Dependence in probabilistic modeling, Dempster-Shafer theory, and probability bounds analysis.

Abstract

LIBRARY VAULTThis report summarizes methods to incorporate information (or lack of information) about inter-variable dependence into risk assessments that use Dempster-Shafer theory or probability bounds analysis to address epistemic and aleatory uncertainty. The report reviews techniques for simulating correlated variates for a given correlation measure and dependence model, computation of bounds on distribution functions under a specified dependence model, formulation of parametric and empirical dependence models, and bounding approaches that can be used when information about the intervariable dependence is incomplete. The report also reviews several of the most pervasive and dangerous myths among risk analysts about dependence in probabilistic models.*The work described in this report was performed for Sandia National Laboratories under Contract No. 19094. 3 AcknowledgmentsDiscussions with Peter Walley were helpful in elucidating the problem of representing dependence in the context of imprecise probabilities and the role of probability boxes and Dempster-Shafer structures as models of imprecise probabilities. 2.5Using inequalities in risk assessments 2.6Caveat: best-possible calculations are NP-hard 2.7Numerical example: fault tree for a pressure tank Sym b o k -Ea is distributed as is an element of is a subset of plus or minus addition, subtraction, etc. under no assumption about the dependence between the operands addition, subtraction, etc. assuming independence addition, subtraction, etc. assuming perfect dependence addition, subtraction, etc. assuming opposite dependence the empty set, i.e., the set having no members probability-box specified by a left side F(x) and a right side E(x) where E(x) I F ( x ) for all x E %, consisting of all non-decreasing functions F from the reals into [O,l] such that E(x) I F(x) I fix). {(sl,ml),. . ., (s,,m,)} an enumeration of the elements of a Dempster-Shafer structure in terms of its focal elements si and their nonzero masses mi beta (v, w) a beta distribution with shape parameters v and w convolve(X, Y,r) convolution (usually addition) assuming that X and Y have correlation r convolve(X, Y,C) convolution (usually addition) assuming the copula C describes the dependence between X and Y covariance between random variables X and Y expectation (mean) of random variable X a functionfwhose domain is the set A and whose range is the set B. In other words, for any element in A, the functionfassigns a value that is in the set B the step function that is zero for all values of x<c and one for all x2c infimum (for a finite set, simply the minimum) the copula min(u,v) corresponding to perfect dependence a normal distribution with mean p and standard deviation o the copula defined by the product uv corresponding to independence probability of event A probability of event A given that event B has occurred probability of conjunction (and) of events A and B probability of disjunction (or) of events A and B infimal convolution for binary operation L between distr...

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