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CHAPTER 8: SUBJECTIVE PROBABILITY BASIC TENET OF DECISION ANALYSIS judgments about uncertainty can be expressed in probabilities UNCERTAINTY AND PERSONAL DECISIONS next job future job opportunities location to live buy or rent medical decisions investment decisions UNCERTAINTY AND PUBLIC POLICY Hurricane warning Earthquake prediction Environmental risk Greenhouse effect Diseases, e.g., AIDS Defense spending 1 PROBABILITY: A FREQUENCY VIEW probability based on long-run frequency - results subjectively extrapolated to current situation results may be one-of-a-kind - probability of nuclear power accident for a new nuclear power plant design PROBABILITY: A SUBJECTIVE INTERPRETATION subjective view: probability represents an individual's belief that an event will occur - ALWAYS based on a state of information - update when the state of information changes probability assessment is art with some scientific basis ASSESSING DISCRETE PROBABILITIES: 2 APPROACHES (3rd in Text) 1. Direct Question - what is the probability you will get an A in MAT 643? - requires DM who is comfortable providing probabilities - can use this method after using the 2nd method a couple times 2 2. Have decision-maker compare with a reference lottery + $1000 A Bet 1 Not A Bet 2 - $ 1000 + $1000 p 1-p - $1000 use probability wheel to find p such that you are indifferent between Bet 1 and 2 - advantage: probabilities won't be even Last Step - check for consistency QUESTIONS: 1. Did our definition of an “A” satisfy the clarity test? 2. How can we check for consistency? ASSESSING CONTINUOUS PROBABILITIES two strategies for assessing continuous CDF 3 consider the age of the a volunteer student 1. Set outcomes and assess probabilities specify age, x, and assess p = P[age < x] Age < =x Bet 1 Age > x Bet 2 + $1000 - $ 1000 + $1000 p - $1000 1-p 2. Set probabilities and determine outcomes specify P[age < x], and determine x Bet 1 + $1000 Age < =x - $ 1000 Age > x Bet 2 + $1000 0.25 - $1000 0.75 alternative approaches 4 - quartiles: 0.0, 0.25, 0.5, 0.75, 1.0 - 0.05 and 0.95 quartiles, 0.25 and 0.75, then 0.5 QUESTION: Which of the two approaches would be the best? CONVERTING CONTINUOUS DISTRIBUTIONS TO DISCRETE DISTRIBUTIONS (DPL NOTE: DPL TAKES CARE OF THIS. BEST TO USE 3-4 OUTCOMES.) see the decision tree for the outcomes and probabilities 1. Extended Pearson-Tukey Method (Keefer & Bodily, 1983) 0.185 0.05 fractile 0.63 0.185 median 0.95 fractile best for symmetric distributions OK for asymmetric distributions requires less assessments 2. Moment Matching Used by DPL 5 Example Normal(10,2) with 3 outcomes Normal_Variable [10] Low .167 Nominal .667 High .167 [6.54] 6.54 [10] 10 [13.5] 13.5 Example: Normal (10,4) with 3 outcomes Normal_Variable [10] Low .167 Nominal .667 High .167 [3.07] 3.07 [10] 10 [16.9] 16.9 QUESTION: What's the difference? Example: Normal (10,4) with 5 outcomes 6 Normal_Variable [10] Lowest .011 [-1.43] -1.43 Lower .222 [4.58] 4.58 Nominal .533 [10] 10 High .222 [15.4] 15.4 Highest .011 [21.4] 21.4 QUESTION: What's the difference? PITFALLS: HEURISTICS AND BIASES probability assessment is not easy heuristics: cognitive techniques we use to make probability assessments - Tversky & Kahneman, 1974 TOM W. EXAMPLE, p. 281 Representativeness make assessments on similarity and ignore base rates use of stereotypes misunderstanding of random processes 7 regression to the mean - extreme cases will be followed by less extreme cases Availability ease by which we can recall similar events from memory recent events may bias our assessments unbalanced reporting can bias assessments illusory correlation: events that happen together may be assumed to be related Anchoring & Adjusting choose initial anchor and then adjust if we begin with median, tend to underadjust - start with 0.05, 0.95, then 0.25, 0.75, then median Motivational Bias people tend to manage our expectations Overcoming Heuristics and Biases individuals can become good at probability assessment awareness of heuristics and biases helps 8 structured techniques help if assessment is difficult, try decomposition DECOMPOSITION & PROBABILITY ASSESSMENT break probability assessment into smaller, more assessable probabilities use influence diagram examples - probability of nuclear accident in a new reactor design - probability of use of a nuclear weapon by a specific country that supports terrorist activities COHERENCE & THE DUTCH BOOK (MONEY PUMP) subjective probabilities must obey axioms & laws of probability - coherent de Finetti's Dutch Book Theorem - if a person's probabilities are not coherent, you can set up a Dutch Book against them - Dutch Book: a series of bets that, in the long run, you are guaranteed to win. The longer you play, the more you win. (Money Pump) 9 EXAMPLE: suppose you assessed P[H]= 0.4 and P[T]=0.5 QUESTION: How many times would you like to play this bet? Bet 1 Not H H Bet 2 + $ 40 - $ 60 + $ 50 T Not T - $ 50 incoherence can be exploited de Finetti showed that only possible not to be exploited if subjective probabilities were coherent SUMMARY subjective probability assessing probabilities - discrete - continuous 10 converting continuous to discrete heuristics & biases - representativeness - availability - anchoring & adjusting - motivational bias decomposition & probability assessment coherence and money pumps 11