Covariance Decompositions for Accurate Computation in Bayesian Scale-Usage Models

Bayesian model updating provides a rigorous framework to account for uncertainty induced by lack of knowledge about engineering systems in their respective mathematical models through updates of the joint probability density function (PDF), the so-called posterior PDF, of the unknown model parameters. The Markov chain Monte Carlo (MCMC) methods are currently the most popular approaches for generating samples from the posterior PDF. However, these methods often found wanting when sampling from difficult distributions (e.g., high-dimensional PDFs, PDFs with flat manifolds, multimodal PDFs, and very peaked PDFs). This paper introduces a new multi-level sampling approach for Bayesian model updating, called Sequential Gauss-Newton algorithm, which is inspired by the Transitional Markov chain Monte Carlo (TMCMC) algorithm. The Sequential Gauss-Newton algorithm improves two aspects of TMCMC to make an efficient and effective MCMC algorithm for drawing samples from difficult posterior PDFs. First, the statistical efficiency of the algorithm is enhanced by use of the systematic resampling scheme. Second, a new MCMC algorithm, called Gauss-Newton MCMC algorithm, is proposed which is essentially an M-H algorithm with a Gaussian proposal PDF tailored to the posterior PDF using the gradient and Hessian information of the negative log posterior. The effectiveness of the proposed algorithm for solving the Bayesian model updating problem is illustrated using three examples with irregularly shaped posterior PDFs.

Hidden conditional random fields (HCRFs) are discriminative latent variable models which have been shown to successfully learn the hidden structure of a given classification problem. An Infinite hidden conditional random field is a hidden conditional random field with a countably infinite number of hidden states, which rids us not only of the necessity to specify a priori a fixed number of hidden states available but also of the problem of overfitting. Markov chain Monte Carlo (MCMC) sampling algorithms are often employed for inference in such models. However, convergence of such algorithms is rather difficult to verify, and as the complexity of the task at hand increases the computational cost of such algorithms often becomes prohibitive. These limitations can be overcome by variational techniques. In this paper, we present a generalized framework for infinite HCRF models, and a novel variational inference approach on a model based on coupled Dirichlet Process Mixtures, the HCRF-DPM. We show that the variational HCRF-DPM is able to converge to a correct number of represented hidden states, and performs as well as the best parametric HCRFs-chosen via cross-validation-for the difficult tasks of recognizing instances of agreement, disagreement, and pain in audiovisual sequences.

A probabilistic tracking model is introduced that identifies storm tracks from feature vectors that are extracted from meteorological analysis data. The model assumes that the genesis and lysis times of each track are unknown and estimates their values along with the track’s position and storm intensity over time. A hidden-state dynamics model (Kalman filter) characterizes the temporal evolution of the storms.The model uses a Bayesian methodology for estimating the unknown lifetimes (genesis–lysis pairs) and tracks of the storms. Prior distributions are placed over the unknown parameters and their posterior distributions are estimated using a Markov Chain Monte Carlo (MCMC) sampling algorithm. The posterior distributions are used to identify and report the most likely storm tracks in the data. This approach provides a unified probabilistic framework that accounts for uncertainty in storm timing (genesis and lysis), storm location and intensity, and the feature detection process. Thus, issues such as missing observations can be accommodated in a statistical manner without human intervention.The model is applied to the field of relative vorticity at the 975-hPa level of analysis from the National Centers for Environmental Prediction Global Forecast System during May–October 2000–02, in the tropical east Pacific. Storm tracks in the National Hurricane Center best-track data (HURDAT) for the same period are used to assess the performance of the storm identification and tracking model.

Sampling across tree space is one of the major challenges in Bayesian phylogenetic inference using Markov chain Monte Carlo (MCMC) algorithms. Standard MCMC tree moves consider small random perturbations of the topology, and select from candidate trees at random or based on the distance between the old and new topologies. MCMC algorithms using such moves tend to get trapped in tree space, making them slow in finding the globally most probable trees (known as “convergence”) and in estimating the correct proportions of the different types of them (known as “mixing”). Here, we introduce a new class of moves, which propose trees based on their parsimony scores. The proposal distribution derived from the parsimony scores is a quickly computable albeit rough approximation of the conditional posterior distribution over candidate trees. We demonstrate with simulations that parsimony-guided moves correctly sample the uniform distribution of topologies from the prior. We then evaluate their performance against standard moves using six challenging empirical data sets, for which we were able to obtain accurate reference estimates of the posterior using long MCMC runs, a mix of topology proposals, and Metropolis coupling. On these data sets, ranging in size from 357 to 934 taxa and from 1740 to 5681 sites, we find that single chains using parsimony-guided moves usually converge an order of magnitude faster than chains using standard moves. They also exhibit better mixing, that is, they cover the most probable trees more quickly. Our results show that tree moves based on quick and dirty estimates of the posterior probability can significantly outperform standard moves. Future research will have to show to what extent the performance of such moves can be improved further by finding better ways of approximating the posterior probability, taking the trade-off between accuracy and speed into account. [Bayesian phylogenetic inference; MCMC; parsimony; tree proposal.]

