Conceptual thinking
about the unknown
Uncertainty: expected
value, sensitivity analysis,
and the value of
information
Review 1
• Your problem is not as unique as you
think it is
• You have more data than you think
you have
• You need less data than you think you
need
• There is a useful measure that is
much simpler than you think it is
Review 2
• If it matters, it is
detectable/observable
• If it can be detected, it can be detected
as an amount (or range of possible
amounts)
• If it can be detected as a range of
possible amounts, it can be measured
Review 3
• Write down a number
• Break it down into pieces
(decomposition)
• Try different decompositions
• Average
Clarifying the measurement
problem
• What is the decision this is supposed
to support?
• What really is the thing being
measured?
• Why does this thing matter to the
decision?
• What do you know about it now?
• What is the value of knowing more?
Calibration
There are two extremes of subjective
confidence:
Over confidence
Under confidence
Uncertainty and Risk
• Uncertainty = The lack of complete
certainty. That is, more than one
outcome is possible, so that the true
outcome/SON/result/value is not
known
• Risk = uncertainty involving hazard.
That is, some outcomes are bad,
involve a loss, or where they are all
bad, some are catastrophic
EXPECTED VALUE ANALYSIS
• Consists of modeling uncertainty as a set of
contingencies that are exhaustive and mutually
exclusive with specific probabilities of occurrence.
• In practice, this means the analyst identifies
representative contingencies and assigns probabilities
to each of them so that they sum to one.
• The probabilities can be based on historically observed
frequencies, subjective assessments, or experts (based
on information, theory, or both).
Calculating the expected value of net
benefits
• Calculate the net benefits of each
contingency
• Multiply by that contingency's probability of
occurrence.
• Sum the weighted benefits
E(NB) = SPi (Bi - Ci)
Representativeness of contingencies
Specification of contingencies
A nnualized
A nnualized
crop value with crop value
irrigation
without
irrigation
$ 4 ,5 0 0 ,0 0 0
4 ,5 0 0 ,0 0 0
4 ,5 0 0 ,0 0 0
4 ,0 0 0 ,0 0 0
3 ,0 0 0 ,0 0 0
A nnualized A nnualized
cost of dam net benef it
&
dist ribution
system
$0
$ 2 0 0 ,0 0 0$4,300,000
2 ,8 0 0 ,0 0 0
2 0 0 ,0 0 0 $1,500,000
3 ,7 0 0 ,0 0 0
2 0 0 ,0 0 0 $600,000
3 ,6 0 0 ,0 0 0
2 0 0 ,0 0 0 $200,000
2 ,8 0 0 ,0 0 0
2 0 0 ,0 0 0
$0
EX(P)
V|P
0 .1 0
$430,000
0 .8 0
$480,000
0 .1 0
$0
$910,000
EX(P)
0 .0 5
0 .1 2
0 .6 6
0 .1 2
0 .0 5
1 .0 0
V|P
215000
$180,000
$396,000
$24,000
$0
$815,000
Decision trees and expected NB
Decision analysis has two stages.
- First, one specifies the logical structure of the
decision problem in terms of sequences of
decisions and realizations of contingencies
using a diagram (called a decision tree) that
links an initial decision to final outcomes.
- Second, one works backwards from final
outcomes to the initial decision, calculating
expected values of net benefits across
contingencies and pruning dominated branches
(ones with lower expected values of net
benefits).
Vaccine example
• Present value of expected net
benefits of the vaccination program is
simply E(CNV) - E(CV) (i.e., the
expected value of the costs when not
implementing the program minus the
expected costs when implementing
the program).
Decision tree for vaccination
program analysis
What’s up for grabs?
• Population at risk
- Total Population (round to
10K)
- Fraction High Risk ?
• Infection Rate ?
• Mortality Rate ?
- Value of Life ?
• Time lost to Flu ?
- Opportunity Cost of Time ?
• Chance of Epidemic
- First Year ?
- Second Year ?
• Total number of people
vaccinated
• Vaccination Rate
• Administrative Costs
- Overheads (Fixed)
- Dose Price (Variable)
• Adverse Reaction Rate ?
• Herd Immunity Rate ?
• Vaccine Effectiveness
Rate?
• Discount Rate ?
Expected net benefits of vaccinations
Expected net benefits of vaccinations
Worst; best case analysis
• What are our maximum downside
risks if we take no action?
• What if we do?
Histogram of realized net benefits
The cost and value of information
• Perfect information?
• Imperfect information?
• Quasi-option value
Exogenous learning
Endogenous learning