Uncertainty about Probability: A Decision Analysis Perspective

The issue of how to think about "uncertainty about probability" is framed and analyzed from the viewpoint of a decision analyst. The failure of nuclear power plants is used as an example. The key idea is to think of probability as describing a state of information on an uncertain event, and to pose the issue of uncertainty in this quantity as uncertainty about a number that would be definitive: it has the property that you would assign it as the probability if you knew it. Logical consistency requires that the probability to assign to a single occurrence in the absence of further information be the mean of the distribution of this definitive number, not the median as is sometimes suggested. Any decision that must be made without the benefit of further information must also be made using the mean of the definitive number's distribution. With this formulation, we find further that the probability of r occurrences in n exchangeable trials will depend on the first n moments of the definitive number's distribution. In making decisions, the expected value of clairvoyance on the occurrence of the event must be at least as great as that on the definitive number. If one of the events in question occurs, then the increase in probability of another such event is readily computed. This means, in terms of coin tossing, that unless one is absolutely sure of the fairness of a coin, seeing a head must increase the probability of heads, in distinction to usual thought. A numerical example for nuclear power shows that the failure of one plant of a group with a low probability of failure can significantly increase the probability that must be assigned to failure of a second plant in the group.     

Probability and Possibility‐Based Representations of Uncertainty in Fault Tree Analysis

Expert knowledge is an important source of input to risk analysis. In practice, experts might be reluctant to characterize their knowledge and the related (epistemic) uncertainty using precise probabilities. The theory of possibility allows for imprecision in probability assignments. The associated possibilistic representation of epistemic uncertainty can be combined with, and transformed into, a probabilistic representation; in this article, we show this with reference to a simple fault tree analysis. We apply an integrated (hybrid) probabilistic‐possibilistic computational framework for the joint propagation of the epistemic uncertainty on the values of the (limiting relative frequency) probabilities of the basic events of the fault tree, and we use possibility‐probability (probability‐possibility) transformations for propagating the epistemic uncertainty within purely probabilistic and possibilistic settings. The results of the different approaches (hybrid, probabilistic, and possibilistic) are compared with respect to the representation of uncertainty about the top event (limiting relative frequency) probability. Both the rationale underpinning the approaches and the computational efforts they require are critically examined. We conclude that the approaches relevant in a given setting depend on the purpose of the risk analysis, and that further research is required to make the possibilistic approaches operational in a risk analysis context.     

Mitigation and Benefits Measures as Policy Tools for Siting Potentially Hazardous Facilities: Determinants of Effectiveness and Appropriateness

How do mitigation and benefits measures affect public acceptance for siting different kinds of potentially hazardous facilities? What kinds of benefits measures are seen as most (or least) appropriate for different kinds of facilities? This study used a nationwide telephone survey consisting of 1,234 interviews with randomly selected respondents to test for the effects of packages of safety and benefits measures for siting a landfill, prison, incinerator and nuclear waste repository. The experimental design used in the survey permits analysis of the fractions of respondents who are willing to change their initial levels of acceptance (or opposition) when presented with a sequence of the safety and benefit measures. The measures vary significantly in their impact on levels of acceptance for the facilities, and some measures that would at face value appear to reassure residents of facility safety turn out to lack credibility and therefore diminish facility acceptance. Ordering of the benefits versus safety measures significantly affects changes in acceptance in surprising ways. The perceived appropriateness of different kinds of benefits measures varies systematically by the type of facility under consideration. It appears that successful benefits packages will directly address the underlying dimensions of concern caused by the facility. These findings point to the importance of further research on “commensurable” benefits measures.     

When Does Information about Probability Count in Choices Under Risk?

