efficient, procedures for sampling approximations of the two-Parameter Poisson-Dirichlet Process and the normalized inverse-Gaussian process. We compare the efficiency of the new approximations to the corresponding stick-breaking approximations, in which we demonstrate a substantial improvement.
Non-Gaussian mixture models are gaining increasing attention for mixture model-based clustering particularly when dealing with data that exhibit features such as skewness and heavy tails. Here, such a mixture distribution is presented, based on the multivariate normal inverse Gaussian (MNIG) distribution. For parameter estimation of the mixture, a Bayesian approach via Gibbs sampler is used; for this, a novel approach to simulate univariate generalized inverse Gaussian random variables and matrix generalized inverse Gaussian random matrices is provided. The proposed algorithm will be applied t…
University of Bern（伯尔尼大学）；University of Lausanne（洛桑大学）
We consider the use of Gaussian process (GP) priors for solving inverse problems in a Bayesian framework. As is well known, the computational complexity of GPs scales cubically in the number of datapoints. We here show that in the context of inverse problems involving integral operators, one faces additional difficulties that hinder inversion on large grids. Furthermore, in that context, covariance matrices can become too large to be stored. By leveraging results about sequential disintegrations of Gaussian measures, we are able to introduce an implicit representation of posterior covariance m…
We propose a parallelizable sparse inverse formulation Gaussian process (SpInGP) for temporal models. It uses a sparse precision GP formulation and sparse matrix routines to speed up the computations. Due to the state-space formulation used in the algorithm, the time complexity of the basic SpInGP is linear, and because all the computations are parallelizable, the parallel form of the algorithm is sublinear in the number of data points. We provide example algorithms to implement the sparse matrix routines and experimentally test the method using both simulated and real data.
We present and describe the GPFDA package for R. The package provides flexible functionalities for dealing with Gaussian process regression (GPR) models for functional data. Multivariate functional data, functional data with multidimensional inputs, and nonseparable and/or nonstationary covariance structures can be modeled. In addition, the package fits functional regression models where the mean function depends on scalar and/or functional covariates and the covariance structure is modeled by a GPR model. In this paper, we present the versatility of GPFDA with respect to mean function and cov…
We consider Bayesian nonparametric density estimation using a Pitman-Yor or a normalized inverse-Gaussian process kernel mixture as the prior distribution for a density. The procedure is studied from a frequentist perspective. Using the stick-breaking representation of the Pitman-Yor process or the expression of the finite-dimensional distributions for the normalized-inverse Gaussian process, we prove that, when the data are replicates from an infinitely smooth density, the posterior distribution concentrates on any shrinking $L^p$-norm ball, $1\leq p\leq\infty$, around the sampling density at…
We introduce optimal transport based resampling in adaptive SMC. We consider elliptic inverse problems of inferring hydraulic conductivity from pressure measurements. We consider two parametrizations of hydraulic conductivity: by Gaussian random field, and by a set of scalar (non-)Gaussian distributed parameters and Gaussian random fields. We show that for scalar parameters optimal transport based SMC performs comparably to monomial based SMC but for Gaussian high-dimensional random fields optimal transport based SMC outperforms monomial based SMC. When comparing to ensemble Kalman inversion w…
This work is concerned with the convergence of Gaussian process regression. A particular focus is on hierarchical Gaussian process regression, where hyper-parameters appearing in the mean and covariance structure of the Gaussian process emulator are a-priori unknown, and are learnt from the data, along with the posterior mean and covariance. We work in the framework of empirical Bayes, where a point estimate of the hyper-parameters is computed, using the data, and then used within the standard Gaussian process prior to posterior update. We provide a convergence analysis that (i) holds for any …
Bayesian inference and uncertainty quantification in a general class of non-linear inverse regression models is considered. Analytic conditions on the regression model $\{\mathscr G(θ): θ\in Θ\}$ and on Gaussian process priors for $θ$ are provided such that semi-parametrically efficient inference is possible for a large class of linear functionals of $θ$. A general semi-parametric Bernstein-von Mises theorem is proved that shows that the (non-Gaussian) posterior distributions are approximated by certain Gaussian measures centred at the posterior mean. As a consequence posterior-based credible …
Bayesian posterior distributions arising in modern applications, including inverse problems in partial differential equation models in tomography and subsurface flow, are often computationally intractable due to the large computational cost of evaluating the data likelihood. To alleviate this problem, we consider using Gaussian process regression to build a surrogate model for the likelihood, resulting in an approximate posterior distribution that is amenable to computations in practice. This work serves as an introduction to Gaussian process regression, in particular in the context of buildin…
