Second-order Probabilities Research Articles

Confidence in Probabilistic Risk Assessment

Abstract Epistemic uncertainties are included in probabilistic risk assessment (PRA) as second-order probabilities that represent the degrees of belief of the scientists that a model is correct. In this article, I propose an alternative approach that incorporates the scientist’s confidence in a probability set for a given quantity. First, I give some arguments against the use of precise probabilities to estimate scientific uncertainty in risk analysis. I then extend the “confidence approach” developed by Brian Hill and Richard Bradley to PRA. Finally, I claim that this approach represents model uncertainty better than the standard (Bayesian) model does.

Monetary policy in a world of radical uncertainty

Monetary policy in a world of radical uncertainty

Ranked set sampling in finite populations with bivariate responses: An application to an osteoporosis study.

The majority of the research on rank-based sampling designs in finite populations has been concerned with univariate situations. In this article, we study design-based estimation using a bivariate ranked set sampling (BIRSS) for finite populations when we have bivariate response variables. We derive the first and second-order inclusion probabilities associated with a BIRSS design. We show that the size of a BIRSS sample is random and propose using a conditional Poisson sampling (CPS) design to rectify this problem. We then use calculated inclusion probabilities to obtain design-based estimators of correlation coefficients between the bone mineral density (BMD) levels at the baseline and followup of a longitudinal BMD study in the province of Manitoba in Canada. We also study the problem of estimating the parameters of a regression model between the followup BMD and easy to obtain auxiliary information from the underlying population. Finally, we study the problem of classifying patients as those with or without osteoporosis using BIRSS and various CPS designs. We show that BIRSS designs are very flexible and can be used to obtain more efficient design-based estimators in sample surveys when dealing with response variables that are hard to measure or expensive to obtain.

Non-Bayesian Social Learning With Uncertain Models

Non-Bayesian social learning theory provides a framework that models distributed inference for a group of agents interacting over a social network. In this framework, each agent iteratively forms and communicates beliefs about an unknown state of the world with their neighbors using a learning rule. Existing approaches assume agents have access to precise statistical models (in the form of likelihoods) for the state of the world. However in many situations, such models must be learned from finite data. We propose a social learning rule that takes into account uncertainty in the statistical models using second-order probabilities. Therefore, beliefs derived from uncertain models are sensitive to the amount of past evidence collected for each hypothesis. We characterize how well the hypotheses can be tested on a social network, as consistent or not with the state of the world. We explicitly show the dependency of the generated beliefs with respect to the amount of prior evidence. Moreover, as the amount of prior evidence goes to infinity, learning occurs and is consistent with traditional social learning theory.

Tracking the implicit acquisition of nonadjacent transitional probabilities by ERPs

The implicit acquisition of complex probabilistic regularities has been found to be crucial in numerous automatized cognitive abilities, including language processing and associative learning. However, it has not been completely elucidated how the implicit extraction of second-order nonadjacent transitional probabilities is reflected by neurophysiological processes. Therefore, this study investigated the sensitivity of event-related brain potentials (ERPs) to these probabilistic regularities embedded in a sequence of visual stimuli without providing explicit information on the structure of the stimulus stream. Healthy young adults (N = 32) performed a four-choice RT task that included a sequential regularity between nonadjacent trials yielding a complex transitional probability structure. ERPs were measured relative to both stimulus and response onset. RTs indicated the rapid acquisition of the sequential regularity and the transitional probabilities. The acquisition process was also tracked by the stimulus-locked and response-locked P3 component: The P3 peak was larger for the sequence than for the random stimuli, while the late P3 was larger for less probable than for more probable short-range relations among the random stimuli. According to the RT and P3 effects, sensitivity to the sequential regularity is assumed to be supported by the initial sensitivity to the transitional probabilities. These results suggest that stimulus–response contingencies on the probabilistic regularities of the ongoing stimulus context are implicitly mapped and constantly revised. Overall, this study (1) highlights the role of predictive processes during implicit memory formation, and (2) delineates a potential to gain further insight into the dynamics of implicit acquisition processes.

A deep forest classifier with weights of class probability distribution subsets

A modification of the Deep Forest or gcForest proposed by Zhou and Feng for solving classification problems is proposed in the paper and called as PM-DF. The main idea for improving classification performance of the Deep Forest is to assign weights to subsets of the class probability distributions at the leaf nodes computed for every training example. The subsets of probability distributions are defined by using Walley’s imprecise pari-mutuel model which compactly divides the unit simplex of probabilities into subsets and allows us to simplify the algorithm of the weight calculation. The weights of the distribution subsets can be viewed in this case as second-order probabilities over subsets of the probability simplex. The optimal weights are computed by solving the standard quadratic optimization problem. The numerical experiments illustrate PM-DF.

On selecting a sample by probability proportional to size with second-order inclusion probabilities and without replacement

Given appropriate sets of first- and second-order inclusion probabilities, the author provides a method that results in samples including units and pairs of units of the universe with the probabilities specified in advance.

