Modeling Uncertainty with Limited Data using Subjective Probability

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

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

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

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

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

9. Eliciting a decision maker’s probability distribution • Estimating 5% percentile of water pump life if we cannot perform tests Reference experiment: wheel of fortune Real life experiment CDF 0.05 Life (hrs) 2000 Ticket 1: Worth $1 only if needle settles in sector  Ticket 1: Worth $1 only if life 5% percentile  2000 hrs P(5% percentile  2000 hrs) = /3600

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: 10 out of 200 pumps failed Data: 1 out of 20 pumps failed

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.