1. 1 Subjective Probability for Modeling Uncertainty with Limited Data Efstratios Nikolaidis The University of Toledo April 2009

2. 2 Risky venture: need to model uncertainty Decision: Irrevocable allocation of resources to achieve desired payoff Outcomes of uncertain events affect payoff … …

3. 3 Selecting best course of action by comparing risk profiles Gas price Probability Return rate Probability Hybrid Diesel

4. 4 Objective probability is inadequate for most practical problems Long-term relative frequency Objective measure Most people understand concept of objective probability There is little data in most practical decisions Too expensive to collect data We cannot conduct a repeatable experiment for one-of- a-kind events –Fuel price in 2011 –Demand for cars in 2011 –Chance for a particular person to die in a car crash

5. 5 Subjective probability can help model uncertainty Principle 1: Probability is a decision maker’s (DM’s) belief that an outcome will materialize Principle 2: DM avoids risky venture that will result in sure loss Belief leads to inclination to act. Elicit it by observing how DM makes choices in the face of uncertainty. Observe inclination to accept gambles in controlled experiments

6. 6 Estimating subjective probability of a candidate winning 2008 U.S. presidential election by using trading data This ticket is worth $1 only if Mr. Obama wins 2008 presidential election Maximum buying price reflected a gambler’s belief that Mr. Obama would win election

7. 7 Trading data from 2008 U.S. presidential election (http://newsfutures.wordpress.com) P(win)=0.5 P(win)=0.8

8. 8 DM is decisive to avoid sure loss Eliciting expert’s probability

9. 9 Eliciting a decision maker’s probability distribution Estimating 5% percentile of water pump life if we cannot perform tests Life (hrs) CDF 0.05 2000 Real life experiment Reference experiment: wheel of fortune P(5% percentile  2000 hrs) =  /360 0 Ticket 1: Worth $1 only if life 5% percentile  2000 hrs Ticket 1: Worth $1 only if needle settles in sector 

10. 10 Combining judgments and data by using Bayes’ rule Example –Judgment: 1 per 10 pumps fail on average –Posterior = likelihood  prior  scaling constant –Subjective probability converges to relative frequency and epistemic uncertainty decreases with amount of data Data: 1 out of 20 pumps failed Data: 10 out of 200 pumps failed

11. 11 Lessons learnt In most practical decisions, we do not have enough data to estimate relative frequencies. Objective probability is inadequate for modeling uncertainty. Subjective probability enables decision-maker to model uncertainty on the basis of both judgment and data. Subjective probability has a solid theoretical justification derived from first principles. Can combine judgment with data by using Bayes’ rule. Subjective probability converges to relative frequency with amount of data increasing. Ambiguity aversion leads to indecision. Some people’s behavior is at odds with the precepts of subjective probability.