Bayesian Inference Procedure Research Articles

Dielectric permittivity-water content relationships in woodchips: Particle size and temperature effects

Information on the amount of water held by woodchips in the field is needed to assess environmental risks when they are kept in stockpiles or for inferring their efficiency as denitrifying bioreactors. Reflectometry is a widely used method for estimating volumetric water content (θ) in permeable media, but material-specific calibration functions should be developed. This study aimed at developing relationships between the bulk dielectric permittivity (εb) and θ in woodchips considering their particle size distribution (PSD), temperature (T) and the dielectric permittivity of the dry material plus air (εsa). The εb of fine, medium and coarse PSDs were measured with CS616-L water content reflectometers up to θ of 0.83 cm3/cm3 and at T (°C) of 10°, 24°, 37°, 55° and 70°. Calibration equations were developed using a multivariate power law function fitted to the data with a hierarchical Bayesian inference procedure. The best fit (R2 = 0.96; RMSE=0.04 cm3/cm3) was obtained for the fine PSD (εsa ∼ 1.09); while medium (εsa ∼ 1.09) and coarse (εsa ∼ 1.03) PSDs were more influenced by T and the presence of large air voids (R2 ∼ 0.92; RMSE∼0.06 cm3/cm3). A modified calibration equation independent of PSD (εsa = 1) reduced only slightly the goodness-of-fit (R2 = 0.90; RMSE=0.07 cm3/cm3). The equations developed in this study are well suited for field applications as they cover a wider range of θ than previously published functions and they are applicable within the range of T found in field applications.

An efficient approach for quantifying parameter uncertainty in the SST turbulence model

In this work, an efficient approach for quantifying the parameter uncertainty for expensive computer models is proposed, which combines the high dimensional model representation (HDMR) technique and the Gaussian process machine learning (GPML) method to construct the surrogate model, then the so-constructed surrogate is used in the Bayesian inference procedure to obtain the posterior distribution of the model parameters. The applications of the proposed approach to simple mathematical functions are investigated, demonstrating its efficiency and accuracy for both continuous function and function with discontinuities. The computer design of the sampling points based on GPML is also proposed and the results show that the proposed method is promising in terms of both efficiency and extensibility to high dimensional problems. After testing the proposed approach with simple mathematical functions, it is applied to the two equation SST turbulence model for hypersonic flow over flat plate with a wide range of Mach numbers. A Bayesian scenario-averaging method based on the flow quantities that can characterize both the flow scenario and the model’s performance in the scenario is proposed and it is employed for the model predictions of new flow scenarios. The results show that the prediction mean values match well with the DNS data and the corresponding uncertainties are well captured.

Multivariate Spatial IV Regression

In this paper, a Multivariate Spatial Regression model with Endogenous Variables is proposed. In order to deal with endogeneity and spatial dependence, the instrumental variables (IV) methodology and an autoregressive spatial structure, frequently used in econometric applications, are implemented. A Bayesian inference procedure based on simulation schemes designed to obtain samples from the posterior distribution of model parameters is developed. Finally, the methodology is illustrated through an application to the impact of broadband access on the economic sectors.

Distinguishing between model- and data-driven inferences for high reliability statistical predictions

Estimating the tails of probability distributions plays a key role in complex engineering systems where the goal is characterization of low probability, high consequence events. When data are collected using physical experimentation, statistical distributional assumptions are often used to extrapolate tail behavior to assess reliability, introducing risk due to extrapolation from an unvalidated (statistical) model. Existing tools to evaluate statistical model fit, such as probability plots and goodness of fit tests, fail to communicate the risk associated with this extrapolation. In this work, we develop a new statistical model validation metric and relate this metric to engineering-driven model validation metrics. The metric measures how consistent the parametric tail estimates are with a more flexible model that makes weaker assumptions about the distribution tails. An extreme-value based generalized Pareto distribution is used for the more flexible model. Models are updated using a Bayesian inference procedure that defaults to reasonably conservative inferences when data are sparse. Properties of the estimation procedure are evaluated in statistical simulation, and the effectiveness of the proposed metrics relative to the standard-of-practice statistical metrics is illustrated using a pedagogical example related to a real, but proprietary, engineering example.

