A probabilistic formulation for empirical population synthesis: sampling methods and tests

Sparsity has become a key concept for solving of high-dimensional inverse problems using variational regularization techniques. Recently, using similar sparsity-constraints in the Bayesian framework for inverse problems by encoding them in the prior distribution has attracted attention. Important questions about the relation between regularization theory and Bayesian inference still need to be addressed when using sparsity promoting inversion. A practical obstacle for these examinations is the lack of fast posterior sampling algorithms for sparse, high-dimensional Bayesian inversion. Accessing the full range of Bayesian inference methods requires being able to draw samples from the posterior probability distribution in a fast and efficient way. This is usually done using Markov chain Monte Carlo (MCMC) sampling algorithms. In this paper, we develop and examine a new implementation of a single component Gibbs MCMC sampler for sparse priors relying on L1-norms. We demonstrate that the efficiency of our Gibbs sampler increases when the level of sparsity or the dimension of the unknowns is increased. This property is contrary to the properties of the most commonly applied Metropolis–Hastings (MH) sampling schemes. We demonstrate that the efficiency of MH schemes for L1-type priors dramatically decreases when the level of sparsity or the dimension of the unknowns is increased. Practically, Bayesian inversion for L1-type priors using MH samplers is not feasible at all. As this is commonly believed to be an intrinsic feature of MCMC sampling, the performance of our Gibbs sampler also challenges common beliefs about the applicability of sample based Bayesian inference.

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.

Comparative assessment of sewer sampling methods for infectious disease surveillance: Insights from transport modeling and simulations of SARS-CoV-2 emissions.

We introduce a data-driven method to generate a large number of plausible, closely interacting 3D human pose-pairs, for a given motion category, e.g., wrestling or salsa dance. With much difficulty in acquiring close interactions using 3D sensors, our approach utilizes abundant existing video data which cover many human activities. Instead of treating the data generation problem as one of reconstruction, either through 3D acquisition or direct 2D-to-3D data lifting from video annotations, we present a solution based on Markov Chain Monte Carlo (MCMC) sampling. Given a motion category and a set of video frames depicting the motion with the 2D pose-pair in each frame annotated, we start the sampling with one or few seed 3D pose-pairs which are manually created based on the target motion category. The initial set is then augmented by MCMC sampling around the seeds, via the Metropolis-Hastings algorithm and guided by a probability density function (PDF) that is defined by two terms to bias the sampling towards 3D pose-pairs that are physically valid and plausible for the motion category. With a focus on efficient sampling over the space of close interactions, rather than pose spaces, we develop a novel representation called interaction coordinates (IC) to encode both poses and their interactions in an integrated manner. Plausibility of a 3D pose-pair is then defined based on the IC and with respect to the annotated 2D pose-pairs from video. We show that our sampling-based approach is able to efficiently synthesize a large volume of plausible, closely interacting 3D pose-pairs which provide a good coverage of the input 2D pose-pairs.

In this paper, we propose a Markov Chain Monte Carlo (MCMC) sampling method with the data-driven proposal distribution for six-degree-of-freedom (6-DoF) SLAM. Recently, visual odometry priors have been widely used as the process model in the SLAM formulation to improve the SLAM performance. However, modeling the uncertainties of incremental motions estimated by visual odometry is especially difficult under challenging conditions, such as erratic motion. For a particle-based model representation, it can represent the uncertainty of the camera motion well under erratic motion compared to the constant velocity model or a Gaussian noise model, but the manner of representing the proposal distribution and sampling the particles is extremely important, as we can maintain only a limited number of particles in the high-dimensional state space. Hence, we propose an effective sampling approach by exploiting MCMC sampling and the data-driven proposal distribution to propagate the particles. We demonstrate the performance of the proposed approach for 6-DoF SLAM using both synthetic and real datasets and compare the performance with those of other sampling methods.

This paper addresses the problem of estimating the lower atmospheric refractivity (M profile) under nonstandard propagation conditions frequently encountered in low‐altitude maritime radar applications. This is done by statistically estimating the duct strength (range‐ and height‐dependent atmospheric index of refraction) from the sea surface reflected radar clutter. These environmental statistics can then be used to predict the radar performance. In previous work, genetic algorithms (GA) and Markov chain Monte Carlo (MCMC) samplers were used to calculate the atmospheric refractivity from returned radar clutter. Although GA is fast and estimates the maximum a posteriori (MAP) solution well, it poorly calculates the multidimensional integrals required to obtain the means, variances, and underlying posterior probability distribution functions of the estimated parameters. More accurate distributions and integral calculations can be obtained using MCMC samplers, such as the Metropolis‐Hastings and Gibbs sampling (GS) algorithms. Their drawback is that they require a large number of samples relative to the global optimization techniques such as GA and become impractical with an increasing number of unknowns. A hybrid GA‐MCMC method based on the nearest neighborhood algorithm is implemented in this paper. It is an improved GA method which improves integral calculation accuracy through hybridization with a MCMC sampler. Since the number of forward models is determined by GA, it requires fewer forward model samples than a MCMC, enabling inversion of atmospheric models with a larger number of unknowns.