Various approaches to stochastic processes exist, noting that key properties such as measurability and continuity are not trivially satisfied. We introduce a new theory for Gaussian processes using improper linear functionals. Using a collection of i.i.d. standard normal variables, we define Gaussian white noise and discuss its properties. This is extended to general Gaussian processes on Hilbert space, where the variance is allowed to be any suitable operator. Our main focus is $L^2$ spaces, and we discuss criteria for Gaussian processes to be continuous in this setting. Finally, we outline a…
This paper studies the infinite-dimensional Bayesian inference method with Hadamard fractional total variation-Gaussian (HFTG) prior for solving inverse problems. First, Hadamard fractional Sobolev space is established and proved to be a separable Banach space under some mild conditions. Afterwards, the HFTG prior is constructed in this separable fractional space, and the proposed novel hybrid prior not only captures the texture details of the region and avoids step effects, but also provides a complete theoretical analysis in the infinite dimensional Bayesian inversion. Based on the HFTG prio…
UC Berkeley（加州大学伯克利分校）；University of Sheffield（谢菲尔德大学）
We propose a new approach to inverse reinforcement learning (IRL) based on the deep Gaussian process (deep GP) model, which is capable of learning complicated reward structures with few demonstrations. Our model stacks multiple latent GP layers to learn abstract representations of the state feature space, which is linked to the demonstrations through the Maximum Entropy learning framework. Incorporating the IRL engine into the nonlinear latent structure renders existing deep GP inference approaches intractable. To tackle this, we develop a non-standard variational approximation framework which…
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 problem, seen as minimisation of the Onsager-Machlup functional of the posterior measure. An open problem in the stability analysis of inverse problems is to establish a relationship between the convergence properties of solutions obtained by the variational approach and by the Bayesian approach. To address this problem, we propose a general convergence theory fo…
The Bayesian methods for linear inverse problems is studied using hierarchical Gaussian models. The problems are considered with different discretizations, and we analyze the phenomena which appear when the discretization becomes finer. A hierarchical solution method for signal restoration problems is introduced and studied with arbitrarily fine discretization. We show that the maximum a posteriori estimate converges to a minimizer of the Mumford--Shah functional, up to a subsequence. A new result regarding the existence of a minimizer of the Mumford--Shah functional is proved. Moreover, we st…
For $\mathcal{O}$ a bounded domain in $\mathbb{R}^d$ and a given smooth function $g:\mathcal{O}\to\mathbb{R}$, we consider the statistical nonlinear inverse problem of recovering the conductivity $f>0$ in the divergence form equation $$ \nabla\cdot(f\nabla u)=g\ \textrm{on}\ \mathcal{O}, \quad u=0\ \textrm{on}\ \partial\mathcal{O}, $$ from $N$ discrete noisy point evaluations of the solution $u=u_f$ on $\mathcal O$. We study the statistical performance of Bayesian nonparametric procedures based on a flexible class of Gaussian (or hierarchical Gaussian) process priors, whose implementation is f…
In nuclear reactor system design and safety analysis, the Best Estimate plus Uncertainty (BEPU) methodology requires that computer model output uncertainties must be quantified in order to prove that the investigated design stays within acceptance criteria. "Expert opinion" and "user self-evaluation" have been widely used to specify computer model input uncertainties in previous uncertainty, sensitivity and validation studies. Inverse Uncertainty Quantification (UQ) is the process to inversely quantify input uncertainties based on experimental data in order to more precisely quantify such ad-h…
The statistical inverse problem of estimating the probability distribution of an infinite-dimensional unknown given its noisy indirect observation is studied in the Bayesian framework. In practice, one often considers only finite-dimensional unknowns and investigates numerically their probabilities. As many unknowns are function-valued, it is of interest to know whether the estimated probabilities converge when the finite-dimensional approximations of the unknown are refined. In this work, the generalized Bayes formula is shown to be a powerful tool in the convergence studies. With the help of…
We solve the inverse problem of deblurring a pixelized image of Jupiter using regularized deconvolution and by sample-based Bayesian inference. By efficiently sampling the marginal posterior distribution for hyperparameters, then the full conditional for the deblurred image, we find that we can evaluate the posterior mean faster than regularized inversion, when selection of the regularizing parameter is considered. To our knowledge, this is the first demonstration of sampling and inference that takes less compute time than regularized inversion in an inverse problems. Comparison to random-walk…
