Effect of Epistemic Uncertainty Modeling Approach on Decision-Making: Example using Equipment Performance Indicator Page: 4 of 8
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Table 1. Utility function (payoff matrix) for equipment unreliability example
Actual State
Decision Green White
Green 100 -50
White -200 100The optimal decision is the one that maximizes expected utility, with the expectation taken over the posterior
distribution ofp [5]. Thus, given x failures are observed in 210 demands, the posterior distribution ofp is
beta(0.5 + x, 262.66 + 210 - x). As an example, the expected utility of a decision that the risk is in the Green
region (R < 10-6) is given by the following:
UGreen =Pr(R < 10-6 )(100)+ Pr(R > 10-6 )(-5O) (5)
Thus, with the values given in the example, the decision-maker whose utility function is described by the
payoff matrix in Table 1 could see no more than 10 failures in 210 demands without deciding to expend
additional resources (i.e., the optimal decision for > 10 failures is that R is White), under this particular
model.
The CNI prior, being a type of maximum entropy prior [6], is intended to be minimally informative, or
"gentle" in the terminology of [7]. However, as discussed in [3], the CNI prior can be relatively "rough"
when updated with sparse data. As discussed by [8] and others, this is actually a problem with conjugate
priors generally, as they tend to have relatively light tails. Recognizing this, [9] examined the use of a so-
called "robust" prior, which in this case is a logistic-Cauchy prior. Using this prior, and proceeding
otherwise analogously to the above process, with parameters determined using the approach described in [9],
the decision-maker could accept at most 5 failures in 210 demands before having to conclude that more
resources should be expended. In other words, the optimal decision is Green for x < 5, and White for x > 5.
3. ALTERNATIVE APPROACHES FOR REPRESENTING EPISTEMIC UNCERTAINTY
It has long been recognized that the choice of prior can influence a decision in the traditional Bayesian
framework. In this framework, a single prior distribution is used to represent epistemic uncertainty in a
quantity such as p in this example. For this reason, it is a tenet of decision-making that sensitivity analysis
should be performed on the choice of prior distribution. However, it seems to be a common belief among
practicing analysts that the form of the prior distribution is less important than the range, as embodied by,
say, an interval from the 5th to the 95th percentile. This belief is sometimes used to pragmatically counter the
argument that it is unreasonable to represent an imprecise state of knowledge with a single, precise
distribution, what [1] refers to as the "dogma of ideal precision."
We now explore two alternatives to the ideal precision of the approximate CNI prior and the heavier-tailed
logistic-Cauchy prior used in this specific example. The first of these will be a type of robust Bayesian
analysis, in the terminology introduced by [8]. The second will be the approach of imprecise probability
described in [1], in which intervals rather than distributions are used to represent epistemic uncertainty.
3.1. Robust Bayes
For the robust Bayes approach, we will model a class of CNI priors, in which we allow the mean constraint
to be uncertain. We adopt the hierarchical approach taken in [10] in which a second-stage distribution is
used to represent uncertainty in the mean value constraint. In this case, we will model the class of beta prior
distributions whose mean is uncertain by a factor of 3 around the value of 1.9 x 10-3, but each with a = 0.5
because each member of the class is an approximate CNI prior.
We will use the OpenBUGS software [11] to carry out the Bayesian inference using Markov chain Monte
Carlo (MCMC) sampling, as described in [12]. The immediate problem is how to encode the epistemic
uncertainty about the mean constraint. We specify that a = 0.5, as discussed above, so that the distributions
will be in the class of (approximate) CNI priors. Our epistemic uncertainty is with respect to the mean value
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Kelly, Dana & Youngblood, Robert. Effect of Epistemic Uncertainty Modeling Approach on Decision-Making: Example using Equipment Performance Indicator, article, June 1, 2012; Idaho Falls, Idaho. (https://digital.library.unt.edu/ark:/67531/metadc834141/m1/4/: accessed March 29, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.