Preposterior Analysis Research Articles

On the Value of Forecasting in Cable Ice Risk Management

This paper addresses the issue of risk management of cable structures with respect to icing events, which may lead to safety issues related to human life, functional disruptions and the associated economic consequences. Emphasis is placed on the value of early warning of icing events, which is based on the monitoring of environmental conditions and short-term forecasting. Decision problems in risk management can be supported by quantifying the value of structural health monitoring (SHM). The approach for the assessment of the value of SHM is based on structural risk assessments together with the Bayesian pre-posterior decision analysis and builds upon the quantification of Value of Information (Vol). Using known probabilistic models, which relate environmental conditions to the events of icing in terms of occurrence, a framework developed earlier is presented for the assessment of the expected value of consequences associated with cable icing events. The consequences are evaluated for different outputs of the probabilistic model to provide a basis for prioritizing risk management decision alternatives. It is shown how forecasting of the environmental characteristics, that is the input to a Bayesian Probabilistic Network model, can be used to update the probability tables of the BPN model and thereby facilitate updating of the ice occurrence probability, which can serve as a decision support tool for the risk management of a cable-supported bridge. The approach is illustrated by an example considering the safety management of a cable-supported bridge subject to the risk of falling ice from the cables. The example shows how the expected value of SHM and forecasting can be assessed, and a sensitivity study concerning the assumptions underlying the consequence modelling is reported.

Physical Experimental Design in Support of Computer Model Development

Computer models of physical systems are often written based on known theory or “first principles” of a system, reflecting substantial knowledge of each component or subsystem, but also the need to use a numerical approach to mimic the more complex behavior of the entire system of interest. However, in some cases, there is insufficient known theory to encode all necessary aspects of the system, and empirical studies are required to generate approximate functional forms. We consider the question of how a physical experiment might be designed to approximate one module or subroutine of a computer model that can otherwise be written from first principles. The concept of preposterior analysis is used to suggest an approach to generating a kind of I-optimal design for this purpose, when the remainder of the computer model is a composition of nonlinear functions that can be directly evaluated as part of the design process. Extensions are then described for situations in which one or more known components must themselves be approximated by metamodels due to the large number of evaluations needed, and for computer models that have iterative structure. A simple “toy” model is used to demonstrate the ideas. Online supplementary material accompanies this article.

SURROGATE PREPOSTERIOR ANALYSES FOR PREDICTING AND ENHANCING IDENTIFIABILITY IN MODEL CALIBRATION

In physics-based engineering modeling and uncertainty quantification, distinguishing the effects of two main sources of uncertainty — calibration parameter uncertainty and model discrepancy — is challenging. Previous research has shown that identifiability, which is quantified by the posterior covariance of the calibration parameters, can sometimes be improved by experimentally measuring multiple responses of the system that share a mutual dependence on a common set of calibration parameters. In this paper, we address the issue of how to select the most appropriate subset of responses to measure experimentally, to best enhance identifiability. We use a preposterior analysis approach that, prior to conducting physical experiments but after conducting computer simulations, can predict the degree of identifiability that will result using different subsets of responses to measure experimentally. It predicts identifiability via the preposterior covariance from a modular Bayesian Monte Carlo analysis of a multi-response spatial random process (SRP) model. Furthermore, to handle the computational challenge in preposterior analysis, we propose a surrogate preposterior analysis based on Fisher information of the calibration parameters. The proposed methods are applied to a simply supported beam example to select two out of six responses to best improve identifiability. The estimated preposterior covariance is compared to the actual posterior covariance to demonstrate the effectiveness of the methods.

