Unbiased Markov chain Monte Carlo for intractable target distributions

Performing numerical integration when the integrand itself cannot be evaluated point-wise is a challenging task that arises in statistical analysis, notably in Bayesian inference for models with intractable likelihood functions. Markov chain Monte Carlo (MCMC) algorithms have been proposed for this setting, such as the pseudo-marginal method for latent variable models and the exchange algorithm for a class of undirected graphical models. As with any MCMC algorithm, the resulting estimators are justified asymptotically in the limit of the number of iterations, but exhibit a bias for any fixed number of iterations due to the Markov chains starting outside of stationarity. This "burn-in" bias is known to complicate the use of parallel processors for MCMC computations. We show how to use coupling techniques to generate unbiased estimators in finite time, building on recent advances for generic MCMC algorithms. We establish the theoretical validity of some of these procedures by extending existing results to cover the case of polynomially ergodic Markov chains. The efficiency of the proposed estimators is compared with that of standard MCMC estimators, with theoretical arguments and numerical experiments including state space models and Ising models.Comment: 40 page

Bayesian computation in imaging inverse problems with partially unknown models

Many imaging problems require solving a high-dimensional inverse problem that is ill-conditioned or ill-posed. Imaging methods typically address this difficulty by regularising the estimation problem to make it well-posed. This often requires setting the value of the so-called regularisation parameters that control the amount of regularisation enforced. These parameters are notoriously difficult to set a priori and can have a dramatic impact on the recovered estimates. In this thesis, we propose a general empirical Bayesian method for setting regularisation parameters in imaging problems that are convex w.r.t. the unknown image. Our method calibrates regularisation parameters directly from the observed data by maximum marginal likelihood estimation, and can simultaneously estimate multiple regularisation parameters. A main novelty is that this maximum marginal likelihood estimation problem is efficiently solved by using a stochastic proximal gradient algorithm that is driven by two proximal Markov chain Monte Carlo samplers, thus intimately combining modern high-dimensional optimisation and stochastic sampling techniques. Furthermore, the proposed algorithm uses the same basic operators as proximal optimisation algorithms, namely gradient and proximal operators, and it is therefore straightforward to apply to problems that are currently solved by using proximal optimisation techniques. We also present a detailed theoretical analysis of the proposed methodology, and demonstrate it with a range of experiments and comparisons with alternative approaches from the literature. The considered experiments include image denoising, non-blind image deconvolution, and hyperspectral unmixing, using synthesis and analysis priors involving the `1, total-variation, total-variation and `1, and total-generalised-variation pseudo-norms. Moreover, we explore some other applications of the proposed method including maximum marginal likelihood estimation in Bayesian logistic regression and audio compressed sensing, as well as an application to model selection based on residuals

Online Structured Sparsity-based Moving Object Detection from Satellite Videos

Inspired by the recent developments in computer vision, low-rank and structured sparse matrix decomposition can be potentially be used for extract moving objects in satellite videos. This set of approaches seeks for rank minimization on the background that typically requires batch-based optimization over a sequence of frames, which causes delays in processing and limits their applications. To remedy this delay, we propose an Online Low-rank and Structured Sparse Decomposition (O-LSD). O-LSD reformulates the batch-based low-rank matrix decomposition with the structured sparse penalty to its equivalent frame-wise separable counterpart, which then defines a stochastic optimization problem for online subspace basis estimation. In order to promote online processing, O-LSD conducts the foreground and background separation and the subspace basis update alternatingly for every frame in a video. We also show the convergence of O-LSD theoretically. Experimental results on two satellite videos demonstrate the performance of O-LSD in term of accuracy and time consumption is comparable with the batch-based approaches with significantly reduced delay in processing

Spatiotemporal modeling of air pollutants and their health effects in the Pittsburgh region

