Convergence of Limiting Cases of Continuous-Time, Discrete-Space Jump Processes to Diffusion Processes for Bayesian Inference

One of the key problems in Reservoir Characterization involves the description and visualization of reservoir heterogeneities (as represented by the spatial variability of properties such as porosity, permeability, thickness, lithofacies types, fracture / fault orientations, sand body geometry, etc). The inherent nonuniqueness associated with this problem has prompted considerable interest in the development and application of stochastic (ie. probabilistic) imaging techniques. Such techniques are designed to generate a family of equiprobable descriptions or stochastic images of these parameters - each image being consistent with all the available quantitative (well-logs, cores, seismic derived constraint intervals, well-test information) and qualitative (geological interpretation) information, and its spatial correlation characteristics. These stochastic images may be viewed as samples from approximations to the "optimal’’ Bayesian posterior distribution on the reservoir parameters. Statistical analysis of the "spread" of this distribution allows for the quantification of risk/uncertainty associated with the spatial variability of these parameters. Also, selected subsets of stochastic images may be "passed" through dynamic flow simulators to assess the distribution of significant production response variables. Such distributions can be incorporated with statistical decision theoretic techniques in order to aid in the optimal forecasting and management of the reservoir. A number of stochastic imaging techniques have been developed in the past few years - indeed this number is rapidly growing. The object of this paper is to assess the current state of the art in stochastic imaging techniques for reservoir characterization, along with associated statistical methodologies for integrating seismic data, and for reservoir performance forecasting and management. The following paragraphs provide an overview of the body of the paper. Section II of the paper provides a comparitive review of stochastic imaging techniques. The reviewed set covers both discrete and continuous single/multivariable methods - these include Boolean algorithms and Marked Point Processes, Indicator methods, (truncated) Gaussian Random Functions. Fractal fields, Simulated Annealing, Markov Random Fields and direct Bayesian Imaging algorithms. Particular attention is given to underlying assumptions* data integration, internal consistency, performance (eg. exactitude, reproduction of spatial correlation structure, quality of approximation to the Bayesian posterior), computational and inferential complexity, and practical limitations. Also, the techniques are compared with respect to their capabilities for incorporating "soft" information (such as inequality constraints), handling anisotropy and trends (ie.lst order non-stationarities), and for imaging vector variables. The potential of seismic data for adding detail to reservoir descriptions "between the wells", is now generally acknowledged. Section III reviews known techniques for integrating seismic data in reservoir descriptions. This includes recent developments in techniques such as External drift, Cokriging, Markov Random Fields, M.A.P. algorithms, Bayesian (Hard/Soft) Inversion, Markov_Bayes algorithms, and ID Stochastic Inversion. Brief descriptions are also provided of methods for conditioning the stochastic images to physics-based "forward models", and to qualitative geological information. Section IV summarizes current techniques for utilizing the stochastic images in performance forecasting and reservoir management. This review emphasizes the use of statistical decision theoretic approaches. Also, current progress in conditioning stochastic reservoir models to production / well-test information is summarized. In section V, illustrative test results are presented of the application of these techniques to both synthetic and where available, "real" reservoir data sets. Reservoir description applications of hybrid multistep approaches are also summarized - here multiple stochastic imaging algorithms are applied in sequence to compute progressively more detailed descriptions. Section VI presents general guidelines for the use of stochastic imaging techniques on specific reservoir characterization problems. The paper concludes with an overview of open problems and current research directions in this field. These include (computationally feasible) multivariable stochastic imaging, incorporation of seismic information, visualization, utilization of stochastic images in dynamic flow simulations, and decision theoretic techniques for reservoir performance forecasting and management.

While empirical Bayes methods thrive in the presence of the thousands of simultaneous hypothesis tests in genomics and other large-scale applications, significance tests and confidence intervals are considered more appropriate for small numbers of tested hypotheses. Indeed, for fewer hypotheses, there is more uncertainty in empirical Bayes estimates of the prior distribution. Confidence intervals have been used to propagate the uncertainty in the prior to empirical Bayes inference about a parameter, but only by combining a Bayesian posterior distribution with a confidence distribution. Combining distributions of both types has also been used to combine empirical Bayes methods and confidence intervals for estimating a parameter of interest. To clarify the foundational status of such combinations, the concept of an evidential model is proposed. In the framework of evidential models, both Bayesian posterior distributions and confidence distributions are special cases of evidential support distributions. Evidential support distributions, by quantifying the sufficiency of the data as evidence, leverage the strengths of Bayesian posterior distributions and confidence distributions for cases in which each type performs well and for cases benefiting from the combination of both. Evidential support distributions also address problems of bioequivalence, bounded parameters, and the lack of a unique confidence distribution.