Environmental protection is one of the themes of today's world. The forest is a recycler of carbon dioxide and natural oxygen bar. Protection of forests, monitoring of forest growth is long-term task of environmental protection. It is very important to automatically statistic the forest coverage rate using optical remote sensing images and the computer, by which we can timely understand the status of the forest of an area, and can be freed from tedious manual statistics. Towards the problem of computational complexity of the global optimization using convexification, this paper proposes a level set segmentation method based on Markov chain Monte Carlo (MCMC) sampling and applies it to forest segmentation in remote sensing images. The presented method needs not to do any convexity transformation for the energy functional of the goal, and uses MCMC sampling method with global optimization capability instead. The possible local minima occurring by using gradient descent method is also avoided. There are three major contributions in the paper. Firstly, by using MCMC sampling, the convexity of the energy functional is no longer necessary and global optimization can still be achieved. Secondly, taking advantage of the data (texture) and knowledge (a priori color) to guide the construction of Markov chain, the convergence rate of Markov chains is improved significantly. Finally, the level set segmentation method by integrating a priori color and texture for forest is proposed. The experiments show that our method can efficiently and accurately segment forest in remote sensing images.

An optimized evaluation method of the bearing capacity of reinforced concrete beam based on the Bayesian theory was proposed in this paper. This evaluation method optimized the traditional Markov Chain-Monte Carlo (MCMC) sampling method, and proposed an improved Metropolis–Hastings (MH) sampling method and a transitive MCMC (TMCMC) sampling method based on the MCMC theory. These two derived sampling methods solved the problem that the traditional MCMC algorithm makes it difficult to achieve convergence when the number of modified parameters is large. Therefore, on the basis of obtaining the measured sample information and the prior information of uncertain parameters, this paper first used multiple “model components” to form a model sample, then carried out a sensitivity analysis based on the relevant response indicators and selected the key parameters that had a great impact on the bearing capacity, carried out static load tests, and extracted and analyzed the experimental data. Then, based on a large amount of analysis data, the improved MH sampling method and TMCMC sampling method were used to establish a posterior probability distribution database. Finally, multiple posterior probability distributions were used to identify and predict the bearing capacity. The results showed that the method was feasible and effective for the evaluation of the bearing capacity of reinforced concrete beam.

Particle swarm optimization–Markov Chain Monte Carlo for accurate visual tracking with adaptive template update

Bayesian inference of channelized section spillover via Markov Chain Monte Carlo sampling

Recently, Monte Carlo (MC) based sampling methods for evaluating high-dimensional posterior integrals have been rapidly developing. Those sampling methods include MC importance sampling (Hammersley and Handscomb 1964; Ripley 1987; Geweke 1989; Wolpert 1991), Gibbs sampling (Geman and Geman 1984; Gelfand and Smith 1990), Hit-and-Run sampling (Smith 1984; Bélislc, Romeijn, and Smith 1993; Chen 1993; Chen and Schmeiser 1993 and 1996), Metropolis-Hastings sampling (Metropolis et al. 1953; Hastings 1970; Green 1995), and hybrid methods (e.g., Müller 1991; Tierney 1994; Berger and Chen 1993). A general discussion of the Gibbs sampler and other Markov chain Monte Carlo (MCMC) methods is given in the Journal of the Royal Statistical Society, Series B (1993), and an excellent roundtable discussion on the practical use of MCMC can be found in Kass et al. (1998). Other discussions or instances of the use of MCMC sampling can be found in Tanner and Wong (1987), Tanner (1996), Geyer (1992), Gelman and Rubin (1992), Gelfand, Smith, and Lee (1992), Gilks and Wild (1992), and many others. Further development of state-of-the-arts MCMC sampling techniques include the accelerated MCMC sampling of Liu and Sabatti (1998, 1999), Liu (1998), and Liu and Wu (1997), and the exact MCMC sampling of Green and Murdoch (1999). Comprehensive accounts of MCMC methods and their applications may also be found in Meyn and Tweedie (1993), Tanner (1996), Gilks, Richardson, and Spiegel-halter (1996), Robert and Casella (1999), and Gelfand and Smith (2000). The purpose of this chapter is to give a brief overview of several commonly used MCMC sampling algorithms as well as to present selectively several newly developed computational tools for MCMC sampling.KeywordsPosterior DistributionMarkov Chain Monte CarloMonte CarloMarkov Chain Monte Carlo AlgorithmMarkov Chain Monte Carlo SamplingThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