Air pollutants have been associated with adverse health outcomes such as cardiovascular and respiratory diseases through epidemiological studies. Spatiotemporal and spatial statistics are widely used in both exposure assessment and health risk estimation of air pollutants. In the current paper, spatiotemporal and spatial models are developed for and applied to four specfic topics about air pollutants: (1) estimating spatiotemporal variations of particulate matter with diameter less than 2.5 um (PM2.5) using monitoring data and satellite aerosol optical depth (AOD) measurements, (2) estimating long-term spatial variations of ozone (O3) using monitoring data and satellite O3 profile measurements, (3) spatiotemporal associating acute exposure of air pollutants to mortality, and (4) spatiotemporal associating chronic air pollution exposure to lung cancer incidence. Environmental, socioeconomic and health data from Allegheny county and the State of Pennsylvania are collected to illustrate these techniques. The public health significance of these studies includes characterizing the exposure level of air pollutants and their health risks for mortality caused by cardiovascular and respiratory diseases and lung cancer incidence in the Pittsburgh region and developing novel spatiotemporal models such as spatiotemporal generalized estimating equations for the regression analysis of spatiotemporal counts data, especially for the massive spatiotemporal data used in epidemiological studies

Estimating Risk Preferences in the Field

We survey the literature on estimating risk preferences using field data. We concentrate our attention on studies in which risk preferences are the focal object and estimating their structure is the core enterprise. We review a number of models of risk preferences—including both expected utility (EU) theory and non-EU models—that have been estimated using field data, and we highlight issues related to identification and estimation of such models using field data. We then survey the literature, giving separate treatment to research that uses individual-level data (e.g., property insurance data) and research that uses aggregate data (e.g., betting market data). We conclude by discussing directions for future research

ISIPTA'07: Proceedings of the Fifth International Symposium on Imprecise Probability: Theories and Applications

B

Perceptually inspired image estimation and enhancement

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Brain and Cognitive Sciences, 2009.Includes bibliographical references (p. 137-144).In this thesis, we present three image estimation and enhancement algorithms inspired by human vision. In the first part of the thesis, we propose an algorithm for mapping one image to another based on the statistics of a training set. Many vision problems can be cast as image mapping problems, such as, estimating reflectance from luminance, estimating shape from shading, separating signal and noise, etc. Such problems are typically under-constrained, and yet humans are remarkably good at solving them. Classic computational theories about the ability of the human visual system to solve such under-constrained problems attribute this feat to the use of some intuitive regularities of the world, e.g., surfaces tend to be piecewise constant. In recent years, there has been considerable interest in deriving more sophisticated statistical constraints from natural images, but because of the high-dimensional nature of images, representing and utilizing the learned models remains a challenge. Our techniques produce models that are very easy to store and to query. We show these techniques to be effective for a number of applications: removing noise from images, estimating a sharp image from a blurry one, decomposing an image into reflectance and illumination, and interpreting lightness illusions. In the second part of the thesis, we present an algorithm for compressing the dynamic range of an image while retaining important visual detail. The human visual system confronts a serious challenge with dynamic range, in that the physical world has an extremely high dynamic range, while neurons have low dynamic ranges.(cont.) The human visual system performs dynamic range compression by applying automatic gain control, in both the retina and the visual cortex. Taking inspiration from that, we designed techniques that involve multi-scale subband transforms and smooth gain control on subband coefficients, and resemble the contrast gain control mechanism in the visual cortex. We show our techniques to be successful in producing dynamic-range-compressed images without compromising the visibility of detail or introducing artifacts. We also show that the techniques can be adapted for the related problem of "companding", in which a high dynamic range image is converted to a low dynamic range image and saved using fewer bits, and later expanded back to high dynamic range with minimal loss of visual quality. In the third part of the thesis, we propose a technique that enables a user to easily localize image and video editing by drawing a small number of rough scribbles. Image segmentation, usually treated as an unsupervised clustering problem, is extremely difficult to solve. With a minimal degree of user supervision, however, we are able to generate selection masks with good quality. Our technique learns a classifier using the user-scribbled pixels as training examples, and uses the classifier to classify the rest of the pixels into distinct classes. It then uses the classification results as per-pixel data terms, combines them with a smoothness term that respects color discontinuities, and generates better results than state-of-art algorithms for interactive segmentation.by Yuanzhen Li.Ph.D

Conjoint probabilistic subband modeling

Thesis (Ph. D.)--Massachusetts Institute of Technology, Program in Media Arts & Sciences, 1997.Includes bibliographical references (leaves 125-133).by Ashok Chhabedia Popat.Ph.D