Prior Distribution Of Parameters Research Articles

Analytic and Stochastic Methods of Structure Parameter Estimation

The paper presents analytic and stochastic methods of structure parameters estimation for a model selection problem. Structure parameters are covariance matrices of parameters of linear and non-linear regression models. To optimize model parameters and structure parameters we maximize a model evidence, a convolution of a data likelihood with a prior distribution of model parameters. The analytic methods are based on the derivatives computation of the approximated model evidence. The stochastic methods are based on the model parameters sampling and data cross-validation. The proposed methods are tested and compared on the synthetic and real data.

Posterior propriety in Bayesian extreme value analyses using reference priors

The Generalized Pareto (GP) and Generalized extreme value (GEV) distributions play an important role in extreme value analyses, as models for threshold excesses and block maxima respectively. For each of these distributions we consider Bayesian inference using "reference" prior distributions (in the general sense of priors constructed using formal rules) for the model parameters, specifically a Jeffreys prior, the maximal data information (MDI) prior and independent uniform priors on separate model parameters. We investigate the important issue of whether these improper priors lead to proper posterior distributions. We show that, in the GP and GEV cases, the MDI prior, unless modified, never yields a proper posterior and that in the GEV case this also applies to the Jeffreys prior. We also show that a sample size of three (four) is sufficient for independent uniform priors to yield a proper posterior distribution in the GP (GEV) case.

Model-based optimal design of experiments —Semidefinite and nonlinear programming formulations

We use mathematical programming tools, such as Semidefinite Programming (SDP) and Nonlinear Programming (NLP)-based formulations to find optimal designs for models used in chemistry and chemical engineering. In particular, we employ local design-based setups in linear models and a Bayesian setup in nonlinear models to find optimal designs. In the latter case, Gaussian Quadrature Formulas (GQFs) are used to evaluate the optimality criterion averaged over the prior distribution for the model parameters. Mathematical programming techniques are then applied to solve the optimization problems. Because such methods require the design space be discretized, we also evaluate the impact of the discretization scheme on the generated design. We demonstrate the techniques for finding D-, A- and E-optimal designs using design problems in biochemical engineering and show the method can also be directly applied to tackle additional issues, such as heteroscedasticity in the model. Our results show that the NLP formulation produces highly efficient D-optimal designs but is computationally less efficient than that required for the SDP formulation. The efficiencies of the generated designs from the two methods are generally very close and so we recommend the SDP formulation in practice.

SIMULATION-BASED METHODOLOGY FOR PREDICTING FOOTBALL MATCH OUTCOMES CONSIDERING EXPERTS' OPINIONS: THE 2010 AND 2014 FOOTBALL WORLD CUP CASES

In this paper we propose a simulation-based method for predicting the 2010 and 2014Football World Cup. Adopting a bayesian perspective, we modeled the number of goals of two opposing teams as a Poisson distribution whose mean is proportional to the relative technical level of opponents. FIFA ratings were taken as the measure of technical level of teams and experts' opinions about scores of matches were taken to construct prior distribution of parameters. Just before each round, tournament simulations were performed in order to estimate probabilities of events of main interest for audience and bettors such as qualifying to the knockout stage, reaching semi-finals, reaching the final match, winningthe tournament, among others.

Hierarchical calibration and validation of computational fluid dynamics models for solid sorbent-based carbon capture

To quantify the predictive confidence of a device scale solid sorbent-based carbon capture design where there is no direct experimental data available, a hierarchical validation methodology is first proposed. In this hierarchy, a sequence of increasingly complex “unit problems” are validated using a statistical calibration framework. This paper describes the computational fluid dynamics (CFD) multi-phase reactive flow simulations and the associated data flows within each unit problem. Each validation requires both simulated and physical data, so the bench-top experiments used in each increasingly complex stage were carefully designed to follow the same operating conditions as the simulation scenarios. A Bayesian calibration procedure is employed and the posterior model parameter distributions obtained at one unit-problem level are used as prior distributions for the same parameters in the next-tier simulations. Overall, the results have demonstrated that the calibrated multiphase reactive flow models within MFIX can be used to capture the bed pressure, temperature, CO2 capture capacity, and kinetics with quantitative accuracy. The CFD modeling methodology and associated uncertainty quantification techniques presented herein offer a solid framework for estimating the predictive confidence in the virtual scale up of a larger carbon capture device.