In the observation of sky-averaged $\mathrm{H\, \small{I}}$ signal from Epoch of Reionization (EoR), model parameter inference can be a computation-intensive work, which makes it hard to perform a direct one-stage model parameter inference by using Markov Chain Monte Carlo (MCMC) sampling method in Bayesian framework. Instead, a two-stage inference is usually used, i.e. the parameters of some characteristic points on the EoR spectrum model are first estimated, which are then used as the input to estimate physical model parameters further. However, some previous works had noticed that this kind of method could bias results, and it could be meaningful to answer the question of whether it is feasible to perform direct one-stage MCMC sampling and obtain unbiased physical model parameter estimations. In this work, we studied this problem and confirmed the feasibility. We find that unbiased estimations to physical model parameters can be obtained with a one-stage direct MCMC sampling method. We also study the influence of some factors that should be considered in practical observations to model parameter inference. We find that a very tiny amplifier gain calibration error (10−5 relative error) with complex spectral structures can significantly bias the parameter estimation; the frequency-dependent antenna beam and geographical position can also influence the results, so that should be carefully handled.

We develop and validate a systematic approach to infer plate boundary strength and rheological parameters in models of mantle flow from surface velocity observations. Based on a realistic rheological model that includes yielding and strain rate weakening from dislocation creep, we formulate the inverse problem in a Bayesian inference framework. To study the distribution of parameters that are consistent with the observations, we compute the maximum a posteriori (MAP) point, Gaussian approximations of the parameter distribution around that MAP point, and employ Markov Chain Monte Carlo (MCMC) sampling methods. The computation of the MAP point and the Gaussian approximation require first and second derivatives of an objective function subject to non-linear Stokes equations; these derivatives are computed efficiently using adjoint Stokes equations. We set up 2-D numerical experiments with many of the elements expected in a global geophysical inversion. This setup incorporates three subduction zones with slab and weak zone (interplate fault) geometry consistent with average seismic characteristics. With these experiments, we demonstrate that when the temperature field is known, we can recover the strength of plate boundaries, the yield stress and strain rate exponent in the upper mantle. When the number of uncertain parameters increases, there are trade-offs between the inferred parameters. These trade-offs depend on how well the observational data represents the surface velocities, and on the weakness of plate boundaries. As the plate boundary coupling drops below a threshold, the uncertainty of the inferred parameters increases due to insensitivity of plate motion to plate coupling. Comparing the trade-offs between inferred rheological parameters found from the Gaussian approximation of the parameter distribution and from MCMC sampling, we conclude that the Gaussian approximation—which is significantly cheaper to compute—is often a good approximation, in particular locally around the MAP point. Thus, the method can be applied to the global problem of inferring non-linear constitutive parameters and plate coupling factors for each subduction zone in a global geophysical inversion with known slab structure.

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.

This discussion is a reply to the comments made by Dr. Jasper Vrugt on the Metropolis‐Hastings (M‐H) algorithm with multiple independent Markov chains proposed by Huang and Merwade (2023a), https://doi.org/10.1029/2023wr034947 concerning the validity of the methodology in estimating Bayesian model averaging (BMA) parameters (weights and variances) of the framework proposed by Raftery et al. (2005), https://doi.org/10.1175/mwr2906.1. In this reply, we address his concerns by emphasizing the motivation of applying the proposed M‐H algorithm to BMA analysis and the applicability of the effective sample size that accounts for the autocorrelation across samples in evaluating the efficiency of Markov chain Monte Carlo sampling. Moreover, the details of sampling procedure for BMA prediction distribution are clarified. On the other hand, we present a fair comparison of the default Expectation‐Maximization, M‐H, and differential evolution adaptive Metropolis (DREAM) algorithms in estimating BMA parameters based on a numerical experiment. Results reinforce the findings obtained from Huang and Merwade (2023a) https://doi.org/10.1029/2023wr034947 and further indicate that the proposed M‐H algorithm is better than the DREAM algorithm in terms of sampling efficiency and prediction accuracy. Accordingly, we raise concerns on the use of DREAM algorithm in BMA analysis and suggest conducting peer reviews on the MODELAVG toolbox.