Games and Decisions in Reliability and Risk (GDRR)
ABSTRACTS
Risk Assessment for Pyroclastic Flows
James Berger
Duke University
Abstract
Risk assessment of rare natural hazards – such as large volcanic block and ash or pyroclastic flows – is addressed. Assessment is approached through a combination of computer modeling, statistical modeling, and extreme-event probability computation. A computer model of the natural hazard is used to provide the needed extrapolation to unseen parts of the hazard space. Statistical modeling of the available data is needed to determine the initializing distribution for exercising the computer model. In dealing with rare events, direct simulations involving the computer model are prohibitively expensive. Solution instead requires a combination of adaptive design of computer model approximations (emulators) and rare event simulation. The techniques that are developed for risk assessment are illustrated on a test-bed example involving pyroclastic flow.
Stakeholder Preference Modeling with Probabilistic Inversion
Roger Cooke
Delft University & Resources for the Future
Abstract
Utilities cannot be updated on observations, and there are no 'utility experts'; the problem of modeling stakeholder preference requires different techniques than structured expert judgment. The goal is to characterize a population of stakeholders via a distribution over utility functions. Probabilistic inversion denotes the operation of inverting a function at a distribution or set of distributions. Solution algorithms are based on variations of the Iterative Proportional Fitting Algorithm. Given discrete choice data ("x% of the population prefer A to B, y% rank C in the 4th place..." we find a distribution over utility functions that optimally reproduce these probabilities. When utility functions are modeled via some functional form, for example multi attribute utilities, then a population of stakeholders is characterized via a distribution over parameters of the utility model. This allows us to validate such utility models against discrete choice data not used for fitting the model. Validation is noticeably absent in methods like AHP and MCDM. Applications to prioritizing marine ecosystem threats and/or valuing health states will be discussed.
Adversarial Risk Analysis: Games and Auctions
David Banks
Duke University
Abstract
Classical game theory has been an unreasonable description for human behavior, and traditional analyses make strong assumptions about common knowledge and fixed payoffs. Classical risk analysis has assumed that the opponent is non-adversarial (i.e., "Nature") and thus is inapplicable to many situations. This work explores Bayesian approaches to adversarial risk analysis, in which each opponent must model the decision process of the other, but there is the opportunity to use human judgment and subjective distributions. The approach is illustrated in the analysis of two important applications: sealed bid auctions and straight poker. The results in these two applications are interestingly different from those found in previous work.
My Current View of DS Inference Introduced via a New Model for Exponentially Distributed Waiting Times
Arthur P. Dempster
Harvard University
Abstract
DS analysis is an inferential calculus based on a set of concepts, models, and operations that taken together amount to a possible major paradigm shift in how probability-based statistical inference is formulated and carried out. Many statisticians who are steeped in the thinking behind one or both of the so-called frequentist and Bayesian schools often find it hard to grasp DS methodology. In part, the difficulty is inevitable, because the underlying ideas, and even notation, are unfamiliar, and do not cohere directly with prevailing textbook norms. Advocates may have been placing too much emphasis on the simplicity and elegance of the formal structure, and too little on motivation. In this talk, I aim to construct a foundation for understanding the formal theory, by introducing it through a natural DS modification of a simple and familiar reliability model, where independent observed exponentially distributed waiting times are used to assess uncertainty about a prospective long run average.
Between Positive and Normative Theories of Choice:
A Mean-Risk Model
Philippe Delquié
INSEAD, France
Abstract
We propose a Risk-Value model derived from on a theory of Disappointment in which there is no single reference point that separates the outcomes as either disappointing or satisfactory. The key behavioral assumption is that individuals facing a risky prospect care about how they will come out relative to the other, more or less favorable, outcomes, not a single benchmark as is commonly assumed in defining risk measures. The model includes some classic measures of risk, such as the variance or Gini mean difference as special cases, although it is distinct from the traditional families of risk measures. The model preserves the essential normative properties of first and second order stochastic dominance for ordering risks, for which necessary and sufficient conditions are provided. It also fulfills the axioms of Convex Risk Measures, widely regarded as compelling for risk management. Without transforming probabilities, the present Mean-Risk model produces a richer set of empirically relevant behaviors than models based on rank dependent probability weighting. This is because in the latter the weights of the outcomes depend exclusively on their ranks, while the implicit weights in our Mean-Risk model depend on their ranks and also on their spacing. In asset trading, the Mean-Risk model can produce EU-type behavior or a no-trading zone as in Rank Dependent Utility (RDU). In asset allocation, investor's behavior can be like in EU (second order risk aversion), or "plunging" like in RDU (first order risk aversion) depending on the performance of the risky asset.