Observation Value Analysis – Integral Part of Bayesian Diagnostics

The decision making process, in general, is understood as a process of selecting one of the available solutions to the problem. One of possible approaches supporting the process is Bayesian statistical decision theory providing a mathematical model to make decisions of a technical nature in conditions of uncertainty. Regarding above, a detailed subject of the research is to analyze the value of the observation, which is a part of preposterior analysis. For the presented network, the main objective was to determine, conducting of which of three tests is the most valuable from the perspective of determining possible need or possibility to omission expensive technical expertise. The basis of verification, which test is the most valuable, is therefore the comparison of expected value of sample information (EVSI) for each of the three tests.The main advantage of preposterior analysis is answering the question which of the considered experiments is potentially the best source of information and what cost needs to be incurred on its execution (price of new information). If the cost of such an examination is relatively small compared to the value of information on the state of nature, this implies a direct recommendation to conduct the experiment.In conclusion, the construction of the decision-making model being a reflection of the state of nature allows to determine the ranking of decisions, including those regarding selection of the optimal test. It is noteworthy, that test result itself can contribute to an increase in the expected value of the decision involved, but on the other hand, the reduction of uncertainty may be considerably outweighed to the necessity to incur expenses for this examination.

Pre-posterior optimization of sequence of measurement and intervention actions under structural reliability constraint

It is common to assess the condition of an existing infrastructure using reliability analysis. When, based on the available information, an existing structure has an estimated failure probability above the admissible level, the default solution often is to either strengthen or replace it. Even if this practice is safe, it may not be the most economical. In order to economically restore and improve our existing infrastructure, the engineering community needs to be able to assess the potential gains associated with reducing epistemic uncertainties using measurements, before opting for costly intervention actions, if they become necessary. This paper provides a pre-posterior analysis framework to (1) optimize sequences of actions minimizing the expected costs and satisfying reliability constraints and (2) quantify the potential gain of making measurements in existing structures. Illustrative examples show that when the failure probability estimated based on the present state of knowledge does not satisfy an admissible threshold, strengthening or replacement interventions can be sub-optimal first actions. The examples also show that significant savings can be achieved by reducing epistemic uncertainties.

Reliability updating and decision analysis for head monitoring of levees

Flood defence levees are crucial for flood protection, especially in riverine and deltaic areas. One of the most critical failure modes of levees is backwards internal erosion or piping. Piping reliability is largely influenced by uncertainties in ground conditions, especially by geohydrological properties such as the hydraulic conductivity of an aquifer. These properties are typically highly uncertain, which can lead to the related probabilities of failure being considerably high. Ways to reduce such uncertainties are by performing additional site investigation, incorporating information from field performance observations or monitoring. After providing some background on uplift and piping, the paper shows how Bayesian posterior analysis can be done using head monitoring observations in aquifers during extreme floods. The approach will be demonstrated by means of an illustrative example using a simplified uplift model showing also how the posterior distributions of the basic random variables change with respect to their priors. Subsequently, it will be illustrated how uncertain future measurements can be used in pre-posterior analysis to support decisions on investments in monitoring. The results suggest that the uncertainties in geohydrological conditions can be reduced significantly and that, if the monitoring cost is low compared to potential retrofitting measures, monitoring can be cost-effective.

Bayesian approach to contaminant source characterization in water distribution systems: adaptive sampling framework

Bayesian analysis can yield a probabilistic contaminant source characterization conditioned on available sensor data and accounting for system stochastic processes. This paper is based on a previously proposed Markov chain Monte Carlo (MCMC) approach tailored for water distribution systems and incorporating stochastic water demands. The observations can include those from fixed sensors and, the focus of this paper, mobile sensors. Decision makers, such as utility managers, need not wait until new observations are available from an existing sparse network of fixed sensors. This paper addresses a key research question: where is the best location in the network to gather additional measurements so as to maximize the reduction in the source uncertainty? Although this has been done in groundwater management, it has not been well addressed in water distribution networks. In this study, an adaptive framework is proposed to guide the strategic placement of mobile sensors to complement the fixed sensor network. MCMC is the core component of the proposed adaptive framework, while several other pieces are indispensable: Bayesian preposterior analysis, value of information criterion and the search strategy for identifying an optimal location. Such a framework is demonstrated with an illustrative example, where four candidate sampling locations in the small water distribution network are investigated. Use of different value-of-information criteria reveals that while each may lead to different outcomes, they share some common characteristics. The results demonstrate the potential of Bayesian analysis and the MCMC method for contaminant event management.