Modeling the health effects of time-varying complex environmental mixtures: Mean field variational Bayes for lagged kernel machine regression.

There is substantial interest in assessing how exposure to environmental mixtures, such as chemical mixtures, affect child health. Researchers are also interested in identifying critical time windows of susceptibility to these complex mixtures. A recently developed method, called lagged kernel machine regression (LKMR), simultaneously accounts for these research questions by estimating effects of time-varying mixture exposures, and identifying their critical exposure windows. However, LKMR inference using Markov chain Monte Carlo methods (MCMC-LKMR) is computationally burdensome and time intensive for large datasets, limiting its applicability. Therefore, we develop a mean field variational Bayesian inference procedure for lagged kernel machine regression (MFVB-LKMR). The procedure achieves computational efficiency and reasonable accuracy as compared with the corresponding MCMC estimation method. Updating parameters using MFVB may only take minutes, while the equivalent MCMC method may take many hours or several days. We apply MFVB-LKMR to PROGRESS, a prospective cohort study in Mexico. Results from a subset of PROGRESS using MFVB-LKMR provide evidence of significant positive association between second trimester cobalt levels and z-scored birthweight. This positive association is heightened by cesium exposure. MFVB-LKMR is a promising approach for computationally efficient analysis of environmental health datasets, to identify critical windows of exposure to complex mixtures.

Flood frequency analysis: The Bayesian choice

After an introduction to the traditional flood frequency analysis methods, this article discusses their limits and the risks associated with their thoughtless use: overconfidence in the estimated values of flood quantiles or return periods and systematic underestimation of risks. The article then presents and illustrates the added value of modern Bayesian flood frequency inference procedures that are statistically consistent, numerically accurate, and now computationally affordable. The implementation of such methods shows that estimated flood frequencies, based on observed samples of limited size, are generally affected by large uncertainties. This acknowledgement should be an incentive for increasing the size of the analyzed samples through a more systematic use of historic information as well as regional approaches in flood frequency analyses. It also clearly points out that the margin of errors should be considered when using inference results for design or risk assessment purposes. Several pieces of software are now available to conduct Bayesian flood frequency analyses relatively straightforwardly. There is no remaining obstacle to the implementation of these modern approaches in operational hydrological studies.This article is categorized under:Science of Water > MethodsScience of Water > Water Extremes

A nonparametric Bayesian approach to copula estimation

ABSTRACTWe propose a novel Dirichlet-based Pólya tree (D-P tree) prior on the copula and based on the D-P tree prior, a nonparametric Bayesian inference procedure. Through theoretical analysis and simulations, we are able to show that the flexibility of the D-P tree prior ensures its consistency in copula estimation, thus able to detect more subtle and complex copula structures than earlier nonparametric Bayesian models, such as a Gaussian copula mixture. Furthermore, the continuity of the imposed D-P tree prior leads to a more favourable smoothing effect in copula estimation over classic frequentist methods, especially with small sets of observations. We also apply our method to the copula prediction between the S&P 500 index and the IBM stock prices during the 2007–08 financial crisis, finding that D-P tree-based methods enjoy strong robustness and flexibility over classic methods under such irregular market behaviours.

Bayesian Nonparametric Causal Inference: Information Rates and Learning Algorithms

We investigate the problem of estimating the causal effect of a treatment on individual subjects from observational data, this is a central problem in various application domains, including healthcare, social sciences, and online advertising. Within the Neyman Rubin potential outcomes model, we use the Kullback Leibler (KL) divergence between the estimated and true distributions as a measure of accuracy of the estimate, and we define the information rate of the Bayesian causal inference procedure as the (asymptotic equivalence class of the) expected value of the KL divergence between the estimated and true distributions as a function of the number of samples. Using Fano method, we establish a fundamental limit on the information rate that can be achieved by any Bayesian estimator, and show that this fundamental limit is independent of the selection bias in the observational data. We characterize the Bayesian priors on the potential (factual and counterfactual) outcomes that achieve the optimal information rate. As a consequence, we show that a particular class of priors that have been widely used in the causal inference literature cannot achieve the optimal information rate. On the other hand, a broader class of priors can achieve the optimal information rate. We go on to propose a prior adaptation procedure (which we call the information based empirical Bayes procedure) that optimizes the Bayesian prior by maximizing an information theoretic criterion on the recovered causal effects rather than maximizing the marginal likelihood of the observed (factual) data. Building on our analysis, we construct an information optimal Bayesian causal inference algorithm.

