Delivering Integrated, Sustainable, Water Resources Solutions Uncertainty, Probability, & Models Robert C. Patev NAD Regional Technical Specialist (978)

Delivering Integrated, Sustainable, Water Resources Solutions 1.What is risk 2.Nature of uncertainty 3.Probability theory 4.Probability models PRESENTATION OVERVIEW

Delivering Integrated, Sustainable, Water Resources Solutions 3 WHAT IS RISK?

Delivering Integrated, Sustainable, Water Resources Solutions 4 WHAT IS RISK? Risk (n.) 1.The possibility of suffering harm or loss; danger. 2.A factor, thing, element, or course involving uncertain danger; a hazard. 3.The danger or probability of loss to an insurer. b. The amount that an insurance company stands to lose. 4.The variability of returns from an investment. b. The chance of nonpayment of a debt. 5.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.

Delivering Integrated, Sustainable, Water Resources Solutions 5 WHAT IS RISK? Risk (n.) 1.The possibility of suffering harm or loss; danger. 2.A factor, thing, element, or course involving uncertain danger; a hazard. 3.The danger or probability of loss to an insurer. b. The amount that an insurance company stands to lose. 4.The variability of returns from an investment. b. The chance of nonpayment of a debt. 5.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.

Delivering Integrated, Sustainable, Water Resources Solutions 6 WHAT IS RISK? Risk (n.) 1.The possibility of suffering harm or loss; danger. 2.A factor, thing, element, or course involving uncertain danger; a hazard. 3.The danger or probability of loss to an insurer. b. The amount that an insurance company stands to lose. 4.The variability of returns from an investment. b. The chance of nonpayment of a debt. 5.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.

NORTH ATLANTIC DIVISION 7 Uncertainty Probability Impact Consequence Risk

Delivering Integrated, Sustainable, Water Resources Solutions 8 WHAT IS RISK? Risk Insurance industry: risk = consequence High Risk Low Risk Consequence

Delivering Integrated, Sustainable, Water Resources Solutions WHAT IS RISK? Risk Public health: risk = incidence (frequency or probability) High Risk Low Risk Probability

Delivering Integrated, Sustainable, Water Resources Solutions 10 WHAT IS RISK? Risk Management & engineering: risk = probability * consequence High Risk Low Risk Probability Consequence

NORTH ATLANTIC DIVISION Federal agency definitions of risk Quantitative Flood Risk C P PxC P

Delivering Integrated, Sustainable, Water Resources Solutions 12 WHAT IS RISK? High Risk Low Risk Probability Consequence High Risk Low Risk Probability Consequence

Delivering Integrated, Sustainable, Water Resources Solutions 13 WHAT IS RISK?

NORTH ATLANTIC DIVISION Federal agency response Quantitative Flood Risk What does “ yellow-and-a-half ” Mean?

Delivering Integrated, Sustainable, Water Resources Solutions WHAT IS RISK? 15

Delivering Integrated, Sustainable, Water Resources Solutions 16 NATURE OF UNCERTAINTY: WHAT DOES IT MEAN TO BE “UNCERTAIN?”

Delivering Integrated, Sustainable, Water Resources Solutions 17 NATURE OF UNCERTAINTY: = ≠

Delivering Integrated, Sustainable, Water Resources Solutions 18 NATURE OF UNCERTAINTY

Delivering Integrated, Sustainable, Water Resources Solutions 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.

Delivering Integrated, Sustainable, Water Resources Solutions 20 NATURE OF UNCERTAINTY AleatoryEpistemic

Delivering Integrated, Sustainable, Water Resources Solutions NATURE OF UNCERTAINTY

Delivering Integrated, Sustainable, Water Resources Solutions 22 WHAT IS PROBABILITY? “The theory of probability is at the bottom nothing but common sense reduced to calculus.” — Laplace 1812

Delivering Integrated, Sustainable, Water Resources Solutions 23 WHAT IS PROBABILITY? Mathematical probability (1654) Axioms of probability (Kolmogorov 1933) Frequency measures Aleatory uncertainty Degree-of-belief measures Epistemic uncertainty

Delivering Integrated, Sustainable, Water Resources Solutions Probability as relative frequency Crop experiments Pharmaceutical testing Games of chance Probability as degree of belief (or truth) Nuclear plant failure Major California earthquake Chance of car crash WHAT IS PROBABILITY? 24

Delivering Integrated, Sustainable, Water Resources Solutions 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)

Delivering Integrated, Sustainable, Water Resources Solutions 26 WHAT IS PROBABILITY? “It’s an excellent proof, but it lacks warmth and feeling”

Delivering Integrated, Sustainable, Water Resources Solutions Frank P. Ramsey 1903 – 1930 PARTIAL BELIEF 27

Delivering Integrated, Sustainable, Water Resources Solutions 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). PARTIAL BELIEF 28

Delivering Integrated, Sustainable, Water Resources Solutions 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.

Delivering Integrated, Sustainable, Water Resources Solutions 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

Delivering Integrated, Sustainable, Water Resources Solutions Verbal DescriptionProbability EquivalentLowHigh virtually impossible very unlikely unlikely fairly unlikely, rather unlikely fair chance, toss-up usually, good chance, probable, likely quite likely very likely, very probably virtually certain 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):

Delivering Integrated, Sustainable, Water Resources Solutions Engineering Risk and Reliability Subjective probability assessment

Delivering Integrated, Sustainable, Water Resources Solutions 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. QUANTIFYING SUBJECTIVE PROBABILITY

Delivering Integrated, Sustainable, Water Resources Solutions 50% 98% 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

Delivering Integrated, Sustainable, Water Resources Solutions PRESENTATION OVERVIEW What is “risk” What is “uncertainty” What is “probability” How well do people assign probabilities?