We consider nonparametric Bayesian estimation inference using a rescaled smooth Gaussian field as a prior for a multidimensional function. The rescaling is achieved using a Gamma variable and the procedure can be viewed as choosing an inverse Gamma bandwidth. The procedure is studied from a frequentist perspective in three statistical settings involving replicated observations (density estimation, regression and classification). We prove that the resulting posterior distribution shrinks to the distribution that generates the data at a speed which is minimax-optimal up to a logarithmic factor, …
Prior distributions for Bayesian inference that rely on the $l_1$-norm of the parameters are of considerable interest, in part because they promote parameter fields with less regularity than Gaussian priors (e.g., discontinuities and blockiness). These $l_1$-type priors include the total variation (TV) prior and the Besov $B^s_{1,1}$ space prior, and in general yield non-Gaussian posterior distributions. Sampling from these posteriors is challenging, particularly in the inverse problem setting where the parameter space is high-dimensional and the forward problem may be nonlinear. This paper ex…
In this article, we propose a new three parameter distribution by compounding negative binomial with reciprocal inverse Gaussian model called negative binomial-reciprocal inverse Gaussian distribution. This model is tractable with some important properties not only in actuarial science but in other fields as well where overdispersion pattern is seen. Some basic properties including recurrence relation of probabilities for computation of successive probabilities have been discussed. In its compound version, when the claims are absolutely continuous, an integral equation for the probability dens…
In this paper, we propose an adaptive matched detector of a signal corrupted by a non-Gaussian noise with an inverse gamma texture. The detector is formed using a set of secondary data measurements, and is analytically shown to have a constant false alarm rate. The analytic performance is validated using Monte Carlo simulations, and the proposed detector is shown to offer preferable performance as compared to the related one-step generalized likelihood ratio test (1S-GLRT) and the adaptive subspace detector (ASD).
In this article, normal inverse Gaussian (NIG) autoregressive model is introduced. The parameters of the model are estimated using Expectation Maximization (EM) algorithm. The efficacy of the EM algorithm is shown using simulated and real world financial data. It is shown that NIG autoregressive model fit very well the considered financial data and hence could be useful in modeling of various real life time-series data.
The resolution of many large-scale inverse problems using MCMC methods requires a step of drawing samples from a high dimensional Gaussian distribution. While direct Gaussian sampling techniques, such as those based on Cholesky factorization, induce an excessive numerical complexity and memory requirement, sequential coordinate sampling methods present a low rate of convergence. Based on the reversible jump Markov chain framework, this paper proposes an efficient Gaussian sampling algorithm having a reduced computation cost and memory usage. The main feature of the algorithm is to perform an a…
Parameter estimation for model-based clustering using a finite mixture of normal inverse Gaussian (NIG) distributions is achieved through variational Bayes approximations. Univariate NIG mixtures and multivariate NIG mixtures are considered. The use of variational Bayes approximations here is a substantial departure from the traditional EM approach and alleviates some of the associated computational complexities and uncertainties. Our variational algorithm is applied to simulated and real data. The paper concludes with discussion and suggestions for future work.
A new class of survival frailty models based on the Generalized Inverse-Gaussian (GIG) distributions is proposed. We show that the GIG frailty models are flexible and mathematically convenient like the popular gamma frailty model. Furthermore, our proposed class is robust and does not present some computational issues experienced by the gamma model. By assuming a piecewise-exponential baseline hazard function, which gives a semiparametric flavour for our frailty class, we propose an EM-algorithm for estimating the model parameters and provide an explicit expression for the information matrix. …
We propose a new class of goodness-of-fit tests for the inverse Gaussian distribution. The proposed tests are weighted $L^2$-type tests depending on a tuning parameter. We develop the asymptotic theory under the null hypothesis and under a broad class of alternative distributions. These results are used to show that the parametric bootstrap procedure, which we employ to implement the test, is asymptotically valid and that the whole test procedure is consistent. A comparative simulation study for finite sample sizes shows that the new procedure is competitive to classical and recent tests, outp…
The autoregressive (AR) models are used to represent the time-varying random process in which output depends linearly on previous terms and a stochastic term (the innovation). In the classical version, the AR models are based on normal distribution. However, this distribution does not allow describing data with outliers and asymmetric behavior. In this paper, we study the AR models with normal inverse Gaussian (NIG) innovations. The NIG distribution belongs to the class of semi heavy-tailed distributions with wide range of shapes and thus allows for describing real-life data with possible jump…
