CHAPTER 8. SUBJECTIVE PROBABILITY
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1 CHAPTER 8. SUBJECTIVE PROBABILITY Frequentist interpretation of probability: Probability = Relative frequency of occurrence of an event Frequentist Definition requires one to specify a repeatable experiment. Example: Throwing a fair coin, Pr(Heads)=0.5 What about following events? Core Meltdown of Nuclear Reactor You being alive at the age of 50? Unsure of the final outcome, though event has occurred. One basketball team beating the other in next day's match Instructor: Dr. J. Rene van Dorp Chapter 8 - Page 202
2 Without doubt the above events are uncertain and we talk about the probability of the above events. These probabilities HOWEVER are NOT Relative Frequencies of Occurrences. What are they? Subjectivist Interpretation of Probability: Probability = Degree of belief in the occurrence of an event By assessing a probability for the events above, one expresses one's DEGREE OF BELIEF. High probabilities coincide with high degree of belief. Low probabilities coincide with low degree of belief. Why do we need it? Frequentist interpretation not always applicable. Allows to model and structure individualistic uncertainty through probability. Instructor: Dr. J. Rene van Dorp Chapter 8 - Page 203
3 ASSESSING SUBJECTIVE DISCRETE PROBABILITIES: Direct Methods: Directly ask for probability assessments. DO NOT WORK WELL IF EXPERTS ARE NOT KNOWLEDGABLE ABOUT PROBABILITIES DO NOT WORK WELL IF PROBABILITIES IN QUESTIONS ARE VERY SMALL (SUCH AS FOR EXAMPLE IN RISK ANALYSES) Indirect Methods: Formulate questions in expert's domain of expertise and extract probability assessment through probability modeling. Examples: Betting Strategies, Reference Lotteries, Paired Comparison Method for Relative Probabilities. Instructor: Dr. J. Rene van Dorp Chapter 8 - Page 204
4 1. Assessing Subjective Discrete Probabilities: Betting Strategies Event: Lakers winning the NBA title this season STEP 1: Offer a person to choose between following the following bets, where X=100, Y=0. Max Profit Bet for Lakers Bet against Lakers Lakers Win Lakers Loose Lakers Win Lakers Loose X -Y -X Y Instructor: Dr. J. Rene van Dorp Chapter 8 - Page 205
5 STEP 2: Offer a person to choose between following the following bets, where X=0, Y=100. (Consistency Check) Max Profit Bet for Lakers Bet against Lakers Lakers Win Lakers Loose Lakers Win Lakers Loose X -Y -X Y Instructor: Dr. J. Rene van Dorp Chapter 8 - Page 206
6 STEP 3: Offer a person to choose between following the following bets, where X=100, Y=100. Max Profit Bet for Lakers Bet against Lakers Lakers Win Lakers Loose Lakers Win Lakers Loose X -Y -X Y Instructor: Dr. J. Rene van Dorp Chapter 8 - Page 207
7 STEP 4: Offer a person to choose between following the following bets, where X=50, Y=100. Max Profit Bet for Lakers Bet against Lakers Lakers Win Lakers Loose Lakers Win Lakers Loose X -Y -X Y CONTINUE UNTIL POINT OF INDIFFERENCE HAS BEEN REACHED. Instructor: Dr. J. Rene van Dorp Chapter 8 - Page 208
8 Assumption: When a person is indifferent between bets the expected payoffs from the bets must be the same. Thus: X Pr(LW) - Y Pr(LL)= -X Pr(LW) + Y Pr(LL) 2 X Pr(LW) - 2 Y (1- Pr(LW) )=0 Pr(LW) = Y X + Y. 2 Example: X=50, Y=100 Pr(LW)= % Instructor: Dr. J. Rene van Dorp Chapter 8 - Page 209
9 2. Assessing Subjective Discrete Probabilities: Reference Loteries Event: Lakers winning the NBA title this season Choose two prices A and B, such that A>>B. Lakers Win Hawaiian Trip Lottery 1 Lottery 2 Lakers Loose (p) Beer Hawaiian Trip (1-p) Beer Instructor: Dr. J. Rene van Dorp Chapter 8 - Page 210
