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ISE 562; Dr. Smith
10/15/2022
Assessing Subjective Probability Distributions
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Decision Theory
There are numerous methods for subjective probability assessment
ISE 562; Dr. Smith
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Fixed consequences, variable probabilities variable consequences, fixed probabilities
Direct assessment (ask for the value) Odds methods
Props (wheels, rulers, software tools) Lottery methods
ISE 562; Dr. Smith
• Generally:
The simpler the method, tendency for lower accuracy with the benefit of speed.
The more complex the method, accuracy tends to increase at the expense of more time
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The method described here falls somewhere between these extremes
Easier for the interviewee (binary choices) Captures probabilities (CDF’s)
Precision of assessment can be selected by the interviewer (3-point, 5-point, 7-point distributions)
ISE 562; Dr. splitting method
• 2 questions to bound the range (zero and 100th percentile.
• 1 question to establish the median (50th percentile).
• A series of questions to assess the (25th percentile).
• A series of questions to assess the (75th percentile).
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Question 1
Question 5
Question 2
Introduction and purpose of interview
Definition of random variable
Assess minimum range value
Question 4
Assess median value
Assess 25th percentile
Assess 75th percentile
Assess maximum range value
ISE 562; Dr. = name of the random variable
1. What is the value of [X] for which there is one chance in a hundred that [X] will be less than that value?
2. What is the value of [X] for which there is one chance in a hundred that [X] will be greater than that value?
3. What is the value of [X] between the above for which there is an equal chance that [X] will be greater than the value and less than the same value? (The median.)
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ISE 562; Dr. s 4,5:
• Respondent is asked to make a choice between A or B where the question is
“Do you think the value of x is: A) “less than x?”
B) “between y1 and y2?”
• If the answer is A (less than x), then the next set
of questions bisects the interval [y1, y2] on the lower side.
• If the answer is B (between), then the next set of questions bisects the interval [y1, y2] on the upper side of the interval.
• So how are the values determined?
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ISE 562; Dr. Smith
• Suppose the answers to questions
1,2,3 were a,c,b
• The questions for the 25th percentile will focus on the lower interval from a to b.
• Compute the midpoint of the lower interval and ask the question:
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25th percentile
Do you think it is more likely the value of x is less than (a+b)/2 or between (a+b)/2 and b?
ISE 562; Dr. Smith
abc Here? Or here?
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25th percentile
If the answer is the lower interval, subdivide the lower interval in half and ask the same question:
(3a+b)/4 (a+b)/2
ISE 562; Dr. ?
Is it more likely that the value is below the midpoint ((3a+b)/4 ) or in the upper interval [(3a+b)/4, b]?
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25th percentile
If the answer is the upper interval, subdivide the upper interval in half and ask the same question:
(a+b)/2 (a+3b)/4
Here? Or here?
Is it more likely that the value is below the midpoint ((a+3b)/4 ) or in the upper interval [(a+3b)/4, b]?
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ISE 562; Dr. SE 562; Dr. Smith
• Repeat this process until the interviewee can’t choose (the differences will be too small)
25th percentile
• The value of x when this occurs will be the 25th percentile value.
• P(X≤x) = 0.25
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75th percentile
Do you think it is more likely the value of x is between b and (b+c)/2 or greater than (b+c)/2?
ISE 562; Dr. Smith
abc Here? Or here?
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75th percentile
If the answer is the lower interval, subdivide the lower interval in half and ask the question:
Here? Or here?
Is it more likely that the value of x is between b and (3b+c)/4 or greater than (3b+c)/4?
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ISE 562; Dr. Smith
75th percentile
If the answer is the upper interval, subdivide the upper interval in half and ask the same question:
Here? Or here?
Is it more likely that the value is between b and (b+3c)/4 or greater than (b+3c)/4?
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ISE 562; Dr. SE 562; Dr. Smith
• Repeat this process until the interviewee can’t choose (the differences will be too small)
75th percentile
• The value of x when this occurs will be the 75th percentile value.
• P(X≤x) = 0.75
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ISE 562; Dr. Smith
• We want to assess the probability distribution of our interviewee’s height in inches
• Questions noted in blue
• Answers noted in red
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ISE 562; Dr. Smith
• “What is the value of my height for which there is one chance in a hundred that my height will be less than that value?”
• “56 inches”
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ISE 562; Dr. Smith
• “What is the value of my height for which there is one chance in a hundred that my height will be more than that value?”
• “Let’s say 64 inches”
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ISE 562; Dr. Smith
• “What is the value of height for which there is an equal chance that height will be greater than the value and less than the same value? (The median.)”
• “62 inches.”
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ISE 562; Dr. Smith
• Do you think it is more likely the value of height is less than 59 inches or between 59 inches and 62 inches?
Here? Or here?
• “I think its more likely it’s less than 59
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25th percentile
ISE 562; Dr. Smith
• Do you think it is more likely the value of height is less than 57.5 inches or between 57.5 inches and 62 inches?
57.5 inches
Here? Or here?
• “I think its more likely it’s between 57.5
and 62 inches.”
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25th percentile
ISE 562; Dr. Smith
• Do you think it is more likely the value of height is less than 58.25 inches or between 58.25 inches and 62 inches?
58.25 inches
Here? Or here?
• “I think its more likely it’s between
58.25 and 62 inches.”
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25th percentile
ISE 562; Dr. Smith
• Do you think it is more likely the value of height is less than 58.625 inches or between 58.625 inches and 62 inches?
58.625 inches
Here? Or here?
• “I can’t really tell; let’s say 58.6.”
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25th percentile
P(X≤58.6) = 0.25
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75th percentile
Do you think it is more likely the value of height is between 62 and 63 inches or greater than 63 inches?
Here? Or here?
“It’s more likely that its lower than 63.”
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ISE 562; Dr. Smith
75th percentile
Do you think it is more likely the value of height is between 62 and 62.5 inches or greater than 62.5 inches?
62.5 inches
Here? Or here?
“I can’t tell beyond this—say 62.25.”
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ISE 562; Dr. (X≤62.25) = 0.75
ISE 562; Dr. • Now we can assemble the values
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1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
54 56 58 60 62 64 66
ISE 562; Dr. the moments • We can use Riemann-Stiejles integration to
calculate the mean and variance
• Redraw the CDF as series of step functions
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56 58.6 62 62.25 64
ISE 562; Dr. Smith
so E[x] 56 g(x) xdF(x) x jk
58.6(.25) 62.0(.25) 62.25(.25) 64.0(.25)
and2 V[x]E[x2]2
58.62 (.25) 62.02 (.25) 62.252 (.25) 64.02 (.25) 61.71252 3.823
or 1.96
bmm g(x)dh(x)limg(x )j g( )j
Computing the moments
a n k1 k1
Like this…
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62 62.25 64 29
What is the value of _____ for which there is one chance in a hundred that it will be less than that value?
What is the value of _____ for which there is one chance in a hundred that it will be greater than that value?
What is the value of _____ between the above for which there is an equal chance that it will be greater than the value and less than the same value?
(The median.)
[0th %-tile]
[100th %-tile]
[50th %-tile]
ISE 562; Dr. Template
[25th %-tile]
[75th %-tile]
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ISE 562; Dr. Smith
Class Exercise:
Assess the probability distribution for Dr. Smith’s weight in pounds or kilograms
Determine the 0, .25, .50, .75, and 1.0 percentile values; write them down
If you choose to use kilograms, convert the percentile values to pounds (multiply by 2.2)
Class Exercise
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