The question addressed in the present research is whether in naturalistic risky decision environments people are sensitive to information about the probability parameter. In Study 1, we showed that in naturalistic scenarios participants generally revealed little interest in obtaining information about the outcomes and probabilities. Moreover, participants asked fewer questions about probabilities for scenarios containing moral considerations. In Study 2, it was shown that, when supplied with information on probabilities, people could be sensitive to this information. This sensitivity depends on two factors. People were less sensitive to probabilities in scenarios perceived as containing ethical considerations. People were also less sensitive to probabilities when they were faced with a single-choice situation than when they were faced with a series of lotteries with different probabilities. This can be accounted for in terms of the evaluability principle.     

Ignorance Is Not Probability

The distinction between ignorance about a parameter and knowing only a probability distribution for that parameter is of fundamental importance in risk assessment. Brief dialogs between a hypothetical decisionmaker and a risk assessor illustrate this point, showing that the distinction has real consequences. These dialogs are followed by a short exposition that places risk analysis in a decision‐theoretic framework, describes the important elements of that framework, and uses these to shed light on Terje Aven's criticism of nonprobabilistic purely “objective” methods. Suggestions are offered concerning a more effective approach to evaluating those methods.     

On The Method of Discrete Probability Distributions in Risk and Reliability Calculations–Application to Seismic Risk Assessment

If the point of view is adopted that in calculations of real-world phenomena we almost invariably have significant uncertainty in the numerical values of our parameters, then, in these calculations, numerical quantities should be replaced by probability distributions and mathematical operations between these quantities should be replaced by analogous operations between probability distributions. Also, practical calculations one way or another always require discretization or truncation. Combining these two thoughts leads to a numerical approach to probabilistic calculations having great simplicity, power, and elegance. The philosophy and technique of this approach is described, some pitfalls are pointed out, and an application to seismic risk assessment is outlined.     

信用评分模型的设计与决策分析

本文对商业银行信贷业务中的贷款者的信用问题,运用决策分析提出了信用评分的设计原理及临界分值的确定方法,给出了最优贷款策略的期望收益和风险的计算公式。最后,以一例说明之。     

Probability Elicitation Under Severe Time Pressure: A Rank‐Based Method

Probability elicitation protocols are used to assess and incorporate subjective probabilities in risk and decision analysis. While most of these protocols use methods that have focused on the precision of the elicited probabilities, the speed of the elicitation process has often been neglected. However, speed is also important, particularly when experts need to examine a large number of events on a recurrent basis. Furthermore, most existing elicitation methods are numerical in nature, but there are various reasons why an expert would refuse to give such precise ratio‐scale estimates, even if highly numerate. This may occur, for instance, when there is lack of sufficient hard evidence, when assessing very uncertain events (such as emergent threats), or when dealing with politicized topics (such as terrorism or disease outbreaks). In this article, we adopt an ordinal ranking approach from multicriteria decision analysis to provide a fast and nonnumerical probability elicitation process. Probabilities are subsequently approximated from the ranking by an algorithm based on the principle of maximum entropy, a rule compatible with the ordinal information provided by the expert. The method can elicit probabilities for a wide range of different event types, including new ways of eliciting probabilities for stochastically independent events and low‐probability events. We use a Monte Carlo simulation to test the accuracy of the approximated probabilities and try the method in practice, applying it to a real‐world risk analysis recently conducted for DEFRA (the U.K. Department for the Environment, Farming and Rural Affairs): the prioritization of animal health threats.     

Probability Information in Risk Communication: A Review of the Research Literature

Communicating probability information about risks to the public is more difficult than might be expected. Many studies have examined this subject, so that their resulting recommendations are scattered over various publications, diverse research fields, and are about different presentation formats. An integration of empirical findings in one review would be useful therefore to describe the evidence base for communication about probability information and to present the recommendations that can be made so far. We categorized the studies in the following presentation formats: frequencies, percentages, base rates and proportions, absolute and relative risk reduction, cumulative probabilities, verbal probability information, numerical versus verbal probability information, graphs, and risk ladders. We suggest several recommendations for these formats. Based on the results of our review, we show that the effects of presentation format depend not only on the type of format, but also on the context in which the format is used. We therefore argue that the presentation format has the strongest effect when the receiver processes probability information heuristically instead of systematically. We conclude that future research and risk communication practitioners should not only concentrate on the presentation format of the probability information but also on the situation in which this message is presented, as this may predict how people process the information and how this may influence their interpretation of the risk.     