Statistical Evaluation of Toxicological Experimental Design for Bayesian Model Averaged Benchmark Dose Estimation with Dichotomous Data

ABSTRACT A method is presented to statistically evaluate toxicity study design for dose– response assessment aimed at minimizing the uncertainty in resulting Benchmark dose (BMD) estimates. Although the BMD method has been accepted as a valuable tool for risk assessment, the traditional no observed adverse effect level (NOAEL)/lowest observed adverse effective level (LOAEL) approach is still the principal basis for toxicological study design. To develop similar protocols for experimental design for BMD estimation, methods are needed that account for variability in experimental outcomes, and uncertainty in dose–response model selection and model parameter estimates. Based on Bayesian model averaging (BMA) BMD estimation, this study focuses on identifying the study design criteria that can reduce the uncertainty in BMA BMD estimates by using a Monte Carlo pre-posterior analysis on BMA BMD predictions. The results suggest that (1) as more animals are tested there is less uncertainty in BMD estimates; (2) one relatively high dose is needed and other doses can then be appropriately spread over the resulting dose scale; (3) placing different numbers of animals in different dose groups has very limited influence on improving BMD estimation; and (4) when the total number of animals is fixed, using more (but smaller) dose groups is a preferred strategy.

Optimal control of a dynamic system with multiple uncertainty in the initial state as based on imperfect measurements of input and output signals

An optimal control problem for a linear nonstationary dynamic system with uncertainty in the initial state is considered. Based on preposterior analysis, an algorithm for implementing an optimal closable output loop is constructed. The control results are illustrated using a numerical example and are compared with those based on two-stage disclosable output loop.

Electronic price-testing scheme for fashion retailing with information updating

Abstract Pricing is a crucial decision for electronic fashion retailers. Motivated by various observed industrial practices in electronic retailing, we study in this paper the optimal Internet pricing schemes which employ price testing with Bayesian information updating following the Bernoulli process. This paper contributes to the literature and advancement of knowledge in a number of ways: (i) we propose an analytical model to study the Internet pricing problem with price-testing and Bayesian information updating for fashion retailers. (ii) We derive the closed-form expressions of the expected value of sampling information (EVSI) and the expected value of perfect information (EVPI) under the price testing scheme. (iii) We conduct the pre-posterior analysis and construct the optimal sampling plan with three different rules. (iv) We develop the optimal posterior pricing policies, with respect to the mean-risk and Value-at-Risk (VaR) objectives. Numerical analyses, which include the studies on EVPI and the efficient frontiers, are presented to generate more insights.

A Bayesian Reliability Approach to Multiple Response Optimization with Seemingly Unrelated Regression Models

This paper presents a Bayesian predictive approach to multiresponse optimization experiments. It generalizes the work of Peterson [33] in two ways that make it more flexible for use in applications. First, a multivariate posterior predictive distribution of seemingly unrelated regression models is used to determine optimum factor levels by assessing the reliability of a desired multivariate response. It is shown that it is possible for optimal mean response surfaces to appear satisfactory yet be associated with unsatisfactory overall process reliabilities. Second, the use of a multivariate normal distribution for the vector of regression error terms is generalized to that of the (heavier tailed) multivariate t-distribution. This provides a Bayesian sensitivity analysis with regard to moderate outliers. The effect of adding design points is also considered through a preposterior analysis. The advantages of this approach are illustrated with two real examples.

Data-worth analysis for multiobjective optimal design of pump-and-treat remediation systems

