22.0 Risk Analysis
• Answer Questions
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• Risk Definitions
• Assessing Probabilities
• Examples: PRA
22.1 Definitions of Risk
The risk of an action d is defined to be the expected loss. Formally, a loss
function L(z, d) determines how much one loses (or gains) if one takes decision
d and the true state of nature is z.
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Using frequentist or Bayesian methods, a statistician makes an inference about
the distribution F (z|d). Then the risk is
Z
L(z, d)f (z|d) dz.
RF (d) = IEF [ L(z, d) ] =
IR
The formalism is not essential; the key idea is that risk is the average amount
one expects to lose if one takes decision d.
For example, suppose one wants to decide whether to install a stoplight at an
intersection. The stoplight has a working lifetime of 10 years, with a start-up
cost of $2000 and an annual maintenance fee of $500.
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Based on previous studies of traffic at that intersection and similar
intersections, we know that the stoplight will save about 1/100 of a life per
year, and prevent 1 accident per year. And suppose a non-fatal accident costs,
on average, $200. Also suppose that the average value of a human life is
$100,000 (we shall discuss the issue of assigning such numbers later).
Then the expected loss of installing the stoplight is:
$2000 + 10 ∗ ($500) = $7, 000.
And the expected loss of not installing the stoplight is:
10 ∗ ($200) + 10 ∗ (1/100) ∗ ($100, 000) = $12, 000.
This formalism conceals a number of practical problems:
• How does one monetize all the different costs in the problem?
• How does one reconcile different views of loss?
• How does one determine the probabilities of accidents at an intersection,
with and without stoplights?
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The monetization problem is controversial. People sometimes say one cannot
put a monetary value on human life. But insurance companies do this all the
time, and transportation managers need some way to put multiple costs on a
common scale. (One approach is to divide GDP by the population size; but in
civil suits law courts often make awards based on non-economic roles, such as
being a parent.)
In the context of a traffic accident, a person can damage their car, break an
arm, and miss their wedding anniversary. Each of these events has a personal
cost, but it is hard to make them comparable and different people would value
these consequences differently.
To complicate the monetization problem, most people have a nonlinear
perception of the value of money.
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Often it becomes the responsibility of the government or the employer to make
these decisions. But that decision should order the outcomes correctly (e.g.,
death costs more than a broken bone), and the value of the money should
reflect a consensus view of the situation.
22.2 Assessing Probabilities
For estimating the probabilities of events, there are two main strategies:
frequentist and Bayesian.
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A frequentist says that the probability of event A (or P [A]) is the proportion
of times that A occurs in a infinite sequence of separate tries. Thus
# times A happens
.
n→∞
n
P [A] = lim
A Bayesian can pick whatever number they prefer for P [A], based on their
own personal experience and intuition, provided that number is consistent
with all of the other probabilities they choose in life.
In the context of risk analysis, the frequentist approach typically leads one to
use historical data, such as the rate of traffic fatalities at intersections without
stoplights. When such data are available, this is generally the preferred
approach.
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But often such data are not available. For example, the decision-maker may
realize that road traffic is increasing, and the mix of vehicles is changing. How
would that affect the fatality rate at an intersection?
In the usual case, one does not have sufficient data to make the frequentist
argument. In those circumstances, one seeks expert opinion, and uses those
subjective probability distributions for the analysis.
But experts often disagree, are notoriously overconfident, and it can be
difficult to express their opinions in terms of distributions F (z|d).
People in general are very poor at making probability judgments. Experts are
only a little better, and only in their specific domain.
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Note that both axes are on the log scale.
People tend to confuse the probability of an event with how quickly the
consequences are realized, and the extent to which the risk is under their
control.
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To further complicate the matter, people have very different attitudes towards
risk. Some are extremely risk averse; others seek out risk. Elke Weber
gathered data shown below:
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The main message is that the risk manager needs some way to assess
probabilities and perceived losses.
Frequentist probabilities are the most easy to justify and do not suffer from
perception biases. Thus they are ideal in for government programs. (However,
citizens can be irrationally fearful, e.g., demanding stronger action on H1N1
at the expense of childhood inoculation programs, and this creates political
pressure to make suboptimal resource investments in risk management.)
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In risk analysis, Bayesian probabilities are matters of expert judgement. When
they are updated with data through Bayes’ Theorem, there is usually little
difficulty. But in realistic applications, one often lacks data that are directly
useful. Expert opinion should be used carefully, with full recognition of the
kinds of biases that can arise.
Expert opinion can be difficult to justify to the public and to ones’ bosses. It
is easy to paint such opinion as partisan or political.
22.3 Examples of Probabilistic Risk Analysis
Based on historical data, suppose the number of children who die each
year from chicken pox has a Poisson distribution with mean 20.5. So the
probability that k children die next year is given by
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λk
exp −λ
IP[X = k] =
k!
where λ = 20.5.
Now suppose that a group of doctors and public health experts say that a
program of mass vaccination would reduce the average number of deaths. But
the program would cost $5,000,000 to administer.
The group of experts will probably have difficulty in predicting how effective
the program will be.
If we say that the life of a child is worth a $500,000, then the program would
need to reduce the average number of deaths from 20.5 to 10.5 in order to
break even.
Suppose that the government decides to experiment with the vaccination
program. In the first year of administration, the number of deaths from
chicken pox is 2. Does this provide evidence that the program is effective?
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The null hypothesis is that the average death rate is greater than or equal to
10.5. The alternative hypothesis is that the average death rate is less than
10.5; i.e.,
H0 : λ∗ ≥ 10.5
vs.
HA : λ∗ < 10.5
where λ∗ is the death rate after the vaccination program has begun.
The significance probability P of the test is the probability of obtaining
results that are as or more supportive of the alternative hypothesis than the
data that are observed, when the null hypothesis is true.
If this is a small number, then the observed outcomes are either a miracle, or
the null hypothesis is incorrect.
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In this example, the significance probability is:
P
= IP[0, 1, or 2 deaths | avg. number of deaths = 10.5 ]
10.50
10.51
10.52
=
exp(−10.5) +
exp(−10.5) +
exp(−10.5)
0!
1!
2!
= 0.0018.
This is strong evidence against the null hypothesis. The program is
cost-effective.
The previous example used frequentist probability—the death rates were
based on historical data (assuming no population growth, no changes in diet,
sanitation, etc.).
As contrast, consider a Bayesian analysis, in which probabilities are placed by
experts on different branches in an event tree.
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In an event tree, experts consider all the possible things that can happen,
as a sequence of events that are conditional on previous outcomes. They
use judgment to assign probabilities at each step, and then multiply those
probabilities to find the chance of each possible final outcome.
The method was widely used, and criticized, in safety analyses of nuclear
power plants. Those analyses were often too optimistic, and overlooked or
underemphasized human error as a failure mode.
This is an event tree for brake failure, with expert probability assessments at
each branch.
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