HARA Frontiers in Portfolio Management
Suleyman. Özekici and E. Çanakoğlu
Koç University, Turkey
Abstract
We consider Black-Scholes type continuous-time models where the market parameters are driven by a modulating discrete-state Markov processes. The states of the market represent the prevailing economic, financial, social and other conditions that affect the deterministic and probabilistic parameters of the model. It describes, in a probabilistic manner, how the regime in the market switches independent of the underlying Brownian motion that drives the stock prices. The investor chooses a portfolio management policy in order to maximize the expected utility of the terminal wealth. The problem is analyzed using dynamic programming and control theory tools for the HARA class of utility functions that include exponential, logarithmic and power functions. We found explicit solutions for the optimal policy and the optimal value function under reasonable assumptions. We also constructed the optimal wealth process explicitly and discussed some of its properties. In particular, it is shown that the optimal policy provides linear risk-return frontiers.
Bayesian Learning in Social Networks
Daron Acemoglu, Munther A. Dahleh, Ilan Lobel, and Asuman Ozdaglar
Massachusetts Institute of Technology
Abstract
We study the (perfect Bayesian) equilibrium of a model of learning over a general social network. Each individual receives a signal about the underlying state of the world, observes the past actions of a stochastically-generated neighborhood of individuals, and chooses one of two possible actions. The stochastic process generating the neighborhoods defines the network topology (social network). The special case where each individual observes all past actions has been widely studied in the literature. We characterize pure-strategy equilibria for arbitrary stochastic and deterministic social networks and characterize the conditions under which there will be asymptotic learning, that is, the conditions under which, as the social network becomes large, individuals converge (in probability) to taking the right action. We show that when private beliefs are unbounded (meaning that the implied likelihood ratios are unbounded), there will be asymptotic learning as long as there is some minimal amount of "expansion in observations". Our main theorem shows that when the probability that each individual observes some other individual from the recent past converges to one as the social network becomes large, unbounded private beliefs are sufficient to ensure asymptotic learning. This theorem therefore establishes that, with unbounded private beliefs, there will be asymptotic learning in almost all reasonable social networks. We also show that for most network topologies, when private beliefs are bounded, there will not be asymptotic learning. In addition, in contrast to the special case where all past actions are observed, asymptotic learning is possible even with bounded beliefs in certain stochastic network topologies.
How does a Bayesian Play Games?
Joseph B. Kadane
Carnegie Mellon University
Abstract
The Bayesian paradigm requires personal probabilities to express and quantify uncertainty. Faced with a decision involving uncertainty about the action of another, a Bayesian player maximizes expected utility, where the expectation is taken with respect to the Bayesian's personal probability. This view is contrasted with the minimax view of zero-sum two-person games, and its extension to Nash equilibrium. It is also contrasted with backward induction as applied classically to iterated prisoner's dilemmas and the "guess two-thirds of the mean" game.
Multidimensional Gamma Fields
Erhan Cinlar
Princeton University
Abstract
Gamma processes are used in modeling stock prices and in theories of reliability and storage. Most such applications require a multi-dimensional version of the classical gamma process. We give the general form of such Levy processes, describe their construction from a sequence of uniform variables, and illustrate their use in defining volatility vector fields.
Partition based Nonparametric Priors to Analyze Data from a Special Class of Dependent Models
Jayaram Sethuraman
Florida State University
Abstract
A special class of dependent data occurs in search, repair and censoring models. In these models the distribution of the first observation is the parameter of interest. The model stipulates that the distribution of successive data depends on this parameter and is restricted to sets depending on previous observations, including covariates. Special cases of such models have been studied in the literature and frequentist methods have been employed for inference. This talk will present a unified general model for such data. It will be shown that the new class of prior distributions called partition based (PD) priors is the natural class to use for Bayesian analysis. A special class of PB priors, namely partition based Dirichlet (PBD) priors have the interesting property that the final calculations are still correct if the partitions present in the prior distribution are based on the data. This feature simplifies the required computations. These results will be illustrated for two special models, namely imperfect repair and censoring models. Bayesian estimates will be compared with frequentist estimates. Bayesian methods will be also illustrated for some censoring models which have not been considered in the literature.