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.

One of the most common types of models that helps us to understand neuron behavior is based on the Hodgkin–Huxley ion channel formulation (HH model). A major challenge with inferring parameters in HH models is non-uniqueness: many different sets of ion channel parameter values produce similar outputs for the same input stimulus. Such phenomena result in an objective function that exhibits multiple modes (i.e., multiple local minima). This non-uniqueness of local optimality poses challenges for parameter estimation with many algorithmic optimization techniques. HH models additionally have severe non-linearities resulting in further challenges for inferring parameters in an algorithmic fashion. To address these challenges with a tractable method in high-dimensional parameter spaces, we propose using a particular Markov chain Monte Carlo (MCMC) algorithm, which has the advantage of inferring parameters in a Bayesian framework. The Bayesian approach is designed to be suitable for multimodal solutions to inverse problems. We introduce and demonstrate the method using a three-channel HH model. We then focus on the inference of nine parameters in an eight-channel HH model, which we analyze in detail. We explore how the MCMC algorithm can uncover complex relationships between inferred parameters using five injected current levels. The MCMC method provides as a result a nine-dimensional posterior distribution, which we analyze visually with solution maps or landscapes of the possible parameter sets. The visualized solution maps show new complex structures of the multimodal posteriors, and they allow for selection of locally and globally optimal value sets, and they visually expose parameter sensitivities and regions of higher model robustness. We envision these solution maps as enabling experimentalists to improve the design of future experiments, increase scientific productivity and improve on model structure and ideation when the MCMC algorithm is applied to experimental data.

Porosity and fluid prediction of Minagish Oolite, Minagish Field of south west of Kuwait was carried out using geostatistical inversion approach. This field has several reservoirs and the oil is accumulated primarily in the Minagish Oolite reservoir of Lower Cretaceous age. This reservoir consists of thick porous carbonate containing oil in unsaturated condition. Geostatistical inversion using Bayesian inference combined with Markov Chain Monte Carlo (MCMC) sampling algorithm integrates the information of well logs, geological constraints, geostatistical parameters and seismic data. Multiple realizations of geostatistical inversion were generated to output highly detailed P-impedance and lithology of the reservoir, which were then used to co-simulate porosity and water saturation. The results of porosity and water saturation of this study are validated through recently drilled wells.

Recent progress using geometry in the design of efficient Markov chain Monte Carlo (MCMC) algorithms have shown the effectiveness of the Fisher Riemannian structure. Furthermore, the theory of the underlying geometry of spaces of statistical models has made an important breakthrough by extending the classical theory on exponential families to their closures, the so-called extended exponential families. This paper looks at the underlying geometry of the Fisher information, in particular its limiting behaviour near boundaries, which illuminates the excellent behaviour of the corresponding geometric MCMC algorithms. Further, the paper shows how Fisher geodesics in extended exponential families smoothly attach the boundaries of extended exponential families to their relative interior. We conjecture that this behaviour could be exploited for trans-dimensional MCMC algorithms.

The Bayesian framework is commonly used to quantify uncertainty in seismic inversion. To perform Bayesian inference, Markov chain Monte Carlo (MCMC) algorithms are regarded as the gold standard technique for sampling from the posterior probability distribution. Consistent MCMC methods have trouble for complex, high-dimensional models, and most methods scale poorly to large datasets, such as those arising in seismic inversion. As an alternative, approximate MCMC methods based on unadjusted Langevin dynamics offer scalability and more rapid sampling at the cost of biased inference. However, when assessing the quality of approximate MCMC samples for characterizing the posterior distribution, most diagnostics fail to account for these biases. In this work, we introduce the kernel Stein discrepancy (KSD) as a diagnostic tool to determine the convergence of MCMC samples for Bayesian seismic inversion. We demonstrate the use of the KSD for measuring sample quality and selecting the optimal Langevin MCMC algorithm for two Gaussian Bayesian inference problems.