An informative prior probability distribution of the gompertz parameters for bayesian approaches in paleodemography.

In paleodemography, the Bayesian approach has been suggested to provide an effective means by which mortality profiles of past populations can be adequately estimated, and thus avoid problems of "age-mimicry" inherent in conventional approaches. In this study, we propose an application of the Gompertz model using an "informative" prior probability distribution by revising a recent example of the Bayesian approach based on an "uninformative" distribution. Life-table data of 134 human populations including those of contemporary hunter-gatherers were used to determine the Gompertz parameters of each population. In each population, we used both raw life-table data and the Gompertz parameters to calculate some demographic values such as the mean life-span, to confirm representativeness of the model. Then, the correlation between the two Gompertz parameters (the Strehler-Mildvan correlation) was re-established. We incorporated the correlation into the Bayesian approach as an "informative" prior probability distribution, and tested its effectiveness using simulated data. Our analyses showed that the mean life-span (≥ age 15) and the proportion of living persons aging over 45 were well-reproduced by the Gompertz model. The simulation showed that using the correlation as an informative prior provides a narrower estimation range in the Bayesian approach than does the uninformative prior. The Gompertz model can be assumed to accurately estimate the mean life-span and/or the proportion of old people in a population. We suggest that the Strehler-Mildvan correlation can be used as a useful constraint in demographic reconstructions of past human populations.

Error probabilities in default Bayesian hypothesis testing

This paper investigates the classical type I and type II error probabilities of default Bayes factors for a Bayesian t test. Default Bayes factors quantify the relative evidence between the null hypothesis and the unrestricted alternative hypothesis without needing to specify prior distributions for the unknown parameters based on one’s prior beliefs. It is shown that in most typical situations in psychological research (when either observing no, small, medium or large effects) default Bayes factors are asymmetric in information, i.e., they result in unequal error probabilities. The tendency to either prefer the null hypothesis or the alternative hypothesis varies for different default Bayes factors. Although this asymmetry in information is a natural property of a Bayes factor, severe cases of asymmetry may be undesirable in a default setting because the underlying default priors are not a translation of one’s prior beliefs. A calibration scheme is presented to make a default Bayes factor symmetric in information under certain conditions.

Sparse and robust normal and t- portfolios by penalized Lq-likelihood minimization

Two important problems arising in traditional asset allocation methods are the sensitivity to estimation error of portfolio weights and the high dimensionality of the set of candidate assets. In this paper, we address both issues by proposing a new criterion for portfolio selection. The new criterion is a two-stage description of the available information, where the q-entropy, a generalized measure of information, is used to code the uncertainty of the data given the parametric model and the uncertainty related to the model choice. The information about the model is coded in terms of a prior distribution that promotes asset weights sparsity. Our approach carries out model selection and estimation in a single step, by selecting a few assets and estimating their portfolio weights simultaneously. The resulting portfolios are doubly robust, in the sense that they can tolerate deviations from both assumed data model and prior distribution for model parameters. Empirical results on simulated and real-world data support the validity of our approach.

Automatic Bayes factors for testing variances of two independent normal distributions

Researchers are frequently interested in testing variances of two independent populations. We often would like to know whether the population variances are equal, whether population 1 has smaller variance than population 2, or whether population 1 has larger variance than population 2. In this article we consider the Bayes factor, a Bayesian model selection and hypothesis testing criterion, for this multiple hypothesis test. Application of Bayes factors requires specification of prior distributions for the model parameters. Automatic Bayes factors circumvent the difficult task of prior elicitation by using data-driven mechanisms to specify priors in an automatic fashion. In this article we develop different automatic Bayes factors for testing two variances: first we apply the fractional Bayes factor (FBF) to the testing problem. It is shown that the FBF does not always function as Occam’s razor. Second we develop a new automatic balanced Bayes factor with equal priors for the variances. Third we propose a Bayes factor based on an adjustment of the marginal likelihood in the FBF approach. The latter two methods always function as Occam’s razor. Through theoretical considerations and numerical simulations it is shown that the third approach provides strongest evidence in favor of the true hypothesis.

