Bayesian methods feature useful properties for solving inverse problems, such as tomographic reconstruction. The prior distribution introduces regularization, which helps solving the ill-posed problem and reduces overfitting. In practice, this often results in a suboptimal posterior temperature and the full potential of the Bayesian approach is not realized. In this paper, we optimize both the parameters of the prior distribution and the posterior temperature using Bayesian optimization. Well-tempered posteriors lead to better predictive performance and improved uncertainty calibration, which we demonstrate for the task of sparse-view CT reconstruction.
相關內容
We revisit the Bayesian Context Trees (BCT) modelling framework for discrete time series, which was recently found to be very effective in numerous tasks including model selection, estimation and prediction. A novel representation of the induced posterior distribution on model space is derived in terms of a simple branching process, and several consequences of this are explored in theory and in practice. First, it is shown that the branching process representation leads to a simple variable-dimensional Monte Carlo sampler for the joint posterior distribution on models and parameters, which can efficiently produce independent samples. This sampler is found to be more efficient than earlier MCMC samplers for the same tasks. Then, the branching process representation is used to establish the asymptotic consistency of the BCT posterior, including the derivation of an almost-sure convergence rate. Finally, an extensive study is carried out on the performance of the induced Bayesian entropy estimator. Its utility is illustrated through both simulation experiments and real-world applications, where it is found to outperform several state-of-the-art methods.
We consider hyper-differential sensitivity analysis (HDSA) of nonlinear Bayesian inverse problems governed by PDEs with infinite-dimensional parameters. In previous works, HDSA has been used to assess the sensitivity of the solution of deterministic inverse problems to additional model uncertainties and also different types of measurement data. In the present work, we extend HDSA to the class of Bayesian inverse problems governed by PDEs. The focus is on assessing the sensitivity of certain key quantities derived from the posterior distribution. Specifically, we focus on analyzing the sensitivity of the MAP point and the Bayes risk and make full use of the information embedded in the Bayesian inverse problem. After establishing our mathematical framework for HDSA of Bayesian inverse problems, we present a detailed computational approach for computing the proposed HDSA indices. We examine the effectiveness of the proposed approach on a model inverse problem governed by a PDE for heat conduction.
We introduce a methodology for robust Bayesian estimation with robust divergence (e.g., density power divergence or {\gamma}-divergence), indexed by a single tuning parameter. It is well known that the posterior density induced by robust divergence gives highly robust estimators against outliers if the tuning parameter is appropriately and carefully chosen. In a Bayesian framework, one way to find the optimal tuning parameter would be using evidence (marginal likelihood). However, we numerically illustrate that evidence induced by the density power divergence does not work to select the optimal tuning parameter since robust divergence is not regarded as a statistical model. To overcome the problems, we treat the exponential of robust divergence as an unnormalized statistical model, and we estimate the tuning parameter via minimizing the Hyvarinen score. We also provide adaptive computational methods based on sequential Monte Carlo (SMC) samplers, which enables us to obtain the optimal tuning parameter and samples from posterior distributions simultaneously. The empirical performance of the proposed method through simulations and an application to real data are also provided.
Physics-informed neural networks (PINNs) have recently been used to solve various computational problems which are governed by partial differential equations (PDEs). In this paper, we propose a multi-output physics-informed neural network (MO-PINN) which can provide solutions with uncertainty distributions for both forward and inverse PDE problems with noisy data. In this framework, the uncertainty arising from the noisy data is first translated into multiple measurements regarding the prior noise distribution using the bootstrap method, and then the outputs of neural networks are designed to satisfy the measurements as well as the underlying physical laws.The posterior estimation of target parameters can be obtained at the end of training, which can be further used for uncertainty quantification and decision making. In this paper, MO-PINNs are demonstrated with a series of numerical experiments including both linear and nonlinear, forward and inverse problems. The results show that MO-PINN is able to provide accurate predictions with noisy data.In addition, we also demonstrate that the prediction and posterior distributions from MO-PINNs are consistent with the solutions from traditional a finite element method (FEM) solver and Monte Carlo methods given the same data and prior knowledge. Finally, we show that additional statistical knowledge can be incorporated into the training to improve the prediction if available.
