Robust Constraint of Luminosity Function Evolution through MCMC Sampling

Global illumination algorithms using Markov chain Monte Carlo (MCMC) sampling are well-known for their efficiency in scenes with complex light transport. Samples in such algorithms are generated as a history of Markov chain states so that they are distributed according to the contributions to the image. The whole process is done based only on the information of the path contributions and user-defined transition probabilities from one state to the others. In light transport simulation, however, there is more information that can be used to improve the efficiency of path sampling. A notable example is multiple importance sampling (MIS) in bidirectional path tracing, which utilizes the probability densities of constructing a given path with different estimators. While MIS is a powerful ordinary Monte Carlo method, how to incorporate such additional information into MCMC sampling has been an open problem. We introduce a novel MCMC sampling framework, primary space serial tempering , which fuses the ideas of MCMC sampling and MIS for the first time. The key idea is to explore not only the sample space using a Markov chain, but also different estimators to generate samples by utilizing the information already available for MIS. Based on this framework, we also develop a novel rendering algorithm, multiplexed Metropolis light transport , which automatically and adaptively constructs paths with appropriate techniques as predicted by MIS. The final algorithm is very easy to implement, yet in many cases shows comparable (or even better) performance than significantly more complex MCMC rendering algorithms.

This article is a review article on the optimal Markov chain Monte Carlo (MCMC) sampling. The focus is on homogeneous Markov chains. This article first reviews the problem of finding the optimal transition matrix, which is defined to minimize the asymptotic variance of MCMC estimators. The article later reviews the locally optimal sampler (LOS), an MCMC sampling that performs local updates based on the optimal transition matrix. We conducted a simulation study to compare the LOS with the Metropolis–Hastings and the Gibbs Sampler. The LOS was shown to provide an improved rate of convergence over these two most popular sampling schemes. The implementation of the LOS requires only minor modifications in existing Gibbs sampling code. WIREs Comput Stat 2013. doi: 10.1002/wics.1265This article is categorized under: Statistical and Graphical Methods of Data Analysis > Markov Chain Monte Carlo (MCMC)

RNA exosomes are large multi-subunit protein complexes involved in controlled and processive 3' to 5' RNA degradation. Exosomes form large molecular chambers and harbor multiple nuclease sites as well as RNA binding regions. This makes a quantitative kinetic analysis of RNA degradation with reliable parameter and error estimates challenging. For instance, recent quantitative biochemical assays revealed that degradation speed and efficiency depend on various factors, such as the type of RNA binding caps and the RNA length. We propose the combination of a differential equation model with Bayesian Markov Chain Monte Carlo (MCMC) sampling for a more robust and reliable analysis of such complex kinetic systems. Using the exosome as a paradigm, it is shown that conventional “best fit” approaches to parameter estimation are outperformed by the MCMC method. The parameter distribution returned by MCMC sampling allows for a reliable and meaningful comparison of the data from different time series. In the case of the exosome, we find that the cap structures of the exosome have a direct effect on the recruitment and degradation of RNA, and that these effects are RNA length-dependent. The described approach can be widely applied to any processive reaction with a similar kinetics like the XRN1-dependent RNA degradation, RNA/DNA synthesis by polymerases, and protein synthesis by the ribosome.

Global fits of physics models require efficient methods for exploring high-dimensional and/or multimodal posterior functions. We introduce a novel method for accelerating Markov Chain Monte Carlo (MCMC) sampling by pairing a Metropolis-Hastings algorithm with a diffusion model that can draw global samples with the aim of approximating the posterior. We briefly review diffusion models in the context of image synthesis before providing a streamlined diffusion model tailored towards low-dimensional data arrays. We then present our adapted Metropolis-Hastings algorithm which combines local proposals with global proposals taken from a diffusion model that is regularly trained on the samples produced during the MCMC run. Our approach leads to a significant reduction in the number of likelihood evaluations required to obtain an accurate representation of the Bayesian posterior across several analytic functions, as well as for a physical example based on a global analysis of parton distribution functions. Our method is extensible to other MCMC techniques, and we briefly compare our method to similar approaches based on normalizing flows. A code implementation can be found at https://github.com/NickHunt-Smith/MCMC-diffusion.