Solar zenith angles for biological research and an expected catch model for diel vertical migration patterns that affect stock size estimates for longfin inshore squid (Doryteuthis pealeii)

Solar zenith angles are useful in diel studies because they are directly related to potential solar irradiance at the point of sampling and can be calculated from location, date, and time of day. We used zenith angles to quantify diel vertical migration effects on both the probability of a positive tow and catch size for longfin squid (Doryteuthis pealeii) using two-stage generalized additive models (GAMs). Zenith angles were better than time of day for modeling diel effects on inshore longfin squid bottom trawl survey catches and were particularly suitable for data collected over large areas and extended time periods. Diel effects were size-specific in most cases. Our expected catch method can be used to account for diel effects when estimating swept-area stock size from research survey data. Differences in observed day–night catches and model results show the potential for bias in swept-area stock size estimates that ignore diel migration effects. Zenith angles may be useful in specifying prior distributions for survey catchability parameters in stock assessment models.

A Bayesian analysis of Generalized Pareto Distribution of runoff minima

AbstractGlobal climate change models have predicted the intensification of extreme events, and these predictions are already occurring. For disaster management and adaptation of extreme events, it is essential to improve the accuracy of extreme value statistical models. In this study, Bayes' Theorem is introduced to estimate parameters in Generalized Pareto Distribution (GPD), and then the GPD is applied to simulate the distribution of minimum monthly runoff during dry periods in mountain areas of the Ürümqi River, Northwest China. Bayes' Theorem treats parameters as random variables and provides a robust way to convert the prior distribution of parameters into a posterior distribution. Statistical inferences based on posterior distribution can provide a more comprehensive representation of the parameters. An improved Markov Chain Monte Carlo (MCMC) method, which can solve high‐dimensional integral computation in the Bayes equation, is used to generate parameter simulations from the posterior distribution. Model diagnosis plots are made to guarantee the fitted GPD is appropriate. Then based on the GPD with Bayesian parameter estimates, monthly runoff minima corresponding to different return periods can be calculated. The results show that the improved MCMC method is able to make Markov chains converge faster. The monthly runoff minima corresponding to 10a, 25a, 50a and 100a return periods are 0.60 m3/s, 0.44 m3/s, 0.32 m3/s and 0.20 m3/s respectively. The lower boundary of 95% confidence interval of 100a return level is below zero, which implies that the Ürümqi River is likely to cease to flow when 100a return level appears in dry periods. Copyright © 2015 John Wiley & Sons, Ltd.

Bayesian analysis of covariance under inverse Gaussian model

This paper considers the problem of analysis of covariance (ANCOVA) under the assumption of inverse Gaussian distribution for response variable from the Bayesian point of view. We develop a fully Bayesian model for ANCOVA based on the conjugate prior distributions for parameters contained in the model. The Bayes estimator of parameters, ANCOVA model and adjusted effects for both treatments and covariates along with predictive distribution of future observations are developed. We also provide the essentials for comparing adjusted treatments effects and adjusted factor effects. A simulation study and a real world application are also performed to illustrate and evaluate the proposed Bayesian model.

Prior Distributions of Material Parameters for Bayesian Calibration of Growth and Remodeling Computational Model of Abdominal Aortic Wall.

For the accurate prediction of the vascular disease progression, there is a crucial need for developing a systematic tool aimed toward patient-specific modeling. Considering the interpatient variations, a prior distribution of model parameters has a strong influence on computational results for arterial mechanics. One crucial step toward patient-specific computational modeling is to identify parameters of prior distributions that reflect existing knowledge. In this paper, we present a new systematic method to estimate the prior distribution for the parameters of a constrained mixture model using previous biaxial tests of healthy abdominal aortas (AAs). We investigate the correlation between the estimated parameters for each constituent and the patient's age and gender; however, the results indicate that the parameters are correlated with age only. The parameters are classified into two groups: Group-I in which the parameters ce,ck1,ck2,cm2,Ghc, and ϕe are correlated with age, and Group-II in which the parameters cm1,Ghm,G1e,G2e, and α are not correlated with age. For the parameters in Group-I, we used regression associated with age via linear or inverse relations, in which their prior distributions provide conditional distributions with confidence intervals. For Group-II, the parameter estimated values were subjected to multiple transformations and chosen if the transformed data had a better fit to the normal distribution than the original. This information improves the prior distribution of a subject-specific model by specifying parameters that are correlated with age and their transformed distributions. Therefore, this study is a necessary first step in our group's approach toward a Bayesian calibration of an aortic model. The results from this study will be used as the prior information necessary for the initialization of Bayesian calibration of a computational model for future applications.