10 Lottery 2 is the REFERENCE LOTTERY and a probability mechanism is specified for lottery 2. Examples of Probability Mechanisms: Throwing a fair coin Ball in an urn Throwing a die, Wheel of fortune Strategy: 1. Specify p 1. Ask which one do you prefer? 2. If Lottery 1 is preferred offer change p i to p 1+i > p i. 3. If Lottery 2 is preferred offer change p i to p 1+i < p i. 4. When indifference point is reached STOP, else goto 2. Instructor: Dr. J. Rene van Dorp Chapter 8 - Page 211
11 Assumption: When Indifference Point has been reached Pr(LW) = p Consistency Checking: Subjective Probabilities must follow the laws of probability Example: If expert specifies Pr(A), Pr(B A) and Pr(A B) then Pr(B A) Pr(A) = Pr(A B) Instructor: Dr. J. Rene van Dorp Chapter 8 - Page 212
12 3. Pairwise Comparisons of Situations (Not on Final Exam) Issaquah class ferry on the Bremerton to Seattle route in a crossing situation within 15 minutes, no other vessels around, good visibility, negligible wind. Other vessel is a navy vessel Other vessel is a product tanker Instructor: Dr. J. Rene van Dorp Chapter 8 - Page 213
13 Question: 1 89 Situation 1 Attribute Situation 2 Issaquah Ferry Class - SEA-BRE(A) Ferry Route - Navy 1st Interacting Vessel Product Tanker Crossing Traffic Scenario 1 st Vessel miles Traffic Proximity 1 st Vessel - No Vessel 2nd Interacting Vessel - No Vessel Traffic Scenario 2 nd Vessel - No Vessel Traffic Proximity 2 nd Vessel - > 0.5 Miles Visibility - Along Ferry Wind Direction - 0 Wind Speed - Likelihood of Collision Situation 1 is worse <===================X==================>> 9: NINE TIMES MORE LIKELY to result in a collision. 7: SEVEN TIMES MORE LIKELY to result in a collision. 5: FIVE TIMES LIKELY to result in a collision. 3: THREE TIMES MORE LIKELY to result in a collision. 1: EQUALY LIKELY to result in a collision. Situation 2 is worse Instructor: Dr. J. Rene van Dorp Chapter 8 - Page 214
14 UNDERLYING PROBABILITY MODEL FOR PAIRWISE COMPARISON QUESTIONNAIRE 1 1 Traffic Scenario 1 = X 2 Traffic Scenario 2 = X Y (X) = Paired Comparison Vector including 2 - way interactions 2 Pr(Accident Propulsion Failure, X ) 1 1 = P e 0 B T Y ( X ) 3 Pr( Accident Pr(Accident Prop. Failure, X Prop. Failure,X 1 2 ) ) = Pe 0 Pe 0 β β T T Y ( X 1 ) Y ( X 2) = e β T 1 2 ( Y ( X ) Y ( X ) ) 4 LN 1 Pr( Accident Prop. Failure, X ) T 2 2 = β Pr(Accident Prop. Failure, X ) ( 1 Y ( X ) Y ( X ) ) Instructor: Dr. J. Rene van Dorp Chapter 8 - Page 215
15 SUBJECTIVE PROBABILITY: ASSESSING CONTINUOUS CDF S. Method 1: Ask for distribution parameters e.g. Normal(µ,σ) Method 2: Ask for distribution quantities and solve for parameters (Chapter 10) Method 3: Ask for shape of CDF e.g. by Assessing a Number of Quantiles Assessing Quantiles Definition: x p is the p-th quantile of random variable X F(x p )=Pr(X x p )=p Terminology: quantile, fractile, percentile, quartile Instructor: Dr. J. Rene van Dorp Chapter 8 - Page 216
16 Assessing CDF is often conducted by assessing a number of quantiles. Method 3A: Use quantile estimates to solve distribution parameters. Method 3B: Connect multiple quantile estimates by straight lines to approximate the CDF Example: Uncertain Event: Current Age of a Movie Actress (e.g..) Instructor: Dr. J. Rene van Dorp Chapter 8 - Page 217