Eliciting ambiguity aversion in unknown and in compound lotteries: a smooth ambiguity model experimental study

Coherent-ambiguity aversion is defined within the (Klibanoff et al., Econometrica 73:1849–1892, 2005) smooth-ambiguity model (henceforth KMM) as the combination of choice-ambiguity and value-ambiguity aversion. Five ambiguous decision tasks are analyzed theoretically, where an individual faces two-stage lotteries with binomial, uniform, or unknown second-order probabilities. Theoretical predictions are then tested through a 10-task experiment. In (unambiguous) tasks 1–5, risk aversion is elicited through both a portfolio choice method and a BDM mechanism. In (ambiguous) tasks 6–10, choice-ambiguity aversion is elicited through the portfolio choice method, while value-ambiguity aversion comes about through the BDM mechanism. The behavior of over 75 % of classified subjects is in line with the KMM model in all tasks 6–10, independent of their degree of risk aversion. Furthermore, the percentage of coherent-ambiguity-averse subjects is lower in the binomial than in the uniform and in the unknown treatments, with only the latter difference being significant. The most part of coherent-ambiguity-loving subjects show a high risk aversion.

Decision Making with Second-Order Imprecise Probabilities

In decision analysis, uncertainty is usually described in the framework of probability. However, a large number of experimental and theoretical studies showed that a single nature of probability does not accurately capture human preferences. To avoid this drawback, they use imprecise probabilities. But, as decision maker is usually uncertain about first-order imprecise probabilities, imprecise hierarchical probability models are used. For most of such models, the second levels are precise. There also exist studies on two-level imprecise hierarchical models, which use imprecise probabilities or possibilities at the second level. Most of these works are based on lower prevision theory leading to a large number of optimization problems. In the present paper, we propose an imprecise hierarchical decision-making model where the first and the second level are described by interval probabilities. The method associates with the construction of a nonadditive measure as a lower prevision and uses this capacity in Choquet integral for constructing a utility function.

The dual theory of the smooth ambiguity model

This paper studies the “dual” theory of the smooth ambiguity model introduced by Klibanoff et al. (Econometrica 73:1849–1892, 2005). Unlike the original model, we characterize attitudes toward ambiguity captured by second-order probabilities. First, we give a set of axioms to derive a dual representation of the smooth ambiguity model. Second, we present a characterization of ambiguity aversion. Last, as an application of our dual model to a portfolio problem, we conduct comparative static predictions which give sufficient conditions to guarantee that an increase in smooth ambiguity aversion decreases the optimal portfolio.

A consistent set of infinite-order probabilities

Some philosophers have claimed that it is meaningless or paradoxical to consider the probability of a probability. Others have however argued that second-order probabilities do not pose any particular problem. We side with the latter group. On condition that the relevant distinctions are taken into account, second-order probabilities can be shown to be perfectly consistent.May the same be said of an infinite hierarchy of higher-order probabilities? Is it consistent to speak of a probability of a probability, and of a probability of a probability of a probability, and so on, ad infinitum? We argue that it is, for it can be shown that there exists an infinite system of probabilities that has a model. In particular, we define a regress of higher-order probabilities that leads to a convergent series which determines an infinite-order probability value. We demonstrate the consistency of the regress by constructing a model based on coin-making machines.

A Horvitz-Thompson Estimator of the Population Mean Using Inclusion Probabilities of Ranked Set Sampling

In this study, we define the Horvitz-Thompson estimator of the population mean using the inclusion probabilities of a ranked set sample in a finite population setting. The second-order inclusion probabilities that are required to calculate the variance of the Horvitz-Thompson estimator were obtained. The Horvitz-Thompson estimator, using the inclusion probabilities of ranked set sample, tends to be more efficient than the classical ranked set sampling estimator especially in a positively skewed population with small sizes. Also, we present a real data example with the volatility of gasoline to illustrate the Horvitz-Thompson estimator based on ranked set sampling.

Eliciting Ambiguity Aversion in Unknown and in Compound Lotteries: A KMM Experimental Approach

We define coherent-ambiguity aversion within the Klibano¤, Marinacci and Mukerji (2005) smooth ambiguity model (henceforth KMM) as the combination of choice-ambiguity aversion and value-ambiguity aversion. We analyze theoretically ?ve ambiguous decision tasks, where a subject faces two-stage lotteries with binomial, uniform or unknown second-order probabilities. We check our theoretical predictions through a 10-task laboratory experiment. In (unambiguous) tasks 1-5, we elicit risk aversion both through a portfolio choice method and through a BDM mechanism. In (ambiguous) tasks 6-10, we elicit choice-ambiguity aversion through the portfolio choice method and value-ambiguity aversion through the BDM mechanism. We ?nd that more than 75% of classi?ed subjects behave according to the KMM model in all tasks 6-10, independent of their degree of risk aversion. Further, the percentage of coherently-ambiguity-averse subjects is lower in the binomial than in the uniform and in the unknown treatment, with only the latter di¤erence being signi?cant. Finally, highly-risk-averse subjects are more prone to coherent-ambiguity.

Truncated Midzuno–Sen Sampling Schemes for Estimating Distribution Functions

We consider a truncated Midzuno–Sen sampling scheme. The proposed method can be used to estimate the distribution function of a study variable assuming that the distribution function of an auxiliary variable is known. The ratio estimator for estimating the distribution function is shown to remain unbiased. We introduce the first- and second-order inclusion probabilities under the truncated Midzuno–Sen sampling scheme. Numerical examples are provided to support our theoretical results.