Optimal Robustness Results for Relative Belief Inferences and the Relationship to Prior-Data Conflict

The robustness to the prior of Bayesian inference procedures based on a measure of statistical evidence is considered. These inferences are shown to have optimal properties with respect to robustness. Furthermore, a connection between robustness and prior-data conflict is established. In particular, the inferences are shown to be effectively robust when the choice of prior does not lead to prior-data conflict. When there is prior-data conflict, however, robustness may fail to hold.

Robust unsupervised cluster matching for network data

Unsupervised cluster matching is a task to find matching between clusters of objects in different domains. Examples include matching word clusters in different languages without dictionaries or parallel sentences and matching user communities across different friendship networks. Existing methods assume that every object is assigned into a cluster. However, in real-world applications, some objects would not form clusters. These irrelevant objects deteriorate the cluster matching performance since mistakenly estimated matching affect on estimation of matching of other objects. In this paper, we propose a probabilistic model for robust unsupervised cluster matching that discovers relevance of objects and matching of object clusters, simultaneously, given multiple networks. The proposed method finds correspondence only for relevant objects, and keeps irrelevant objects unmatched, which enables us to improve the matching performance since the adverse impact of irrelevant objects is eliminated. With the proposed method, relevant objects in different networks are clustered into a shared set of clusters by assuming that different networks are generated from a common network probabilistic model, which is an extension of stochastic block models. Objects assigned into the same clusters are considered as matched. Edges for irrelevant objects are assumed to be generated from a noise distribution irrespective of cluster assignments. We present an efficient Bayesian inference procedure of the proposed model based on collapsed Gibbs sampling. In our experiments, we demonstrate the effectiveness of the proposed method using synthetic and real-world data sets, including multilingual corpora and movie ratings.

A Bayesian Markov-Switching Correlation Model for Contagion Analysis on Exchange Rate Markets

This article develops a new Markov-switching vector autoregressive (VAR) model with stochastic correlation for contagion analysis on financial markets. The correlation and the log-volatility dynamics are driven by two independent Markov chains, thus allowing for different effects such as volatility spill-overs and correlation shifts with various degrees of intensity. We outline a suitable Bayesian inference procedure based on Markov chain Monte Carlo algorithms. We then apply the model to some major and Asian-Pacific cross rates against the U.S. dollar and find strong evidence supporting the existence of contagion effects and correlation drops during crises, closely in line with the stylized facts outlined in the contagion literature. A comparison of this model with its closest competitors, such as a time-varying parameter VAR, reveals that our model has a better predictive ability. Supplementary materials for this article are available online

Neglected chaos in international stock markets: Bayesian analysis of the joint return–volatility dynamical system

We use a novel Bayesian inference procedure for the Lyapunov exponent in the dynamical system of returns and their unobserved volatility. In the dynamical system, computation of largest Lyapunov exponent by traditional methods is impossible as the stochastic nature has to be taken explicitly into account due to unobserved volatility. We apply the new techniques to daily stock return data for a group of six countries, namely USA, UK, Switzerland, Netherlands, Germany and France, from 2003 to 2014, by means of Sequential Monte Carlo for Bayesian inference. The evidence points to the direction that there is indeed noisy chaos both before and after the recent financial crisis. However, when a much simpler model is examined where the interaction between returns and volatility is not taken into consideration jointly, the hypothesis of chaotic dynamics does not receive much support by the data (“neglected chaos”).