17 STEP 1: You know age of actress is between 30 and 65 STEP 2: Consider Reference Lottery Age 46 Hawaiian Trip Lottery 1 Lottery 2 Age > 46 (p) Beer Hawaiian Trip (1-p) Beer You decide you are indifferent for p=0.5 Pr(Age 46)=0.5 Instructor: Dr. J. Rene van Dorp Chapter 8 - Page 218
18 STEP 3: Consider Reference Lottery Age 50 Hawaiian Trip Lottery 1 Lottery 2 Age > 50 (p) Beer Hawaiian Trip (1-p) Beer You decide you are indifferent for p=0.8 Pr(Age 50)=0.8 Instructor: Dr. J. Rene van Dorp Chapter 8 - Page 219
19 STEP 4: Consider Reference Lottery Age 40 Hawaiian Trip Lottery 1 Lottery 2 Age > 40 (p) Beer Hawaiian Trip (1-p) Beer You decide you are indifferent for p=0.05 Pr(Age 40)=0.05 Instructor: Dr. J. Rene van Dorp Chapter 8 - Page 220
20 STEP 5: Approximate Cumulative Distribution Function Pr(Age < Years) Years Instructor: Dr. J. Rene van Dorp Chapter 8 - Page 221
21 Use of REFERENCE LOTTERIES to assess quantiles: 1. Fix Horizontal Axes: Age 46 Hawaiian Trip Lottery 1 Lottery 2 Age > 46 (p) Beer Hawaiian Trip (1-p) Beer Strategy: Adjust Probability p until indifference point has been reached by using a probability mechanism. Instructor: Dr. J. Rene van Dorp Chapter 8 - Page 222
22 2. Fix Vertical Axes: Age x Hawaiian Trip Lottery 1 Lottery 2 Age > x (0.35) Beer Hawaiian Trip (0.65) Beer Strategy: Adjust the Age x until indifference point has been reached. Instructor: Dr. J. Rene van Dorp Chapter 8 - Page 223
23 OVERALL STRATEGY TO ASSESS CONTINUOUS CUMULATIVE DISTRIBUTION FUNCTION: STEP 1: Ask for the Median (50% Quantile). STEP 2: Ask for Extreme Values (0% quantile, 100% quantile). STEP 3: Ask for High/Low Values (5% Quantile, 95% Quantile). STEP 4: Ask for 1 st and 3 rd Quartile (25% Quantile, 75% Quantile). STEP 5 A: Approximate CDF through straight line technique STEP 5 B: Model CDF between assessed points STEP 5 C: Calculate Best Fit in a family of CDF s. Instructor: Dr. J. Rene van Dorp Chapter 8 - Page 224
24 USING CONTINUOUS CDF S IN DECISION TREES Advanced Approach: Monte Carlo Simulation (Chapter 11) Simple Approach: Use Discrete Approximation that well approximates the Expected value of the underlying continuous distributions 1. Extended Pearson Tukey-Method: A Continuous Fan Node is replaced by a Three Branch Uncertainy Node Extended Pearson Tukey-Method specifies what Three Outcomes to choose and which Three Probabilities to assign to these outcomes. Works well for Symmetric Continuous Distribution Functions Instructor: Dr. J. Rene van Dorp Chapter 8 - Page 225
25 95% Quantile Median 5% Quantile Pr(Age < Years) Years Age=30 Age= 40 (0.185) Age = 46 (0.630) Age = 65 Continuous Fan Age = 61 (0.185) Discrete Approximation Instructor: Dr. J. Rene van Dorp Chapter 8 - Page 226
26 Next, calculate The Expected Value of The Discrete Approximation: Age Pr(Age) Age*Pr(Age) E[Age] Four Point Bracket Median Method: A Continuous Fan Node is replaced by a Four Branch Uncertainy node STEP 1: Divide total range in four equally likely intervals STEP 2: Determine bracket median in each interval STEP 3: Assing equal probabilitities in to all bracket medians (0.25 in this case) Instructor: Dr. J. Rene van Dorp Chapter 8 - Page 227
27 Pr(Age < Years) Years Age=30 Age= 41 (0.25) Age= 44 (0.25) Age = 65 Continuous Fan Age= 48 (0.25) Age = 53 (0.25) Discrete Approximation Instructor: Dr. J. Rene van Dorp Chapter 8 - Page 228
28 Next, calculate The Expected Value of The Discrete Approximation: Age Pr(Age) Age*Pr(Age) E[Age] 46.5 Accuracy of Bracket Median method can be improved by using a five point approximation, a six point approximation, etc. until the approximated expected value does not change any more (beyond a specified accuracy level). Instructor: Dr. J. Rene van Dorp Chapter 8 - Page 229