Process-based design flood estimation in ungauged basins by conditioning model parameters on regional hydrological signatures

The use of rainfall–runoff models constitutes an alternative to statistical approaches (such as at-site or regional flood frequency analysis) for design flood estimation and represents an answer to the increasing need for synthetic design hydrographs associated with a specific return period. Nevertheless, the lack of streamflow observations and the consequent high uncertainty associated with parameters estimation usually pose serious limitations to the use of process-based approaches in ungauged catchments, which in contrast represent the majority in practical applications. This work presents a Bayesian procedure that, for a predefined rainfall–runoff model, allows for the assessment of posterior parameters distribution, using limited and uncertain information available about the response of ungauged catchments, i.e. the regionalized first three L-moments of annual streamflow maxima. The methodology is tested for a catchment located in southern Italy and used within a Monte Carlo scheme to obtain design flood values and simulation uncertainty bands through both event-based and continuous simulation approaches. The obtained results highlight the relevant reduction in uncertainty bands associated with simulated peak discharges compared to those obtained considering a prior uniform distribution for model parameters. A direct impact of uncertainty in regional estimates of hydrological signatures on posterior parameters distribution is also evident. For the selected case study, continuous simulation, generally, better matches the estimates of the statistical flood frequency analysis.

Estimating performance of Feynman's ratchet with limited information

We estimate the performance of Feynman’s ratchet at given values of the ratio of cold to hot reservoir temperatures (θ) and the figure of merit (efficiency in the case of engine and coefficienct of performance in the case of refrigerator). The latter implies that only the ratio of two intrinsic energy scales is known to the observer, but their exact values are completely uncertain. The prior probability distribution for the uncertain energy parameters is argued to be Jeffreys prior. We define an average measure for performance of the model by averaging, over the prior distribution, the power output (heat engine) or the χ-criterion (refrigerator) which is the product of rate of heat absorbed from the cold reservoir and the coefficient of performance (COP). We observe that the figure of merit, at optimal performance close to equilibrium, is reproduced by the prior-averaging procedure. Further, we obtain the well-known expressions of finite-time thermodynamics for the efficiency at optimal power and the COP at optimal χ-criterion, given by and respectively. This analogy is explored further and we point out that the expected heat flow from and to the reservoirs, behaves as an effective Newtonian flow. We also show, in a class of quasi-static models of quantum heat engines, how Curzon–Ahlborn efficiency emerges in asymptotic limit with the use of Jeffreys prior.

Flood and drought hydrologic monitoring: the role of model parameter uncertainty

Abstract. Land surface modeling, in conjunction with numerical weather forecasting and satellite remote sensing, is playing an increasing role in global monitoring and prediction of extreme hydrologic events (i.e., floods and droughts). However, uncertainties in the meteorological forcings, model structure, and parameter identifiability limit the reliability of model predictions. This study focuses on the latter by assessing two potential weaknesses that emerge due to limitations in our global runoff observations: (1) the limits of identifying model parameters at coarser timescales than those at which the extreme events occur, and (2) the negative impacts of not properly accounting for model parameter equifinality in the predictions of extreme events. To address these challenges, petascale parallel computing is used to perform the first global-scale, 10 000 member ensemble-based evaluation of plausible model parameters using the VIC (Variable Infiltration Capacity) land surface model, aiming to characterize the impact of parameter identifiability on the uncertainty in flood and drought predictions. Additionally, VIC's global-scale parametric sensitivities are assessed at the annual, monthly, and daily timescales to determine whether coarse-timescale observations can properly constrain extreme events. Global and climate type results indicate that parameter uncertainty remains an important concern for predicting extreme events even after applying monthly and annual constraints to the ensemble, suggesting a need for improved prior distributions of the model parameters as well as improved observations. This study contributes a comprehensive evaluation of land surface modeling for global flood and drought monitoring and suggests paths forward to overcome the challenges posed by parameter uncertainty.