最大化生存概率的投资策略

本文研究最大化生存概率准则下的最优投资问题。假设投资者面临着不可对冲的随机风险,市场是不完备的,任何投资策略都不能完全消除财富总量的下行风险。本文主要结果是:假设无风险资产利率大于零,分别研究了无借贷约束和有借贷约束条件下基于最大化生存概率准则的最优投资问题,运用随机控制理论,通过求解HJB方程,获得了最优投资策略及相应最大生存概率的闭式解,给出了数值算例,通过比较静态分析揭示了生存概率和投资策略与各参数之间的数量关系。结果表明,风险资产最优投资比例随财富总量的增加而减少,企业的生存概率随财富总量的增加而增加,随法定水平的增加而减少。     

顾客购买行为影响因素分析及重购概率的预测

顾客购买行为是市场营销研究的重要内容之一。目前已有一些文献通过引入几个属性变量,构建起描述顾客购买行为或预测购买概率的随机模型。本文在前人研究的基础上,首先分析了对顾客购买行为影响较大的顾客购买决策、前后两次购买间隔时间、顾客重购行为和顾客逃逸等因素,而后将这些因素综合考虑,给出预测日用消费品重购概率的一种方法,并用实际数据对预测方法进行检验,预测的结果与实际数据非常接近。本方法的特点一是预测精度较高,二是具有一般性,适用于一般的日用消费品。     

Direct Inference, Probability, and a Conceptual Gulf in Risk Communication

Differences in the conceptual frameworks of scientists and nonscientists may create barriers to risk communication. This article examines two such conceptual problems. First, the logic of "direct inference" from group statistics to probabilities about specific individuals suggests that individuals might be acting rationally in refusing to apply to themselves the conclusions of regulatory risk assessments. Second, while regulators and risk assessment scientists often use an "objectivist" or "relative frequency" interpretation of probability statements, members of the public are more likely to adopt a "subjectivist" or "degree of confidence" interpretation when estimating their personal risks, and either misunderstand or significantly discount the relevance of risk assessment conclusions. If these analyses of inference and probability are correct, there may be a conceptual gulf at the center of risk communication that cannot be bridged by additional data about the magnitude of group risk. Suggestions are made for empirical studies that might help regulators deal with this conceptual gulf.     

Modeling Correlated Discrete Uncertainties in Event Trees with Copulas

Modeling the dependence between uncertainties in decision and risk analyses is an important part of the problem structuring process. We focus on situations where correlated uncertainties are discrete, and extend the concept of the copula‐based approach for modeling correlated continuous uncertainties to the representation of correlated discrete uncertainties. This approach reduces the required number of probability assessments significantly compared to approaches requiring direct estimates of conditional probabilities. It also allows the use of multiple dependence measures, including product moment correlation, rank order correlation and tail dependence, and parametric families of copulas such as normal copulas, t‐copulas, and Archimedean copulas. This approach can be extended to model the dependence between discrete and continuous uncertainties in the same event tree.     

