Subjective Probability - Human Decision Making - Lecture Slides, Slides of Human-Computer Interaction Design

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Subjective Probability

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Subjective Interpretation of Probability

 Frequentist Interpretation of Probability

 Long-term frequency of occurrence of an event  Requires repeatable experiments

 What about the Following Events?

 Core meltdown of a nuclear reactor  You being healthy at the age of 70  Oregon beating Stanford in their 1970 football game (unsure about the outcome although the event has happened)

Subjective interpretation of probability: an individual’s degree of belief in the occurrence of an event

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Measures of Uncertainty

 Probability

 Odds (For or Against)

 Odds for: θ(A)  Pr(Pr(A)A)

Pr(A)

Pr(A) θ(A )

log( (A))log(Pr(A))log(Pr(A))

log( (A))log(Pr(A))log(Pr(A))

 Odds against:

 Log Odds (For or Against)

 Log odds for:  Log odds against:

 Words

 Fuzzy qualifiers  e.g. common, likely, unusual, rare, etc.

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Assessing Discrete Probabilities

 Direct Methods

 Directly ask for probability assessment  Simple to implement  The decision maker may not be able to give a direct answer or place little confidence in the answer  Do not work well if the probabilities in questions are small

 Indirect Methods

 Formulate questions in the decision maker’s domain of expertise and then extract probability assessment through probability modeling  Betting approach  Reference lottery

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Football Example (Cont’d)

Step 1: set X =100, Y =0 (or other numbers which make the preference obvious)

Step 2: set X =0, Y =100 (a consistency check; the decision should be switched)

Step 3: set X =100, Y =100 (the comparison is not obvious anymore)

Continue till the decision maker is indifferent between the two bets Assumption: when the decision maker is indifferent between bets, the expected payoffs from the bets are the same

X ∙Pr(Ohio State wins) – Y ∙Pr(Michigan wins) = – X ∙Pr(Ohio State wins)+ Y ∙Pr(Michigan wins)  Pr(Ohio State wins) = Y /( X + Y )

e.g. Point of indifference at X =70, Y =100, Pr(Ohio State wins)=0.

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Betting Approach (Cont’d)

 Pro

 A straightforward approach

 Cons

 Many people do not like the idea of betting  Most people dislike the prospect of losing money (risk-averse)  Does not allow choosing any other bets to protect from losses

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Reference Lottery (Cont’d)

 Steps

 Step 1: Start with some p1 and ask which lottery the decision maker prefers  Step 2: If lottery 1 is preferred, then choose p2 higher than p1; if the reference lottery is preferred, then choose p2 less than p  Continue till the decision maker is indifferent between the two lotteries

Assumption: when the decision maker is indifferent between lotteries, Pr(Ohio State wins)=p

 Cons

 Some people have difficulties making assessments in hypothetical games  Some people do not like the idea of a lottery-like game

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Consistence and The Dutch Book

 Subjective probabilities must follow the laws of probability. If they

don’t, the person assessing the probabilities is inconsistent

 Dutch Book

 A combination of bets which, on the basis of deductive logic, can be shown to entail a sure loss

Ohio State will play Michigan in football this year. Your friend says that Pr(Ohio State wins) = 40% and Pr(Michigan wins) = 50%. You note that the probabilities do not add up to 1, but your friend stubbornly refuses to change his estimates. You think, “Great! Let us set up two bets.”

Football Example:

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Assessing Continuous Probabilities

 If the distribution of the uncertain event is known a priori (more in

Chapter 9)

 Assess the distribution parameters directly  Assess distribution quantities and then solve for the parameters

 If the distribution is not known a priori, we can assess several

cumulative probabilities and then use these probabilities to draft a

CDF

 Pick a few values and then assess their cumulative probabilities (fix x -axis)  Pick a few cumulative probabilities and then assess their corresponding x values (fix F( x )-axis)

X 0.25 is lower quartile X 0.5 is median X 0.75 is upper quartile

p

Xp^ x

F ( x )=Pr( X ≤ x )

( p fractile or 100 p th^ percentile)

X 0.3 is 0.3 fractile or 30th^ percentile

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Assess the current age (unknown), A, of a movie star

Method one: pick a few possible ages (say A=29, 40, 44, 50, and 65) and then assess their corresponding cumulative probabilities using the techniques for assessing discrete probabilities (such as the reference lottery approach)

Suppose the following assessments were made: Pr(A≤29)=0, Pr(A≤40)=0.05, Pr(A≤44)=0.5, Pr(A≤50)=0.80, Pr(A≤65)=

Reference Lottery for Assessing Pr(A≤40)

Age of Movie Start Example

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Method two: pick a few cumulative probabilities and then assess their corresponding values using the reference lottery approach

Typically, we assess extreme values (5th^ percentile and 95th^ percentile) , lower quartile (25th^ percentile), upper quartile (75th^ percentile) and median (50th percentile)

Reference Lottery for Assessing the Value Whose Cumulative Prob. is 0. (the 35th^ percentile)

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Using Continuous CDF in Decision Trees

 Monte Carlo Simulation (Not covered in this course; Chapter 11 of the

textbook)

 Approximate With a Discrete Distribution

 Use a few representative points in the distribution  Extended Pearson – Tukey method  Bracket median method

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Continuous chance node

Discrete approximation

CDF of the Age of the Movie Star

Age of Movie Star Example (Cont’d)

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Bracket Median Method

 The bracket median of an interval of CDF corresponding to [a,b] is a

value m* between a and b such that Pr(a≤ X ≤ m)=Pr(m≤ X ≤ b)

 Can approximate virtually any kind of distribution

 Steps

 Step 1: Divide the entire range of cumulative probabilities [0,100%] into several equally likely intervals (typically three, four, or five intervals; the more intervals, the better the approximation yet more computations)  Step 2: Assess the bracket median for each interval  Step 3: Assign equal probability to each bracket median (=100%/# of intervals )