On Large Deviations
S. R. S. Varadhan
Courant Institute of Mathematical Sciences, NYU
Abstract
There has been some interest recently on ergodic theorems for averages of the form 1⁄n∑f(x1i,x2i,...,xki). We will examine the large deviation behavior of these averages, leads to variations of the usual large deviation results, i.e. behavior of 1⁄n log E[exp(∑ ai f(xi))] for suitable sequences {ai}.
Bayesian Modeling of Train Doors Reliability
Fabrizio Ruggeri and Antonio Pievatolo
CNR-IMATI, Italy
Abstract
We analyze failures of train doors in a European underground line over a nine-year period and develop Bayesian models considering both scales (days and kilometers) at which failures are recorded. Aims of the research are both description of current failures and forecast of new ones, in a context of reliability checking before expiration of warranty time.
Game Theory and Warranties
Simon Wilson
Trinity College, Ireland
Abstract
Warranties can be thought of as insurance against early failure for the buyer of a product. In this talk, the statistical modeling, decision theoretic and game theoretic aspects of warranties are discussed. Specifically, game theory arises because of several different competitive elements of setting warranty conditions. One element is the so-called moral hazard problem, where more generous warranties lead to reckless product use by the consumer and hence higher chance of in-warranty failure; a game between seller and buyer. Another is the more classic game theory problem of setting warranties in the context of competition between sellers. Appropriate statistical models or warranty analysis are also discussed, with emphasis on models for two-dimensional warranties where failure is measured by time and a measure of usage, such as mileage for an automobile.
Triangulation of Sample, Parameter and Predictive Information
Nader Ebrahimi, Northern Illinois University
Ehsan S. Soofi, University of Wisconsin-Milwaukee
Refik Soyer, George Washington University
Abstract
Triangulation of sample, parameter, and prediction brings together the measure of sample information about the parameter, known as Lindley's measure, and the measure of predictive in-formation. For the case of conditional independence of the sample and the future outcomes given the parameter, decompositions of joint parameter-predictive information characterize Lindley's measure as the sample information about the parameter and prediction jointly, show that Lindley's measure dominates predictive information, and quantify the predictive share. For prediction of an order statistic based on the preceding ones, the conditional independence no longer holds and the joint parameter-predictive information exceeds the parameter information by an amount due to the Markovian property of order statistics. These results map the trace of information flow from the sample to the predictive distribution. Implications include that the reference posterior distribution corresponding to the prior that maximizes the sample information about the parameter remains optimal according to the joint parameter-predictive information. More specific relationships between the parameter and predictive information are explored for a broad subfamily of the exponential family. Information provided by failure times and survival times are compared and sufficient conditions are given for observing failures to be more or less informative than survivals about prediction of a lifetime.
Randomized Discontinuation Design
Peter Mueller
M. D. Anderson Cancer Center
Abstract
We discuss an application of Bayesian decision theory and risk evaluation to the optimal planning of a clinical trial design known as random discontinuation designs (RDD). RDDs proceed in two stages. During the first stage all patients are treated with the experimental therapy. A subgroup of patients who show evidence of response during the first stage are then randomized to control and treatment in a second stage. The intention of the design is to identify in the first stage a subpopulation of patients who could potentially benefit from the treatment, and carry out the comparison in the second stage only in that identified subgroup. Most applications are to oncology phase II trials for cytostatic agents. The design is characterized by several design parameters. We discuss an optimal choice of the tuning parameters based on a Bayesian decision theoretic framework. We define a probability model for putative cytostatic agents and specify a suitable utility function. The utility function formalizes the investigator's preferences for different actions under hypothetical future outcomes and assumed parameters.
Particle-based Accelerated Life Test Designs
Nicholas Polson, University of Chicago
Refik Soyer, George Washington University
Abstract
In this paper we consider a Bayesian decision theoretic setup for optimal design of accelerated life tests. In so doing, we discuss difficulties in evaluating preposterior losses in adopting the Bayesian framework and present simulation-based design algorithms. More specifically, we provide a particle-based approach to design of accelerated life tests and illustrate the implementation of the approach in one and two stage problems.