Many functions have approximately-known upper and/or lower bounds, potentially aiding the modeling of such functions. In this paper, we introduce Gaussian process models for functions where such bounds are (approximately) known. More specifically, we propose the first use of such bounds to improve Gaussian process (GP) posterior sampling and Bayesian optimization (BO). That is, we transform a GP model satisfying the given bounds, and then sample and weight functions from its posterior. To further exploit these bounds in BO settings, we present bounded entropy search (BES) to select the point gaining the most information about the underlying function, estimated by the GP samples, while satisfying the output constraints. We characterize the sample variance bounds and show that the decision made by BES is explainable. Our proposed approach is conceptually straightforward and can be used as a plug in extension to existing methods for GP posterior sampling and Bayesian optimization.
We present Posterior Temperature Optimized Bayesian Inverse Models (POTOBIM), an unsupervised Bayesian approach to inverse problems in medical imaging using mean-field variational inference with a fully tempered posterior. Bayesian methods exhibit useful properties for approaching inverse tasks, such as tomographic reconstruction or image denoising. A suitable prior distribution introduces regularization, which is needed to solve the ill-posed problem and reduces overfitting the data. In practice, however, this often results in a suboptimal posterior temperature, and the full potential of the Bayesian approach is not being exploited. In POTOBIM, we optimize both the parameters of the prior distribution and the posterior temperature with respect to reconstruction accuracy using Bayesian optimization with Gaussian process regression. Our method is extensively evaluated on four different inverse tasks on a variety of modalities with images from public data sets and we demonstrate that an optimized posterior temperature outperforms both non-Bayesian and Bayesian approaches without temperature optimization. The use of an optimized prior distribution and posterior temperature leads to improved accuracy and uncertainty estimation and we show that it is sufficient to find these hyperparameters per task domain. Well-tempered posteriors yield calibrated uncertainty, which increases the reliability in the predictions. Our source code is publicly available at github.com/Cardio-AI/mfvi-dip-mia.
In black-box function optimization, we need to consider not only controllable design variables but also uncontrollable stochastic environment variables. In such cases, it is necessary to solve the optimization problem by taking into account the uncertainty of the environmental variables. Chance-constrained (CC) problem, the problem of maximizing the expected value under a certain level of constraint satisfaction probability, is one of the practically important problems in the presence of environmental variables. In this study, we consider distributionally robust CC (DRCC) problem and propose a novel DRCC Bayesian optimization method for the case where the distribution of the environmental variables cannot be precisely specified. We show that the proposed method can find an arbitrary accurate solution with high probability in a finite number of trials, and confirm the usefulness of the proposed method through numerical experiments.
We propose two generic methods for improving semi-supervised learning (SSL). The first integrates weight perturbation (WP) into existing "consistency regularization" (CR) based methods. We implement WP by leveraging variational Bayesian inference (VBI). The second method proposes a novel consistency loss called "maximum uncertainty regularization" (MUR). While most consistency losses act on perturbations in the vicinity of each data point, MUR actively searches for "virtual" points situated beyond this region that cause the most uncertain class predictions. This allows MUR to impose smoothness on a wider area in the input-output manifold. Our experiments show clear improvements in classification errors of various CR based methods when they are combined with VBI or MUR or both.
This paper is concerned with data-driven unsupervised domain adaptation, where it is unknown in advance how the joint distribution changes across domains, i.e., what factors or modules of the data distribution remain invariant or change across domains. To develop an automated way of domain adaptation with multiple source domains, we propose to use a graphical model as a compact way to encode the change property of the joint distribution, which can be learned from data, and then view domain adaptation as a problem of Bayesian inference on the graphical models. Such a graphical model distinguishes between constant and varied modules of the distribution and specifies the properties of the changes across domains, which serves as prior knowledge of the changing modules for the purpose of deriving the posterior of the target variable $Y$ in the target domain. This provides an end-to-end framework of domain adaptation, in which additional knowledge about how the joint distribution changes, if available, can be directly incorporated to improve the graphical representation. We discuss how causality-based domain adaptation can be put under this umbrella. Experimental results on both synthetic and real data demonstrate the efficacy of the proposed framework for domain adaptation. The code is available at //github.com/mgong2/DA_Infer .
Amortized inference has led to efficient approximate inference for large datasets. The quality of posterior inference is largely determined by two factors: a) the ability of the variational distribution to model the true posterior and b) the capacity of the recognition network to generalize inference over all datapoints. We analyze approximate inference in variational autoencoders in terms of these factors. We find that suboptimal inference is often due to amortizing inference rather than the limited complexity of the approximating distribution. We show that this is due partly to the generator learning to accommodate the choice of approximation. Furthermore, we show that the parameters used to increase the expressiveness of the approximation play a role in generalizing inference rather than simply improving the complexity of the approximation.
注冊地址： 北京市海淀區羊坊店路18號2幢3層301-191