International audience

In the context of Bayesian inversion for scientific and engineering modeling, Markov chain Monte Carlo (MCMC) sampling strategies have become the benchmark due to their flexibility and robustness in dealing with arbitrary posterior probability density functions (PDFs). However, these algorithms have been shown to be inefficient when sampling from high-dimensional posterior distributions or exhibit multimodality and/or strong parameter correlations. In such contexts, transitional MCMC (TMCMC) provides a more efficient alternative. Despite the recent applicability for Bayesian updating and model selection across a variety of disciplines, TMCMC may require a prohibitive number of tempering stages when the prior pdf is significantly different from the target posterior. Furthermore, the need to start with an initial set of samples from the prior distribution may present a challenge when dealing with implicit priors, e.g., based on feasible regions. Finally, TMCMC cannot be used for inverse problems with improper prior PDFs that represent a lack of prior knowledge on all or a subset of parameters. A generalization of TMCMC is proposed that alleviates such challenges and limitations toward providing a more robust and efficient tempering sampling strategy. We present convergence analysis, proving that the distance between the intermediate distributions and the target posterior distribution monotonically decreases as the algorithm proceeds. We also demonstrate the advantages of the proposed generalization through a series of test problems and an engineering application in the oil and gas industry.

This chapter provides a brief introduction to the use of Bayesian graphical models in R. In these models, parameters are treated as random quantities on an equal footing with the random variables. This allows complex stochastic systems to modeled, often using Markov chain Monte Carlo (MCMC) sampling methods. We first consider a series of examples, ranging from simple repeated sampling, linear regression models, random coefficient regression models, to the chest clinic example of Chap. 3. We formulate these as Bayesian graphical models, and represent them graphically in a compact form using plates. We then describe a special case, in which the unknown parameters are all discrete, and explain how probability propagation methods described in Chap. 3 may be used to compute the posterior distributions. We then turn to the general case, when MCMC sampling is required, and explain the computations involved in Metropolis-Hastings sampling and Gibbs sampling. Finally we illustrate a Bayesian linear regression analysis using the R2WinBUGS package.

Combining autoencoder neural network and Bayesian inversion to estimate heterogeneous permeability distributions in enhanced geothermal reservoir: Model development and verification

Entity resolution is the process of merging and removing duplicate records from multiple data sources, often in the absence of unique identifiers. Bayesian models for entity resolution allow one to include a priori information, quantify uncertainty in important applications, and directly estimate a partition of the records. Markov chain Monte Carlo (MCMC) sampling is the primary computational method for approximate posterior inference in this setting, but due to the high dimensionality of the space of partitions, there are no agreed upon standards for diagnosing nonconvergence of MCMC sampling. In this article, we review Bayesian entity resolution, with a focus on the specific challenges that it poses for the convergence of a Markov chain. We review prior methods for convergence diagnostics, discussing their weaknesses. We provide recommendations for using MCMC sampling for Bayesian entity resolution, focusing on the use of modern diagnostics that are commonplace in applied Bayesian statistics. Using simulated data, we find that a commonly used Gibbs sampler performs poorly compared with two alternatives.

This paper presents a new methodology for reliability-based inspection planning focusing on robust and accurate computational strategies for fatigue-reliability updating using inspection results. The core of the proposed strategy is a conditioned sampling-based method, implemented by a Fast Probability Analyzer (FPA) software where efficiency is achieved by using the importance sampling principal. For a single component or limit state, FPA first generates Markov-Chain Monte Carlo (MCMC) samples in the failure domain, then applies an adaptive stratified importance sampling (ASIS) method to compute probability of failure (PoF) with error control. Once the MCMC samples have been created, solving a reliability updating problem is fairly straightforward and computationally robust relative to the conventional system reliability methods that rely on linearization of the limit states. The new approach is demonstrated using examples including stiffened panels of a ship-shaped vessel where reliability is updated using inspection results from 100 panel connections.

Sampling technique is one of the asymptotically unbiased estimation approaches for inference in Bayesian probabilistic models. Markov chain Monte Carlo (MCMC) is a kind of sampling methods, which is widely used in the inference of complex probabilistic models. However, current MCMC methods can incur high autocorrelation of samples, which means that the samples generated by MCMC samplers are far from independent. In this paper, we propose the neural networks Langevin Monte Carlo (NNLMC) which makes full use of the flexibility of neural networks and the high efficiency of the Langevin dynamics sampling to construct a new MCMC sampling method. We propose the new update function to generate samples and employ appropriate loss functions to improve the performance of NNLMC during the process of sampling. We evaluate our method on a large diversity of challenging distributions and real datasets. Our results show that NNLMC is able to sample from the target distribution with low autocorrelation and rapid convergence, and outperforms the state-of-the-art MCMC samplers.

