Uncertainty, Probability, & Models Robert C. Patev NAD Regional Technical Specialist

1. Uncertainty, Probability, & Models • Robert C. Patev • NAD Regional Technical Specialist • (978) 318-8394

2. PRESENTATION OVERVIEW • What is risk • Nature of uncertainty • Probability theory • Probability models

3. WHAT IS RISK?

4. WHAT IS RISK? • Risk (n.) • The possibility of suffering harm or loss; danger. • A factor, thing, element, or course involving uncertain danger; a hazard. • The danger or probability of loss to an insurer. b. The amount that an insurance company stands to lose. • The variability of returns from an investment. b. The chance of nonpayment of a debt. • One considered with respect to the possibility of loss. • tr.v. risked, risk·ing, risks • 1. To expose to a chance of loss or damage; hazard. See Synonyms at endanger. 2. To incur the risk of: His action risked a sharp reprisal.

5. WHAT IS RISK? • Risk (n.) • The possibility of suffering harm or loss; danger. • A factor, thing, element, or course involving uncertain danger; a hazard. • The danger or probability of loss to an insurer. b. The amount that an insurance company stands to lose. • The variability of returns from an investment. b. The chance of nonpayment of a debt. • One considered with respect to the possibility of loss. • tr.v. risked, risk·ing, risks • 1. To expose to a chance of loss or damage; hazard. See Synonyms at endanger. 2. To incur the risk of: His action risked a sharp reprisal.

6. WHAT IS RISK? • Risk (n.) • The possibility of suffering harm or loss; danger. • A factor, thing, element, or course involving uncertain danger; a hazard. • The danger or probability of loss to an insurer. b. The amount that an insurance company stands to lose. • The variability of returns from an investment. b. The chance of nonpayment of a debt. • One considered with respect to the possibility of loss. • tr.v. risked, risk·ing, risks • 1. To expose to a chance of loss or damage; hazard. See Synonyms at endanger. 2. To incur the risk of: His action risked a sharp reprisal.

7. Uncertainty Impact Probability Consequence Risk

8. WHAT IS RISK? • Risk • Insurance industry: risk = consequence High Risk Consequence Low Risk

9. WHAT IS RISK? • Risk • Public health: risk = incidence (frequency or probability) High Risk Low Risk Probability

10. WHAT IS RISK? • Risk • Management & engineering: • risk = probability * consequence High Risk Consequence Low Risk Probability

11. Quantitative Flood Risk Federal agency definitions of risk C P PxC PxC P

12. WHAT IS RISK? High Risk High Risk Consequence Consequence Low Risk Low Risk Probability Probability

13. WHAT IS RISK?

14. Quantitative Flood Risk Federal agency response What does “ yellow-and-a-half ” Mean?

15. WHAT IS RISK?

16. NATURE OF UNCERTAINTY: WHAT DOES IT MEAN TO BE “UNCERTAIN?”

17. NATURE OF UNCERTAINTY: ≠ =

18. NATURE OF UNCERTAINTY

19. NATURE OF UNCERTAINTY • Aleatory vs. epistemic uncertainty • Aleatory uncertainty is that of random or stochastic occurrences; natural variation in space or time; irresolvable. • Epistemic uncertainty is that due to limited knowledge or information.

20. NATURE OF UNCERTAINTY Aleatory Epistemic

21. NATURE OF UNCERTAINTY

22. WHAT IS PROBABILITY? • “The theory of probability is at the bottom nothing but common sense reduced to calculus.” • — Laplace 1812

23. WHAT IS PROBABILITY? • Mathematical probability (1654) • Axioms of probability (Kolmogorov 1933) • Frequency measures Aleatory uncertainty • Degree-of-belief measures Epistemic uncertainty

24. WHAT IS PROBABILITY? • Probability as degree of belief (or truth) • Nuclear plant failure • Major California earthquake • Chance of car crash • Probability as relative frequency • Crop experiments • Pharmaceutical testing • Games of chance

25. PARTIAL BELIEF • Axioms of probability • Any ‘meaning’ that adheres to the axioms is, de facto, ‘probability’ • Relative frequency • Classical frequency (DeMoivre) • Limiting frequency (Venn) • Infinite trials (Fisher, Gibbs) • Degree of belief • Necessary (rational) degree of belief (H.Jeffreys) • Subjective degree of belief (Ramsey, Savage, deFinetti) • “Radical probabilism” (R.Jeffrey)

26. WHAT IS PROBABILITY? “It’s an excellent proof, but it lacks warmth and feeling”

27. PARTIAL BELIEF • Frank P. Ramsey 1903 – 1930

28. PARTIAL BELIEF • Subjectivist view: • Probabilities are simply degree-of-beliefs by rational individuals, without basis in objective reality. • Two people may have different probabilities for the same uncertain event, and both be correct. • In practice, • it's hard to get people to tell us what their degrees of belief are, and they probably don’t know anyway • To get round this, Ramsey postulated that people need to 'put their money where their probabilities are'. • Fundamental meaning of subjective (Ramsey) probability: • the willingness to take action based on the belief (to bet).

29. PARTIAL BELIEF • Ramsey’s theory • Ramsey shows that these assumptions lead to a measure that satisfies the axioms of probability • Thus, partial belief measured in this way is a ‘probability’ and thus adheres to the mathematical properties of probabilities.

30. QUANTIFYING SUBJECTIVE PROBABILITY • Introspection • Ask the decision maker directly, what’s his or her belief about the probability that the event will occurs. • Might include surveys • Need to incorporate possibly several estimates • Betting approach • How many dollars would it take to make the decision maker indifferent about which side to bet on? • Can solve for the probability

31. QUANTIFYING SUBJECTIVE PROBABILITY Source: Lichtenstein, S., and J.R. Newman (1967). "Empirical scaling of common verbal phrases associated with numerical probabilities," Psychon. Sci., 9(10): 563-564.

32. Engineering Risk and Reliability Subjective probability assessment

33. QUANTIFYING SUBJECTIVE PROBABILITY • Difficulties … • People don’t have well-formed subjective probabilities • Many people do not like to bet, hard to get their opinion in this way • Risk-averse people had to make the bets small enough to rule out risk-aversion • Some people have difficulty grasping the hypothetical game that is asked of them. • Some may fail to take the measuring seriously.

34. QUANTIFYING SUBJECTIVE PROBABILITY • How good are we at quantifying uncertainty? • How may ships passed through the Panama Canal last year? • Best estimate • 25-75% bounds • 1-99% bounds 98% 50%

35. PRESENTATION OVERVIEW • What is “risk” • What is “uncertainty” • What is “probability” • How well do people assign probabilities?