29 PITFALLS: HEURISTICS AND BIASES Thinking probabilistically is not easy!!!! When eliciting expert judgment, expert use primitive cognitive techniques to make their assessments. These techniques are in general simple and intuitively appealing, however they may result in a number of biases. Representative Bias: Probability estimate is made on the basis of Similarities within a Group. One tend to ignore relevant information such as incidence/base rate. Instructor: Dr. J. Rene van Dorp Chapter 8 - Page 230
30 Example: X is the event that a person is sloppy dressed In your judgement Managers (M) are well dressed: Pr(X M)=0.1. In your judgement Computer Scientist are badly dressed. Pr(X C)=0.8. At a conference with 90% attendence of managers and 10% attendance of computer scientist you Observe a Person and notices that he dresses (particularly) sloppy. What do think is more likely?: "The person is a computer scientist" or "The persion is a manager" Instructor: Dr. J. Rene van Dorp Chapter 8 - Page 231
31 WHAT THE ANSWER SHOULD BE? Pr( C Pr( M X ) X ) = Pr( X C) Pr( C) Pr( X ) Pr( X M ) Pr( M ) Pr( X ) = Pr( X C) Pr( C) Pr( X M ) Pr( M ) = * 0.9 < 1 IN OTHER WORDS: It is more likely that this person is a manager than a computer scientist. Availability Bias: Probability estimate is made according to the ease with which one can retrieve similar events. Instructor: Dr. J. Rene van Dorp Chapter 8 - Page 232
32 Anchoring Bias: One makes first assessment (anchor) and make subsequent assessments relative to this anchor. Motivational Bias: Incentives are always present, such that people do not really say what they believe. DECOMPOSITION AND PROBABILITY ASSESSMENTS Break down problem into finer detail using probability laws until you have reached a point at which experts are comfortable in making the assessment in a meaning full manner. Next, aggregate the detail assessment using probability laws to obtain probability estimates at a lower level of detail. Instructor: Dr. J. Rene van Dorp Chapter 8 - Page 233
33 Stock Market Example Stock Price Market Stock Price Stock Price Up Stock Price Up Market Up Stock Price Down Stock Price Down Market Down Stock Price Up Stock Price Down Instructor: Dr. J. Rene van Dorp Chapter 8 - Page 234
34 EXPERT JUDGEMENT ELICITATION PROCEDURE STRUCTURED APPROACH TO CAPTURING AN EXPERTS KNOWLEDGE BASE AND CONVERT HIS\HER KNOWLEDGE BASE INTO QUANTITATIVE ASSESSMENTS. MODELERS SKILLED IN DECOMPOSITION AND AGGREGATION OF ASSESSMENTS NORMATIVE ELICITATION PROCESS = MULTIPLE CYCLES (AT LEAST 2) 1. DECOMPOSITION OF EVENT OF INTEREST TO A MEANINGFULL LEVEL FOR SUBSTANTIVE EXPERT EXPERTS SUBSTANTIVE 2. ELICITATION OF JUDGMENT OF SUBSTANTIVE EXPERT FACILI- TATED BY NORMATIVE EXPERT KNOWLEDGABLE ABOUT THE SUBJECT MATTER AND EXTENSIVE EXPERIENCE 3. AGGREGATION OF JUDGEMENTS BY NORMATIVE EXPERT Instructor: Dr. J. Rene van Dorp Chapter 8 - Page 235
35 Nuclear Regulatory Example Electrical Power Failure? Control System Failure? Cooling System Failure? Accident? Which probability estimates do we need to calculate: The Probability of An Accident? Instructor: Dr. J. Rene van Dorp Chapter 8 - Page 236
36 A = Accident, L = Cooling System Failure, N = Control System Failure, E = Electrical System Failure. STAGE 1: WORK BACKWORDS STEP 1: Assess the four conditional probabilities: Pr( A L, N), Pr( A L, N), Pr( A L, N), Pr( A L, N) STEP 2: Assess the two conditional probabilities: Pr( L E),Pr( L E) STEP 3: Assess the two conditional probabilities Pr( N E),Pr( N E) Instructor: Dr. J. Rene van Dorp Chapter 8 - Page 237