Variable selection is very important in many fields, and for its resolution many procedures have been proposed and investigated. Among them are Bayesian methods that use Markov chain Monte-Carlo (MCMC) sampling algorithms. A problem with MCMC sampling, however, is that it cannot guarantee that the samples are exactly from the target distributions. This drawback is overcome by related methods known as perfect sampling algorithms. In this paper, we propose the use of two perfect sampling algorithms to perform variable selection within the Bayesian framework. They are the sandwiched coupling from the past (CFTP) algorithm and the Gibbs coupler. We focus our attention to scenarios where the model coefficients and noise variance are known. We indicate the condition under which the sandwiched CFTP can be applied. Most importantly, we design a detailed scheme to adapt the Gibbs coupler algorithm to variable selection. In addition, we discuss the possibilities of applying perfect sampling when the model coefficients and noise variance are unknown. Test results that show the performance of the algorithms are provided.

A D-Wave quantum annealer (QA) having a 2048 qubit lattice, with no missing qubits and couplings, allowed embedding of a complete graph of a restricted Boltzmann machine (RBM). A handwritten digit OptDigits dataset having 8 × 7 pixels of visible units was used to train the RBM using classical contrastive divergence. Embedding of the classically trained RBM into the D-Wave lattice was used to demonstrate that the QA offers a high-efficiency alternative to the classical Markov chain Monte Carlo (MCMC) for reconstructing missing labels of the test images as well as a generative model on a classically trained RBM. At any training iteration, the D-Wave-based classification had classification error less than half of that in MCMC. The main goal of this study was to investigate the quality of the QA sample from the RBM model probability distribution and compare it to a classical MCMC during the Gibbs sampling traditionally used in RBM training. For the OptDigits dataset, the states in the D-Wave sample belonged to about twice as many local valleys (LVs) as in the MCMC sample. All the lowest energy (the highest joint probability), local minima in the MCMC sample were also found by the D-Wave. The D-Wave missed many of the higher-energy LVs, while also finding many “new” LVs consistently missed by the MCMC. It was established that the “new” LVs that the D-Wave finds are important for the model distribution in terms of the energy of the corresponding local minima, the width of the LVs and the height of the escape barrier.

I describe algorithms for drawing from distributions using adaptive Markov chain Monte Carlo (MCMC) methods; I introduce a Mata function for performing adaptive MCMC, amcmc(); and I present a suite of functions, amcmc_ *(), that allows an alternative implementation of adaptive MCMC. amcmc() and amcmc_ *() can be used with models set up to work with Mata's moptimize( ) (see [M-5] moptimize( )) or optimize( ) (see [M-5] optimize( )) or with standalone functions. To show how the routines can be used in estimation problems, I give two examples of what Chernozhukov and Hong (2003, Journal of Econometrics 115: 293–346) refer to as quasi-Bayesian or Laplace-type estimators—simulation-based estimators using MCMC sampling. In the first example, I illustrate basic ideas and show how a simple linear model can be fit by simulation. In the next example, I describe simulation-based estimation of a censored quantile regression model following Powell (1986, Journal of Econometrics 32: 143–155); the discussion describes the workings of the command mcmccqreg. I also present an example of how the routines can be used to draw from distributions without a normalizing constant and used in Bayesian estimation of a mixed logit model. This discussion introduces the command bayesmixedlogit.

We present a novel tracking system that adaptively selects a shape of the posterior over time, where the selection is efficiently performed by the uncertainty calibrated Markov Chain Monte Carlo (UCMCMC) sampler. In conventional trackers, the posterior is typically described by a single prior distribution. On the other hand, our tracker allows the posterior to have multiple prior distributions, namely normal and Student’s t distribution, and to choose an appropriate distribution according to the tracking environment. The optimal distribution is determined by the UCMCMC sampler in the process of reducing the uncertainty of the shape. Experimental results demonstrate that the proposed multi-shape posterior helps improve the tracking performance in terms of accuracy.