New insights into Europa's surface using Galileo/NIMS data and MCMC modeling

A multistage single arm phase II trial with binary endpoint is considered. Bayesian posterior probabilities are used to monitor futility in interim analyses and efficacy in the final analysis. For a beta-binomial model, decision rules based on Bayesian posterior probabilities are converted to "traditional" decision rules in terms of number of responders among patients observed so far. Analytical derivations are given for the probability of stopping for futility and for the probability to declare efficacy. A workflow is presented on how to select the parameters specifying the Bayesian design, and the operating characteristics of the design are investigated. It is outlined how the presented approach can be transferred to statistical models other than the beta-binomial model.

Two basic formulas, for the mean and variance of the number of effects in an epidemiological cohort, are de- rived. The formula for variance shows extra-binomial variation or overdispersion when there is correlated uncertainty of the probability of an effect. The formulas were validated by a numerical Monte Carlo study. The method of including epistemic uncertainty discussed by Hofer (E. Hofer, Health Physics, 2007) is generalized to include separately uncer- tainty from a Bayesian posterior distribution when the prior is known, and uncertainty of the prior. In this note, two basic formulas, for the mean and vari- ance of the number of effects in an epidemiological cohort, are derived. It is conventionally assumed that the number of effects has either a Poisson or binomial distribution (1). The formula for the variance given here reduces to the binomial result in the case of no correlations of the probability of an effect, but shows extra-binomial variation or overdisper- sion when there are correlations. In practice, the variance could be calculated using the method advocated by Hofer (2), where, for each individual in an epidemiological cohort, some number j = 1…M of al- ternate realizations of the dose and hence the probability of an effect, taking into account possible correlations, are gen- erated using Monte Carlo. This method is generalized here to include separately uncertainty from a Bayesian posterior distribution when the prior is known, and uncertainties caused by lack of knowledge of the prior. This is because, within a linear dose-effect response model, the average num- ber of effects is proportional to the posterior-average- collec- tive dose, and the important uncertainty is that of the poste- rior-average-collective dose caused by lack of knowledge of the prior.

A key problem in Reservoir Characterization concerns the description and visualization of reservoir heterogeneities as represented by properties such as porosity, permeability, etc. The inherent nonuniqueness associated with this problem has prompted considerable interest in the development and application of Stochastic Imaging techniques. Such techniques are designed to generate a family of equiprobable images of these properties - each image being consistent with all the available quantitative (well-logs, cores, seismic derived constraint intervals, well-test information) and qualitative (geological interpretations) information, and its spatial correlation characteristics. These stochastic images may be viewed as samples from approximations to the "optimal" Bayesian posterior distribution on the unknown properties. Statistical analysis of the "spread" of this distribution allows for the quantification of risk/uncertainty associated with the spatial variability of these properties. Such distributions can be used in dynamic flow simulations and statistical decision theoretic techniques for optimal forecasting and management of the reservoir.

ABSTRACTThe jumping model has been used as an effective tool in tracking and detecting changes for continuous statistics in various applications. In this article, we extend the current jumping model from the continuous case to the discrete case to track and monitor the changes in attribute data. In this method, the jumping model–based posterior distribution of the process mean is constructed with attribute data and prior knowledge of the process. The posterior distribution consists of several components that account for the weights of the process to be “in-control” or “out-of-control.” Using the component representing the in-control weight as the monitoring index, a jumping model–based control chart is developed to monitor the attribute data process. The proposed chart is further extended to cover different out-of-control modes. The performance of the jumping model–based chart is investigated and compared to conventional control charts through numerical studies and a real-world data set. The results demonstrate the effectiveness of the proposed chart.

This article presents a Bayesian track-before-detect (TBD) method based on Gaussian message passing to detect and track a target in the low SNR scene. Removing the threshold brings great computation demanding to TBD methods. Because of the distributive law and computation consistency, message passing can reduce the calculation load efficiently. Also, as a probabilistic inference algorithm, message passing can perform the TBD method as a Bayesian posterior distribution. Meanwhile, the close form of multivariate Gaussian message passing is derived, with the adoption of Taylor series expansion to treat the nonlinearity in observation equation. The simulation results demonstrate the effectiveness the proposed algorithm.

Reducing estimation risk using a Bayesian posterior distribution approach: Application to stress testing mortgage loan default

In the analysis of metastable diffusion processes, Transition Path Theory (TPT) provides a way to quantify the probability of observing a given transition between two disjoint metastable subsets of state space. However, many TPT-based methods for diffusion processes compute the primary objects from TPT, such as the committor and probability current, by solving partial differential equations. The computational performance of these methods is limited by the need for mesh-based computations, the need to estimate the coefficients of the stochastic differential equation that defines the diffusion process, and the use of Markovian processes to approximate the diffusion process. We propose a Monte Carlo method for approximating the primary objects from TPT from sample trajectory data of the diffusion process, without estimating drift or diffusion coefficients. We discretise the state space of the diffusion process using Voronoi tessellations and construct a non-Markovian jump process on the dual Delaunay graph. For the jump process, we define committors, probability currents, and streamlines, and use these to define piecewise constant approximations of the corresponding objects from TPT for diffusion processes. Rigorous error bounds and convergence theorems establish the validity of our approach. A comparison of our method with TPT for Markov chains (Metzner et al., Multiscale Model Simul. 2009) on a triple-well 2D potential provides proof of principle.