The Known Unknowns: Neural Representation of Second-Order Uncertainty, and Ambiguity

Predictions provided by action-outcome probabilities entail a degree of (first-order) uncertainty. However, these probabilities themselves can be imprecise and embody second-order uncertainty. Tracking second-order uncertainty is important for optimal decision making and reinforcement learning. Previous functional magnetic resonance imaging investigations of second-order uncertainty in humans have drawn on an economic concept of ambiguity, where action-outcome associations in a gamble are either known (unambiguous) or completely unknown (ambiguous). Here, we relaxed the constraints associated with a purely categorical concept of ambiguity and varied the second-order uncertainty of gambles continuously, quantified as entropy over second-order probabilities. We show that second-order uncertainty influences decisions in a pessimistic way by biasing second-order probabilities, and that second-order uncertainty is negatively correlated with posterior cingulate cortex activity. The category of ambiguous (compared with nonambiguous) gambles also biased choice in a similar direction, but was associated with distinct activation of a posterior parietal cortical area; an activation that we show reflects a different computational mechanism. Our findings indicate that behavioral and neural responses to second-order uncertainty are distinct from those associated with ambiguity and may call for a reappraisal of previous data.

Analytical procedures for weekly hydrological droughts: a case of Canadian rivers

Two entities of importance in hydrological droughts, viz. the longest duration, LT , and the largest magnitude, MT (in standardized terms) over a desired time period (which could also correspond to a specific return period) T, have been analysed for weekly flow sequences of Canadian rivers. Analysis has been carried out in terms of week-by-week standardized values of flow sequences, designated as SHI (standardized hydrological index). The SHI sequence is truncated at the median level for identification and evaluation of expected values of the above random variables, E(LT ) and E(MT ). SHI sequences tended to be strongly autocorrelated and are modelled as autoregressive order-1, order-2 or autoregressive moving average order-1,1. The drought model built on the theorem of extremes of random numbers of random variables was found to be less satisfactory for the prediction of E(LT ) and E(MT ) on a weekly basis. However, the model has worked well on a monthly (weakly Markovian) and an annual (random) basis. An alternative procedure based on a second-order Markov chain model provided satisfactory prediction of E(LT ). Parameters such as the mean, standard deviation (or coefficient of variation), and lag-1 serial correlation of the original weekly flow sequences (obeying a gamma probability distribution function) were used to estimate the simple and first-order drought probabilities through closed-form equations. Second-order probabilities have been estimated based on the original flow sequences as well as SHI sequences, utilizing a counting method. The E(MT ) can be predicted as a product of drought intensity (which obeys the truncated normal distribution) and E(LT ) (which is based on a mixture of first- and second-order Markov chains). Citation Sharma, T. C. & Panu, U. S. (2010) Analytical procedures for weekly hydrological droughts: a case of Canadian rivers. Hydrol. Sci. J. 55(1), 79–92.

Ambiguity and the value of information

The value of information is studied in a non-expected utility model of ambiguity with second-order probabilities. Information that reduces ambiguity has a positive value for ambiguity-averse decision makers, and the value of information that resolves ambiguity increases with greater ambiguity and with greater ambiguity aversion. Since information that resolves risk is valuable, and must also resolve ambiguity, the value of such information for ambiguity averters increases with greater ambiguity and with greater ambiguity aversion.

On how to define, understand and describe risk

The prevailing perspectives and definitions of risk, at least in the engineering community, are based on probabilities. In this paper we argue that such perspectives and definitions are too narrow. The probability component of the risk concept should be replaced by uncertainty. By jumping directly into probabilities, important uncertainty aspects could easily be overlooked or truncated. In the paper we point at several extended risk definitions, and a formal structure for the various perspectives and definitions is developed. Fundamental concepts such as second-order probabilities and uncertainties are discussed. Examples are provided showing the importance of the choice of risk perspective in a risk assessment and decision-making context. The examples cover offshore operations, security and market price risks.

On representing second-order uncertainty in multi-state systems via moments of mixtures of Dirichlet distributions†

Second-order probabilities have been proposed as representations of the uncertainty in the parameters of probabilistic models such as Bayesian belief networks. We investigate conditions under which second-order probabilities can be represented in terms of their marginal moments. We show that certain combinations of marginal means and variances do not correspond to any valid second-order joint distribution. By fitting a Dirichlet mixture to marginal mean and variance information, we derive sufficient conditions for a valid second-order joint distribution to exist. †LA-UR-07-1330

Do We Need Second-Order Probabilities?

Although it has often been claimed that all the information contained in second-order probabilities can be contained in first-order probabilities, no practical recipe for the elimination of second-order probabilities without loss of information seems to have been presented. Here, such an elimination method is introduced for repeatable events. However, its application comes at the price of losses in cognitive realism. In spite of their technical eliminability, second-order probabilities are useful because they can provide models of important features of the world that are cognitively more plausible than those that can be obtained with single-level probabilities.