Bayesian model updating has received considerable attention and has been extensively used in structural damage detection. It provides a rigorous statistical framework for realizing structural system identification and characterizing uncertainties associated with modeling and measurements. The Markov Chain Monte Carlo (MCMC) is a promising tool for inferring the posterior distribution of model parameters to avoid the intractable evaluation of multi-dimensional integration. However, the efficacy of most MCMC techniques suffers from the curse of parameter dimension, which restricts the application of Bayesian model updating to the damage detection of large-scale systems. In addition, there are several MCMC techniques that require users to properly choose application-specific models, based on the understanding of algorithm mechanisms and limitations. As seen in the literature, there is a lack of comprehensive work that investigates the performances of various MCMC algorithms in their application of structural damage detection. In this study, the Differential Evolutionary Adaptive Metropolis (DREAM), a multi-chain MCMC, is explored and adapted to Bayesian model updating. This paper illustrates how DREAM is used for model updating with many uncertainty parameters (i.e., 40 parameters). Furthermore, the study provides a tutorial to users who may be less experienced with Bayesian model updating and MCMC. Two advanced single-chain MCMC algorithms, namely, the Delayed Rejection Adaptive Metropolis (DRAM) and Transitional Markov Chain Monte Carlo (TMCMC), and DREAM are elaborately introduced to allow practitioners to understand better the concepts and practical implementations. Their performances in model updating and damage detection are compared through three different engineering applications with increased complexity, e.g., a forty-story shear building, a two-span continuous steel beam, and a large-scale steel pedestrian bridge.

Markov Chain Monte Carlo (MCMC) algorithms are employed for accurate uncertainty assessments in non-linear geophysical inverse problems. However, one of their main drawbacks is the considerable number of sampled models needed to attain stable posterior estimations, especially in high-dimensional parameter spaces. We use the Discrete Cosine Transform (DCT) to reparametrize a Bayesian Electrical Resistivity Tomography (ERT) inversion solved through an MCMC sampling. In this framework, the unknown parameters become the series of coefficients associated with the retained DCT base functions. We employ the Differential Evolution Markov Chain (DEMC) algorithm that guarantees a more accurate and rapid sampling of the posterior density than more standard MCMC algorithms (such as the random walk Metropolis). To draw essential conclusions about the reliability of the implemented algorithm, we focus on inversions of a synthetic subsurface block model. We assess the benefits provided by the DCT compression of the model space by comparing the outcomes of the implemented inversion approach with those provided by a DEMC algorithm running in the full, un-reduced model space. Although preliminary, our results are promising and prove that the implemented inversion approach guarantees rapid convergence toward the stationary regime, thereby preserving an accurate sampling of the posterior model.

We describe the importance and widespread use of Markov chain Monte Carlo (MCMC) algorithms, with an emphasis on the ways in which theoretical analysis can help with their practical implementation. In particular, we discuss how to achieve rigorous quantitative bounds on convergence to stationarity using the coupling method together with drift and minorisation conditions. We also discuss recent advances in the field of adaptive MCMC, where the computer iteratively selects from among many different MCMC algorithms. Such adaptive MCMC algorithms may fail to converge if implemented naively, but they will converge correctly if certain conditions such as Diminishing Adaptation are satisfied.

In this paper, the authors outline the general principles behind an approach to Bayesian system identification and highlight the benefits of adopting a Bayesian framework when attempting to identify models of nonlinear dynamical systems in the presence of uncertainty. It is then described how, through a summary of some key algorithms, many of the potential difficulties associated with a Bayesian approach can be overcome through the use of Markov chain Monte Carlo (MCMC) methods. The paper concludes with a case study, where an MCMC algorithm is used to facilitate the Bayesian system identification of a nonlinear dynamical system from experimentally observed acceleration time histories.

In this paper, we derive a Markov chain Monte Carlo (MCMC) algorithm supported by a neural network. In particular, we use the neural network to substitute derivative calculations made during a Metropolis adjusted Langevin algorithm (MALA) step with inexpensive neural network evaluations. Using a complex, high-dimensional blood coagulation model and a set of measurements, we define a likelihood function on which we evaluate the new MCMC algorithm. The blood coagulation model is a dynamic model, where derivative calculations are expensive and hence limit the efficiency of derivative-based MCMC algorithms. The MALA adaptation greatly reduces the time per iteration, while only slightly affecting the sample quality. We also test the new algorithm on a 2-dimensional example with a non-convex shape, a case where the MALA algorithm has a clear advantage over other state of the art MCMC algorithms. To assess the impact of the new algorithm, we compare the results to previously generated results of the MALA and the random walk Metropolis Hastings (RWMH).

There is an increasing use of Markov chain Monte Carlo (MCMC) algorithms for fitting statistical models in psychometrics, especially in situations where the traditional estimation techniques are very difficult to apply. One of the disadvantages of using an MCMC algorithm is that it is not straightforward to determine the convergence of the algorithm. Using the output of an MCMC algorithm that has not converged may lead to incorrect inferences on the problem at hand. The convergence is not one to a point, but that of the distribution of a sequence of generated values to another distribution, and hence is not easy to assess; there is no guaranteed diagnostic tool to determine convergence of an MCMC algorithm in general. This article examines the convergence of MCMC algorithms using a number of convergence diagnostics for two real data examples from psychometrics. Findings from this research have the potential to be useful to researchers using the algorithms. For both the examples, the number of iterations required (suggested by the diagnostics) to be reasonably confident that the MCMC algorithm has converged may be larger than what many practitioners consider to be safe.