The Bayesian approach to inverse problems, in which the posterior probability distribution on an unknown field is sampled for the purposes of computing posterior expectations of quantities of interest, is starting to become computationally feasible for partial differential equation (PDE) inverse problems. Balancing the sources of error arising from finite-dimensional approximation of the unknown field, the PDE forward solution map and the sampling of the probability space under the posterior distribution are essential for the design of efficient computational Bayesian methods for PDE inverse problems. We study Bayesian inversion for a model elliptic PDE with an unknown diffusion coefficient. We provide complexity analyses of several Markov chain Monte Carlo (MCMC) methods for the efficient numerical evaluation of expectations under the Bayesian posterior distribution, given data δ. Particular attention is given to bounds on the overall work required to achieve a prescribed error level ε. Specifically, we first bound the computational complexity of ‘plain’ MCMC, based on combining MCMC sampling with linear complexity multi-level solvers for elliptic PDE. Our (new) work versus accuracy bounds show that the complexity of this approach can be quite prohibitive. Two strategies for reducing the computational complexity are then proposed and analyzed: first, a sparse, parametric and deterministic generalized polynomial chaos (gpc) ‘surrogate’ representation of the forward response map of the PDE over the entire parameter space, and, second, a novel multi-level Markov chain Monte Carlo strategy which utilizes sampling from a multi-level discretization of the posterior and the forward PDE. For both of these strategies, we derive asymptotic bounds on work versus accuracy, and hence asymptotic bounds on the computational complexity of the algorithms. In particular, we provide sufficient conditions on the regularity of the unknown coefficients of the PDE and on the approximation methods used, in order for the accelerations of MCMC resulting from these strategies to lead to complexity reductions over ‘plain’ MCMC algorithms for the Bayesian inversion of PDEs.

Reversible-jump Markov chain Monte Carlo (RJ-MCMC) is a technique for simultaneously evaluating multiple related (but not necessarily nested) statistical models that has recently been applied to the problem of phylogenetic model selection. Here we use a simulation approach to assess the performance of this method and compare it to Akaike weights, a measure of model uncertainty that is based on the Akaike information criterion. Under conditions where the assumptions of the candidate models matched the generating conditions, both Bayesian and AIC-based methods perform well. The 95% credible interval contained the generating model close to 95% of the time. However, the size of the credible interval differed with the Bayesian credible set containing approximately 25% to 50% fewer models than an AIC-based credible interval. The posterior probability was a better indicator of the correct model than the Akaike weight when all assumptions were met but both measures performed similarly when some model assumptions were violated. Models in the Bayesian posterior distribution were also more similar to the generating model in their number of parameters and were less biased in their complexity. In contrast, Akaike-weighted models were more distant from the generating model and biased towards slightly greater complexity. The AIC-based credible interval appeared to be more robust to the violation of the rate homogeneity assumption. Both AIC and Bayesian approaches suggest that substantial uncertainty can accompany the choice of model for phylogenetic analyses, suggesting that alternative candidate models should be examined in analysis of phylogenetic data. [AIC; Akaike weights; Bayesian phylogenetics; model averaging; model selection; model uncertainty; posterior probability; reversible jump.].

Uncertainty in systemic risks rankings: Bayesian and frequentist analysis

ABSTRACTThe predictive distribution is a mixture of the original distribution model and is used for predicting a future observation. Therein, the mixing distribution is the posterior distribution of the distribution parameters in the Bayesian inference. The mixture can also be computed for the frequentist inference because the Bayesian posterior distribution has the same meaning as a frequentist confidence interval. I present arguments against the concept of predictive distribution. Examples illustrate these. The most important argument is that the predictive distribution can depend on the parameterization. An improvement of the theory of the predictive distribution is recommended.

The nonlinear response of molecular systems undergoing Markovian stochastic reorientations is calculated up to fifth order in the amplitude of the external field. Time-dependent perturbation theory is used to compute the relevant response functions as in earlier treatments (G. Diezemann, Phys. Rev. E{\bf 85}, 051502 (2012), Phys. Rev. E{\bf 96}, 022150 (2017)). Here, we consider the reorientational motion of isolated molecules and extend the existing calculations for the model of isotropic rotational diffusion to the model of anisotropic rotational diffusion and to the model of rotational random jumps. Depending on the values of some model parameters, we observe a hump in the modulus of the nonlinear susceptibility for either of these models. Interestingly, for the model of rotational random jumps, the appearance of this hump depends on the way the coupling to the external field is modelled in the master equation approach. If the model of anisotropic rotational diffusion is considered, the orientation of the diffusion tensor relative to the molecular dipole moment and additionally the amount of anisotropy in the rotational diffusion constants determine the detailed shape of the nonlinear response. In this case, the height of the observed hump is found to increase with increasing 'diffusional anisotropy'. We discuss our results in relation to the features observed experimentally in supercooled liquids.