Estimating People Flow from Spatiotemporal Population Data via Collective Graphical Mixture Models

Thanks to the prevalence of mobile phones and GPS devices, spatiotemporal population data can be obtained easily. In this article, we propose a mixture of collective graphical models for estimating people flow from spatiotemporal population data. The spatiotemporal population data we use as input is the number of people in each grid cell area over time, which is aggregated information about many individuals; to preserve privacy, they do not contain trajectories of each individual. Therefore, it is impossible to directly estimate people flow. To overcome this problem, the proposed model assumes that transition populations are hidden variables and estimates the hidden transition populations and transition probabilities simultaneously. The proposed model can handle changes of people flow over time by segmenting time-of-day points into multiple clusters, where different clusters have different flow patterns. We develop an efficient variational Bayesian inference procedure for the collective graphical mixture model. In our experiments, the effectiveness of the proposed method is demonstrated by using four real-world spatiotemporal population datasets in Tokyo, Osaka, Nagoya, and Beijing.

Biomass: an r package for estimating above‐ground biomass and its uncertainty in tropical forests

Summary Estimating forest above‐ground biomass (AGB), or carbon (AGC), in tropical forests has become a major concern for scientists and stakeholders. However, AGB assessment procedures are not fully standardized and even more importantly, the uncertainty associated with AGB estimates is seldom assessed. Here, we present an r package designed to compute both AGB/AGC estimate and its associated uncertainty from forest plot datasets, using a Bayesian inference procedure. The package builds upon previous work on pantropical and regional biomass allometric equations and published datasets by default, but it can also integrate unpublished or complementary datasets in many steps. BIOMASS performs a number of standard tasks on input forest tree inventories: (i) tree species identification, if available, is automatically corrected; (ii) wood density is estimated from tree species identity; (iii) if height data are available, a local height–diameter allometry may be built; else height is inferred from pantropical or regional models; (iv) finally, AGB/AGC are estimated by propagating the errors associated with all the calculation steps up to the final estimate. R code is given in the paper and in the appendix for the purpose of illustration. The BIOMASS package should contribute to improved standards for AGB calculation for tropical forest stands, and will encourage users to report the uncertainties associated with stand‐level AGB/AGC estimates in future studies.

A non-iterative (trivial) method for posterior inference in stochastic volatility models

We propose a new non-iterative, very simple but accurate, Bayesian inference procedure for the stochastic volatility model. The only requirement of our approach is to solve a large, sparse linear system which we avoid by iteration.

Estimation of Cointegrated Spaces: A Numerical Case Study on Efficiency, Accuracy and Influence of the Model Noise

In this paper we develop an analysis of multivariate time series that exhibit reduced rank cointegration, implying that a lower dimensional linear projection of the process can be obtained in which the projected process becomes stationary. Detection of the rank and basis upon which to project the process for stationarity to hold is a critical problem when working with such settings in practice. There is a range of practice when performing estimation of in such multivariate time series settings. In this paper we provide a review of a few selected different models and estimation techniques for these multivariate time series. Having presented an overview of important new directions with regard to estimation of cointegration relationships we then turn our attention to the performance of a range of estimation procedures. In particular we design a range of numerical studies in order to assess some of these approaches in terms of efficiency and accuracy. In particular, we study the question related to examining the robustness of these classes of estimation procedure in Bayesian and non-parametric estimation approaches to the influence of the model noise in the estimation of the cointegration space. In this context we develop a novel Bayesian inference procedure not previously studied in cointegration models to estimate the cointegration space. This is based on a Markov Chain Monte Carlo sampling method, that consists of a novel extension of Hamiltonian and Geodesic Monte Carlo for the present problem. We will illustrate the performance of this method numerically and show that it produces results on par with an efficient Gibbs Sampler.