The design and the management of pump-and-treat (PAT) remediation systems for contaminated aquifers under uncertain hydrogeological settings and parameters often involve decisions that trade off cost optimality against reliability. Both design objectives can be improved by planning site characterization programs that reduce subsurface parameter uncertainty. However, the cost for subsurface investigation often weighs heavily upon the budget of the remedial action and must thus be taken into account in the trade-off analysis. In this paper, we develop a stochastic data-worth framework with the purpose of estimating the economic opportunity of subsurface investigation programs. Since the spatial distribution of hydraulic conductivity is most often the major source of uncertainty, we focus on the direct sampling of hydraulic conductivity at prescribed locations of the aquifer. The data worth of hydraulic conductivity measurements is estimated from the reduction of the overall management cost ensuing from the reduction in parameter uncertainty obtained from sampling. The overall cost is estimated as the expected value of the cost of installing and operating the PAT system plus penalties incurred due to violations of cleanup goals and constraints. The crucial point of the data-worth framework is represented by the so-called pre-posterior analysis. Here, the tradeoff between decreasing overall costs and increasing site-investigation budgets is assessed to determine a management strategy proposed on the basis of the information available at the start of remediation. The goal of the pre-posterior analysis is to indicate whether the proposed management strategy should be implemented as is, or re-designed on the basis of additional data collected with a particular site-investigation program. The study indicates that the value of information is ultimately related to the estimates of cleanup target violations and decision makers’ degree of risk-aversion.

Bayesian portfolio selection with multi-variate random variance models

We consider multi-period portfolio selection problems for a decision maker with a specified utility function when the variance of security returns is described by a discrete time stochastic model. The solution of these problems involves a dynamic programming formulation and backward induction. We present a simulation-based method to solve these problems adopting an approach which replaces the preposterior analysis by a surface fitting based optimization approach. We provide examples to illustrate the implementation of our approach.

A Bayesian approach to randomized controlled trials in children utilizing information from adults: the case of Guillain-Barre

Guillain-Barré syndrome (GBS) is a rare neurologic disease that occurs at all ages, causing a progressive, ascending paralysis that usually resolves over weeks or months. The disease appears to be identical in children and adults, except that children recover more quickly, with fewer residua. For patients who lose the ability to walk independently, the main treatment options are plasmapheresis or intravenous immune globulin (IVIg), treatments that have shown to have identical effectiveness in adults in two large RCTs involving 388 patients. The effectiveness of the treatments in children has only been studied in small, poorly controlled studies. If one could capture all eligible patients in the United States, only about 100-300 children would be available for a trial annually. The goal of this case was to demonstrate how Bayesian methods could be used to incorporate prior information on treatment efficacy from adults to design a randomized noninferiority trial of IVIg versus plasmapheresis in children. A Bayesian normal-normal model on the hazard ratio of time to independent walking was implemented. An evidence-based prior was constructed that was equivalent to 72 children showing exact equivalence between the therapies. A design was constructed based on a Bayesian normal-normal model on the hazard ratio, yielding a sample size of 160 children, with a preposterior analysis demonstrating a "Type I" error rate of 5% and a power of 77%. This case study illustrates a rational approach to constructing an evidence-based prior that would allow information from adults to formally augment data from children to minimize unnecessary pediatric experimentation. The frequentist properties of a Bayesian design can be evaluated and reported as they would be for a standard design. Discussion of the appropriate prior for such designs is both a necessary and desirable feature of Bayesian trials.

Framework to evaluate the worth of hydraulic conductivity data for optimal groundwater resources management in ecologically sensitive areas

We propose a framework that combines simulation optimization with Bayesian decision analysis to evaluate the worth of hydraulic conductivity data for optimal groundwater resources management in ecologically sensitive areas. A stochastic simulation optimization management model is employed to plan regionally distributed groundwater pumping while preserving the hydroecological balance in wetland areas. Because predictions made by an aquifer model are uncertain, groundwater supply systems operate below maximum yield. Collecting data from the groundwater system can potentially reduce predictive uncertainty and increase safe water production. The price paid for improvement in water management is the cost of collecting the additional data. Efficient data collection using Bayesian decision analysis proceeds in three stages: (1) The prior analysis determines the optimal pumping scheme and profit from water sales on the basis of known information. (2) The preposterior analysis estimates the optimal measurement locations and evaluates whether each sequential measurement will be cost‐effective before it is taken. (3) The posterior analysis then revises the prior optimal pumping scheme and consequent profit, given the new information. Stochastic simulation optimization employing a multiple‐realization approach is used to determine the optimal pumping scheme in each of the three stages. The cost of new data must not exceed the expected increase in benefit obtained in optimal groundwater exploitation. An example based on groundwater management practices in Florida aimed at wetland protection showed that the cost of data collection more than paid for itself by enabling a safe and reliable increase in production.