Risk Preferences,Probability Weighting,and Strategy Tradeoffs in Wildfire Management

Wildfires present a complex applied risk management environment, but relatively little attention has been paid to behavioral and cognitive responses to risk among public agency wildfire managers. This study investigates responses to risk, including probability weighting and risk aversion, in a wildfire management context using a survey‐based experiment administered to federal wildfire managers. Respondents were presented with a multiattribute lottery‐choice experiment where each lottery is defined by three outcome attributes: expenditures for fire suppression, damage to private property, and exposure of firefighters to the risk of aviation‐related fatalities. Respondents choose one of two strategies, each of which includes “good” (low cost/low damage) and “bad” (high cost/high damage) outcomes that occur with varying probabilities. The choice task also incorporates an information framing experiment to test whether information about fatality risk to firefighters alters managers' responses to risk. Results suggest that managers exhibit risk aversion and nonlinear probability weighting, which can result in choices that do not minimize expected expenditures, property damage, or firefighter exposure. Information framing tends to result in choices that reduce the risk of aviation fatalities, but exacerbates nonlinear probability weighting.     

Development of a Standard Soil-to-Skin Adherence Probability Density Function for Use in Monte Carlo Analyses of Dermal Exposure

It has recently been suggested that "standard" data distributions for key exposure variables should be developed wherever appropriate for use in probabilistic or "Monte Carlo" exposure analyses. Soil-on-skin adherence estimates represent an ideal candidate for development of a standard data distribution: There are several readily available studies which offer a consistent pattern of reported results, and more importantly, soil adherence to skin is likely to vary little from site-to-site. In this paper, we thoroughly review each of the published soil adherence studies with respect to study design, sampling, and analytical methods, and level of confidence in the reported results. Based on these studies, probability density functions (PDF) of soil adherence values were examined for different age groups and different sampling techniques. The soil adherence PDF developed from adult data was found to resemble closely the soil adherence PDF based on child data in terms of both central tendency (mean = 0.49 and 0.63 mg-soil/cm²-skin, respectively) and 95th percentile values (1.6 and 2.4 mg-soil/cm²-skin, respectively). Accordingly, a single, "standard" PDF is presented based on all data collected for all age groups. This standard PDF is lognormally distributed; the arithmetic mean and standard deviation are 0.52 ± 0.9 mg-soil/cm²-skin. Since our review of the literature indicates that soil adherence under environmental conditions will be minimally influenced by age, sex, soil type, or particle size, this PDF should be considered applicable to all settings. The 50th and 95th percentile values of the standard PDF (0.25 and 1.7 mg-soil/cm²-skin, respectively) are very similar to recent U.S. EPA estimates of "average" and "upper-bound" soil adherence (0.2 and 1.0 mg-soil/cm²-skin, respectively).     

Communicating Low Risk Magnitudes: Incidence Rates Expressed as Frequency Versus Rates Expressed as Probability

The study investigated the effects of incidence rates stated as a probability (e.g., 0006) and incidence rate information expressed in terms of frequency (e.g., 600 in 1,000,000) on risk-avoidant behavior. Subjects were informed about the risks associated with an old and a new, improved medication. They were asked how much they were willing to pay for the safer medicine. Risk information was given either in a frequency or a probability format. The second factor manipulated was the level of risk, either high or low. As expected, analysis of variance yielded a significant interaction. Subjects confronted with high risk in the frequency format were willing to pay the highest prices for the improved medication. The choice between frequency or probability format can be made according to the goal of the communication of risk.     

基于条件破产概率的保险公司财务预警与资本分配

保险公司需要在可能出现的破产事件之前建立一种适当的预警系统。为优化资本效率,本文提出了保险公司财务预警系统理念并给出了预警时刻定义;针对离散型和连续性损失程度分布,数值模拟了保险公司破产分布密度和预警发出时刻。对不同的初始资本和不同的资本补充方案,计算了多重预警时刻的变动状况,最后提出了预警系统的调整方向以及保险公司资本补充的内部分配原则。本文成果有助于保险公司提高资本效率,构建资金分散化的适当策略以提高生存能力。     

On The Quantitative Definition of Risk

A quantitative definition of risk is suggested in terms of the idea of a "set of triplets". The definition is extended to include uncertainty and completeness, and the use of Bayes' theorem is described in this connection. The definition is used to discuss the notions of "relative risk", "relativity of risk", and "acceptability of risk".