Probabilistic inference of fatigue damage propagation with limited and partial information

A general method of probabilistic fatigue damage prognostics using limited and partial information is developed. Limited and partial information refers to measurable data that are not enough or cannot directly be used to statistically identify model parameter using traditional regression analysis. In the proposed method, the prior probability distribution of model parameters is derived based on the principle of maximum entropy (MaxEnt) using the limited and partial information as constraints. The posterior distribution is formulated using the principle of maximum relative entropy (MRE) to perform probability updating when new information is available and reduces uncertainty in prognosis results. It is shown that the posterior distribution is equivalent to a Bayesian posterior when the new information used for updating is point measurements. A numerical quadrature interpolating method is used to calculate the asymptotic approximation for the prior distribution. Once the prior is obtained, subsequent measurement data are used to perform updating using Markov chain Monte Carlo (MCMC) simulations. Fatigue crack prognosis problems with experimental data are presented for demonstration and validation.

An Application of Bayesian Posterior Analysis for Disc Drive Annual Failure Rate (AFR) Estimate

ABSTRACT As an important reliability metric in disc drive industries, the annual failure rate (AFR), estimated from reliability demonstration tests (RDTs), is often used to make business decisions. Due to limited test sample size and short test duration, estimating the AFR from direct test data using Weibull distribution fit may result in a wide confidence interval, raising concerns regarding the uncertainty of the AFR estimate. To improve the confidence of the estimate, this article presents a Bayesian posterior analysis approach to estimate the AFR. Prior distributions of the Weibull distribution's shape and scale parameters are presented based on historic test data. An analysis of real RDT data demonstrates that the Bayesian posterior estimate of the AFR can produce a significantly narrower confidence interval than direct Weibull fitting, promoting higher confidence to accept the AFR estimate. To evaluate the significance of the prior distribution parameters on the AFR estimate, a sensitivity analysis is presented with respect to each individual parameter. The results indicate that the standard deviation value of either β or η has a more significant impact on the AFR estimate than the mean value.

Fundamental two-stage formulation for Bayesian system identification, Part I: General theory

Structural system identification is concerned with the determination of structural model parameters (e.g., stiffness, mass) based on measured response data collected from the subject structure. For linear structures, one popular strategy is to adopt a ‘two-stage’ approach. That is, modal identification (e.g., frequency, mode shape) is performed in Stage I, whose information is used for inferring the structural parameters in Stage II. Different variants of Bayesian two-stage formulations have been proposed in the past. A prediction error model is commonly introduced to build a link between Stages I and II, treating the most probable values of the natural frequencies and mode shapes identified in Stage I as ‘data’ for Stage II. This type of formulation, which casts a prediction error model through descriptive statistics, involves heuristics that distort the fundamental nature of the Bayesian approach, although it has appeared to be inevitable. In this paper, a fundamental theory is developed for the Bayesian two-stage problem. The posterior distribution of structural parameters is derived rigorously in terms of the information available in the problem, namely the prior distribution of structural parameters, the posterior distribution of modal parameters in Stage I and the distribution of modal parameters conditional on the structural parameters that connects Stages I and II. The theory reveals a fundamental principle that ensures no double-counting of prior information in the two-stage identification process. Mathematical statements are also derived that provide insights into the role of the structural modeling error. Beyond the original structural model identification problem that motivated the work, the developed theory can be applied in more general settings. In the companion paper, examples with synthetic and real experimental data are provided to illustrate the proposed theory.

Probabilistic modelling of macroseismic attenuation and forecast of damage scenarios

According to the idea now widespread that macroseismic intensity should be expressed in probabilistic terms, a beta-binomial model has been proposed in the literature to estimate the probability of the intensity at site in the Bayesian framework and a clustering procedure has been adopted to define learning sets of macroseismic fields required to assign prior distributions of the model parameters. This article presents the results concerning the learning sets obtained by exploiting the large Italian macroseismic database DBM1I11 (Locati et al. in DBMI11, the 2011 version of the Italian Macroseismic Database, 2011. http://emidius.mi.ingv.it/DBMI11/ ) and discusses the problems related to their use in probabilistic modelling of the attenuation in seismic regions of the European countries partners of the UPStrat-MAFA project (2012), namely South Iceland, Portugal, SE Spain and Mt Etna volcano area (Italy). Anisotropy and the presence of offshore earthquakes are some of the problems faced. All the work has been carried out in the framework of the Task B of the project.