Fused L1/2 prior for large scale linear inverse problem with Gibbs bouncy particle sampler

Markov random fields (MRFs) have been widely used as prior models in various inverse problems such as tomographic reconstruction. While MRFs provide a simple and often effective way to model the spatial dependencies in images, they suffer from the fact that parameter estimate is difficult. In practice, this means that MRFs typically have very simple structure that cannot completely capture the subtle characteristics of complex images. In this paper, we present a novel Gaussian mixture Markov random field model (GM-MRF) that can be used as a very expressive prior model for inverse problems such as denoising and reconstruction. This method forms a global image model by merging together individual Gaussian-mixture models for image patches. Moreover, we present a novel analytical framework for computing MAP estimates with the GM-MRF prior model through the construction of exact surrogate functions that result in a sequence of quadratic optimizations. We demonstrate the value of the approach with some simple applications to denoising of dual-energy CT images.

In this paper, we consider the super-resolution image reconstruction problem. We propose a Markov chain Monte Carlo (MCMC) approach to find the maximum a posterior probability (MAP) estimation of the unknown high-resolution image. Firstly, Gaussian Markov random field (GMRF) is exploited for modeling the prior probability distribution of the unknown high-resolution image. Then, a MCMC technique (in particular, the Gibbs sampler) is introduced to generate samples from the posterior probability distribution to compute the MAP estimation of the unknown high-resolution image, which is obtained as the mean of the samples. Moreover, we derive a bound on the convergence time of the proposed MCMC approach. Finally, the experimental results are presented to verify the superior performance of the proposed approach and the validity of the proposed bound.

Markov Chain Monte Carlo Models, Gibbs Sampling, & Metropolis Algorithm for High-Dimensionality Complex Stochastic Problems

Approximate Bayesian inference for hierarchical Gaussian Markov random field models

This paper presents 3-D brain tissue classification schemes using three recent promising energy minimization methods for Markov random fields: graph cuts, loopy belief propagation and tree-reweighted message passing. The classification is performed us ng the well known finite Gaussian mixture Markov Random Field model. Results from the above methods are compared with widely used iterative conditional modes algorithm. The evaluation is per formed on a dataset containing simulated Tl-weighted MR brain volumes with varying noise and intensity non-uniformities. The comparisons are performed in terms of energies as well as based on ground truth segmentations, using various quantitative metrics.

Hierarchical models for repeated binary data using the IBF sampler

Variable selection in high dimensional data is a challenging problem due to the exponential number of variable combinations, and Markov Chain Monte Carlo (MCMC) methods represent the state of the art to solve it. With genomics data this problem becomes even more difficult because there are generally more dimensions (variables) than points (records) leading to slow convergence and numerically unstable solutions. On the other hand, despite many alternative prototypes and languages, R remains a popular system to compute machine learning models. Unfortunately, R can be particularly slow with heavy matrix computations and the high number of iterations required by MCMC methods. Moreover, making R scale to large matrices, possibly beyond RAM, requires careful system integration. Recently, array DBMSs have opened the possibility of manipulating matrices of unlimited size. With such motivation in mind, we present algorithmic optimizations to accelerate the computation of variable selection in linear regression with the Gibbs sampler, a fundamental MCMC method. Such optimizations have the potential to accelerate other models. We study how to leverage the speed and scalability of the array DBMS to exploit our optimizations in R. We present a comprehensive experimental evaluation to assess time efficiency and model quality with a cancer data set containing RNA and miRNA variables to predict survival time. We show our optimized algorithm combining DBMS and R processing is significantly faster than R alone. We show our system allows fast joint analysis of RNA and miRNA variables, instead of analyzing them separately. Finally, we confirm our algorithm finds medically significant variables already identified in the biomedical literature. Our optimized MCMC method for the array DBMS can be easily called from R, leaving the final model within R runtime in RAM for further interpretation.

SUMMARY The analysis of infectious disease data presents challenges arising from the dependence in the data and the fact that only part of the transmission process is observable. These difficulties are usually overcome by making simplifying assumptions. The paper explores the use of Markov chain Monte Carlo (MCMC) methods for the analysis of infectious disease data, with the hope that they will permit analyses to be made under more realistic assumptions. Two important kinds of data sets are considered, containing temporal and non-temporal information, from outbreaks of measles and influenza. Stochastic epidemic models are used to describe the processes that generate the data. MCMC methods are then employed to perform inference in a Bayesian context for the model parameters. The MCMC methods used include standard algorithms, such as the Metropolis–Hastings algorithm and the Gibbs sampler, as well as a new method that involves likelihood approximation. It is found that standard algorithms perform well in some situations but can exhibit serious convergence difficulties in others. The inferences that we obtain are in broad agreement with estimates obtained by other methods where they are available. However, we can also provide inferences for parameters which have not been reported in previous analyses.