Preposterior analysis for option pricing

The second partial derivative of a European-style vanilla option with respect to the current price of the underlying asset—the option gamma—defines a probability density function for the current underlying price. By use of entropy maximization it is possible to obtain an option gamma, from which the associated option pricing formula can be recovered by integration. A number of pricing formulae are obtained in this manner, corresponding to different specifications of the constraints. When the available market information consists solely of a set of traded option prices, the entropic formulation leads to a model-independent calibration procedure. The result thus obtained also allows one to recover the relevant Greeks.

Simulation-based designs for accelerated life tests

Abstract In this paper we present a Bayesian decision theoretic approach to the design of accelerated life tests (ALT). We discuss computational issues regarding the evaluation of expectation and optimization steps in the solution of the decision problem. We illustrate how Monte Carlo methods can be used in preposterior analysis to find optimal designs and how the required computational effort can be avoided by using curve-fitting techniques. In so doing, we adopt the recent Monte-Carlo-based approaches of Muller and Parmigiani (1995. J. Amer. Statist. Assoc. 90, 503–510) and Muller (2000. Bayesian Statistics 6, forthcoming) to develop optimal Bayesian designs. These approaches facilitate the preposterior analysis by replacing it with a sequence of scatter plot smoothing/regression techniques and optimization of the corresponding fitted surfaces. We present our development by considering single and multiple-point fixed, as well as, sequential design problems when the underlying life model is exponential, and illustrate the implementation of our approach with some examples.

Hypothesis testing as risk behaviour with regard to beliefs

In this paper hypothesis-testing behaviour is compared to risk-taking behaviour. It is proposed that choosing a suitable test for a given hypothesis requires making a preposterior analysis of two aspects of such a test: the probability of obtaining supporting evidence and the evidential value of this evidence. This consideration resembles the one a gambler makes when choosing among bets, each having a probability of winning and an amount to be won. A confirmatory testing strategy can be defined within this framework as a strategy directed at maximizing either the probability or the value of a confirming outcome. Previous theories on testing behaviour have focused on the human tendency to maximize the probability of a confirming outcome. In this paper, two experiments are presented in which participants tend to maximize the confirming value of the test outcome. Motivational factors enhance this tendency dependent on the context of the testing situation. Both this result and the framework are discussed in relation to other studies in the field of testing behaviour. Copyright © 2000 John Wiley & Sons, Ltd.

Optimal Bayesian two-phase designs

In this paper we present a Bayesian decision theoretic approach to the two-phase design problem. The solution of such sequential decision problems is usually difficult to obtain because of their reliance on preposterior analysis. In overcoming this problem, we adopt the Mont-Carlo-based approach of Müller and Parmigiani (1995) and develop optimal Bayesian designs for two-phase screening tests. A rather attractive feature of the Monte-Carlo approach is that it facilitates the preposterior analysis by replacing it with a sequence of scatter plot smoothing/regression techniques and optimization of the corresponding fitted surfaces. The method is illustrated for depression in adolescents using data from past studies.

On the Efficacy of Bayesian Inference for Nonidentifiable Models

Although classical statistical methods are inapplicable in point estimation problems involving nonidentifiable parameters, a Bayesian analysis using proper priors can produce a closed form, interpretable point estimate in such problems. The question of whether, and when, the Bayesian approach produces worthwhile answers is investigated. In contrast to the preposterior analysis of this question offered by Kadane, we examine the question conditionally, given the information provided by the experiment. An important initial insight on the matter is that posterior estimates of a nonidentifiable parameter can actually be inferior to the prior (no-data) estimate of that parameter, even as the sample size grows to infinity. In general, our goal is to characterize, within the space of prior distributions, classes of priors that lead to posterior estimates that are superior, in some reasonable sense, to one's prior estimate. This goal is shown to be feasible through a detailed examination of a particular two-parameter Binomial model. Our results support the proposition that a Bayesian analysis tends to be efficacious in the estimation problem studied, and suggest that Bayesian updating can produce useful answers in the presence of nonidentifiability.