37 STEP 4: Assess the probability: Pr(E ) STAGE 2: AGGREGATE DETAILED PROBABILITY ESTIMATES TO ASSESS THE PROBABILITY OF AN ACCIDENT STEP 5: Apply Law of Total Probability Pr( A) = Pr( A L, N) Pr( L, N) + Pr( A L, N) Pr( L, N) + Pr( A L, N) Pr( L, N ) + Pr( A L, N ) Pr( L, N) Instructor: Dr. J. Rene van Dorp Chapter 8 - Page 238
38 STEP 6: Apply Law of Total Probability Pr( L, N) = Pr( L, N E) Pr( E) + Pr( L, N E) Pr( E) Do the same with: Pr( L, N), Pr( L, N),Pr( L, N) STEP 7: Apply Conditional Independence assumption Pr( L, N E) = Pr( L E) Pr( N E) STEP 8: Go Back to STEP 5 and substitute the appropriate value to calculate the probability of an accident. Instructor: Dr. J. Rene van Dorp Chapter 8 - Page 239
39 EXPERT JUDGMENT ELICITATION PRINCIPLES (Source: Experts in Uncertainty ISBN: by Roger M. Cooke) 1. Reproducibility: It must be possible for Scientific peers to review and if necessary reproduce all calculations. This entails that the calculational model must be fully specified and the ingredient data must be made available. 2. Accountability: The source of Expert Judgment must be identified (who do they work for and what is their level of expertise). Instructor: Dr. J. Rene van Dorp Chapter 8 - Page 240
40 3. Empirical Control: Expert probability assessment must in principle be susceptible to empirical control. 4. Neutrality: The method for combining/evaluating expert judgements should encourage experts to state true opinions. 5. Fairness: All Experts are treated equally, prior to processing the results of observation Instructor: Dr. J. Rene van Dorp Chapter 8 - Page 241
41 PRACTICAL EXPERT JUDGMENT ELICITATION GUIDELINES 1. The questions must be clear. Prepare an attractive format for the questions and graphic format for the answers. 2. Perform a dry run. Be prepared to change questionnaire format. 3. An Analyst must be present during the elicitation. 4. Prepare a brief explanation of the elicitation format and of the model for processing the responses. 5. Avoid Coaching. (You are not the Expert) 6. The elicitation session should not exceed 1 hour. Instructor: Dr. J. Rene van Dorp Chapter 8 - Page 242
42 COHERENCE AND THE DUTCH BOOK Subjective probabilities must follow The Laws of Probability. If they do not, the person assessing the probabilities is incoherent. Incoherence Possibility of a Dutch Book Dutch Book: A series of bets in which the opponent is guaranteed to loose and you win. Instructor: Dr. J. Rene van Dorp Chapter 8 - Page 243
43 Example: There will be a Basketball Game tonight between Lakers and Celtics and your friend says that Pr(Lakers Win)=40% and Pr(Celtic Win)=50%. You note that the probabilities do not add up to 1, but your friend stubbornly refuses to change his initial estimates. You think, "GREAT!" let's set up a series of bets! Instructor: Dr. J. Rene van Dorp Chapter 8 - Page 244
44 BET 1 BET 2 Max Profit Max Profit Lakers Win (0.4) $60 Celtics Loose (0.5) -$50 Lakers Loose (0.6) -$40 Celtics Win (0.5) $50 Note that, EMV of both bets equal $0 according to his probability assessments and can thus be considered fair and he should be willing to engage in both. Lakers Win : Bet 1 - You win $60, Bet 2: You Loose $50, Net Profit: $10 Lakers Loose: Bet 1 - You loose $40, Bet 2: You win $50, Net Profit: $10 Instructor: Dr. J. Rene van Dorp Chapter 8 - Page 245
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