Multilevel modeling is a modern approach to deal with hierarchical or a nested data structure which can assess the variability between clusters. Bayesian Markov Chain Monte Carlo (MCMC) methods of estimations are advanced methods applicable for estimating multilevel models. However, these estimation methods are not as yet tested to identify its’ performances as well as the properties associated with these estimation methods. This study targets to conduct a comparison of Bayesian MCMC methods which are developed for multilevel models where the response is normally distributed. The comparison is based upon extensive simulations and an application to a real-life dataset. The performance of Gibbs sampling (GS) and Metropolis Hastings (MH) methods are compared using a simulation study and additionally the factors which can affect the performance of both MCMC methods are identified. Practicality of these methods in real world scenario is confirmed through the application of MCMC method to a dataset. In the simulations though the Metropolis Hastings (MH) shows slightly better performance than Gibbs, there is no evidence to indicate that significant differences exist between these methods except for small samples where MH is superior. The results from the example are not as clear as from the simulations.

In this paper, we propose a model-free volumetric Next Best View (NBV) algorithm for accurate 3D reconstruction using a Markov Chain Monte Carlo method for high-mix-low-volume objects in manufacturing. The volumetric information gain based Next Best View algorithm can in real-time select the next optimal view that reveals the maximum uncertainty of the scanning environment with respect to a partially reconstructed 3D Occupancy map, without any priori knowledge of the target. Traditional Occupancy grid maps make two independence assumptions for computational tractability but suffer from the overconfident estimation of the occupancy probability for each voxel leading to less precise surface reconstructions. This paper proposes a special case of the Markov Chain Monte Carlo (MCMC) method, the Gibbs sampler, to accurately estimate the posterior occupancy probability of a voxel by randomly sampling from its high-dimensional full posterior occupancy probability given the entire volumetric map with respect to the forward sensor model with a Gaussian distribution. Numerical experiments validate the performance of the MCMC Gibbs sampler algorithm under the ROS-Industry framework to prove the accuracy of the reconstructed Occupancy map and the completeness of the registered point cloud. The proposed MCMC Occupancy mapping could be used to optimise the tuning parameters of the online NBV algorithms via the inverse sensor model to realise industry automation.

We address the solution of large-scale statistical inverse problems in the framework of Bayesian inference. The Markov chain Monte Carlo (MCMC) method is the most popular approach for sampling the posterior probability distribution that describes the solution of the statistical inverse problem. MCMC methods face two central difficulties when applied to large-scale inverse problems: first, the forward models (typically in the form of partial differential equations) that map uncertain parameters to observable quantities make the evaluation of the probability density at any point in parameter space very expensive; and second, the high-dimensional parameter spaces that arise upon discretization of infinite-dimensional parameter fields make the exploration of the probability density function prohibitive. The challenge for MCMC methods is to construct proposal functions that simultaneously provide a good approximation of the target density while being inexpensive to manipulate. Here we present a so-called Stoch...

MCMC Methods for Continuous-Time Financial Econometrics

CHAPTER 13 - MCMC Methods for Continuous-Time Financial Econometrics

In fields such as statistics, econometrics, physics, and biology, Markov chain Monte Carlo (MCMC) methods have been widely recognized as powerful computing tools to analyze complex probability distributions and estimate unsolvable integrations. However, evidences have shown that a naive application of basic MCMC methods is not feasible for problems involved in high dimensional probability distributions. In this thesis, we focus on understanding and improving MCMC methods in high dimensional problems. We first consider improving existing MCMC methods for three commonly used Bayesian shrinkage models, namely, the Bayesian group lasso, the Bayesian sparse group lasso, and the Bayesian fused lasso models. We propose two-block Gibbs samplers for sampling from their posterior distributions. Our new algorithms effectively alleviate the impact of high dimensionality and posterior dependence using a blocking technique. We prove their convergence and spectral properties, and show their computational efficiency through simulated and real data examples. We also provide a clearer theoretical understanding towards establishing Markov chain central limit theorems (CLT) based on convergence rates in Wasserstein distance. Existing works on Wasserstein distance based CLTs either convert Wasserstein bounds into total variation bounds or assume geometric contraction in Wasserstein bounds. Alternatively, we obtain two CLTs that directly depend on (sub-geometric) convergence rates in Wasserstein distance. Our CLTs hold for Lipschitz functions under certain moment conditions. We further show that the geometric contraction in Wasserstein bounds is a stronger condition than that used in our theorems. Finally, we apply these CLTs to four sets of Markov chain examples including an Exponential Integrator version of the Metropolis Adjusted Langevin Algorithm (EI-MALA), an unadjusted Langevin algorithm (ULA), a class of nonlinear autoregressive processes and a special autoregressive model that generates reducible chains.

Localization for MCMC: sampling high-dimensional posterior distributions with local structure