Probabilistic-based assessment of a masonry arch bridge considering inferential procedures

Considering the safety assessment requirements of masonry arch bridges, different levels of reliability, based on uncertainty, may be distinguished, whose core objectives are to accurately analyse the ultimate load-carrying capacity and the serviceability structural response. Within this framework, a simplified full-probabilistic methodology for the safety assessment of existing masonry arch bridges is proposed, which combines both structural analysis and Bayesian inference procedures. The proposed framework aims to determine the ultimate load-carrying capacity (Ultimate Limit State) of masonry arch bridges, by using probabilistic procedures and Limit States principles. Geometric, material and load characterization, as well as inherent uncertainties will be also considered. In order to determine the ultimate load-carrying capacity, a limit analysis approach, based on the mechanism method, will be employed. Due to the high computational costs required by a probabilistic safety assessment framework, a sensitivity analysis will then be introduced. The incorporation of new information from monitoring and/or testing will be performed by the application of Bayesian inference methodologies. Based on the information collected, two reliability indexes will be computed and compared, one with data collected from design documentation and literature and the other with data collected from testing, emphasizing the importance of testing and the advantages of Bayesian inference procedures. The probabilistic framework developed is tested and validated in a Portuguese railway masonry arch bridge from the 19th century.

A simulation assessment of methods to infer cultural transmission on dark networks

The social transmission of beliefs, behaviors, and technologies is a central function of dark networks, just as it is in legitimate networks. One motivation for disrupting dark networks is to break the flow of information and learning. It is often unclear, however, which network should be targeted for disruption because individuals inhabit multiple and correlated networks, and the most relevant network for a given cultural process must be inferred from limited empirical data. Three analytic methods potentially are able to distinguish among alternative network diffusion processes: autoregression, dyadic regression with permutations, and dyadic regression with or random effects. All three rely on having measureable cultural outcomes and network or tree-like connections among the data points. We tested the ability of each method to infer cultural diffusion correctly within 4000 simulated datasets generated on two historical networks that linked violent and pacifist Anabaptist religious groups. Under both frequentist and Bayesian inference procedures, regression of dyadic matrices with random effects exhibited the best statistical performance. We found similar results in a more comprehensive search of the network parameter space that simulated both network structures and the diffusion of traits.

Bayesian inference for agreement measures

ABSTRACTThe agreement of different measurement methods is an important issue in several disciplines like, for example, Medicine, Metrology, and Engineering. In this article, some agreement measures, common in the literature, were analyzed from a Bayesian point of view. Posterior inferences for such agreement measures were obtained based on well-known Bayesian inference procedures for the bivariate normal distribution. As a consequence, a general, simple, and effective method is presented, which does not require Markov Chain Monte Carlo methods and can be applied considering a great variety of prior distributions. Illustratively, the method was exemplified using five objective priors for the bivariate normal distribution. A tool for assessing the adequacy of the model is discussed. Results from a simulation study and an application to a real dataset are also reported.

Inference of reaction rate parameters based on summary statistics from experiments

We present the results of an application of Bayesian inference and maximum entropy methods for the estimation of the joint probability density for the Arrhenius rate parameters of the rate coefficient of the H2/O2-mechanism chain branching reaction H+O2→OH+O. Available published data is summary statistics in terms of nominal values and error bars of the rate coefficient of this reaction at a number of temperature values obtained from shock-tube experiments. Our approach relies on generating data, in this case OH concentration profiles, consistent with the given summary statistics, using Approximate Bayesian Computation methods and a Markov chain Monte Carlo procedure. The approach permits the forward propagation of parametric uncertainty through the computational model in a manner that is consistent with the published statistics. A consensus joint posterior on the parameters is obtained by pooling the posterior parameter densities given each consistent data set. To expedite this process, we construct efficient surrogates for the OH concentration using a combination of Padé and polynomial approximants. These surrogate models adequately represent forward model observables and their dependence on input parameters and are computationally efficient to allow their use in the Bayesian inference procedure. We also utilize Gauss–Hermite quadrature with Gaussian proposal probability density functions for moment computation resulting in orders of magnitude speedup in data likelihood evaluation. Despite the strong non-linearity in the model, the consistent data sets all result in nearly Gaussian conditional parameter probability density functions. The technique also accounts for nuisance parameters in the form of Arrhenius parameters of other rate coefficients with prescribed uncertainty. The resulting pooled parameter probability density function is propagated through stoichiometric hydrogen–air auto-ignition computations to illustrate the need to account for correlation among the Arrhenius rate parameters of one reaction and across rate parameters of different reactions.