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

Delivering Integrated, Sustainable, Water Resources Solutions

Uncertainty, Probability, & Models

Robert C. PatevNAD Regional Technical Specialist

(978) 318-8394


1. What is risk

2. Nature of uncertainty

3. Probability theory

4. Probability models

PRESENTATION OVERVIEW


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WHAT IS RISK?


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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.


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WHAT IS RISK?


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.




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WHAT IS RISK?


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.



NORTH ATLANTIC DIVISION

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Uncertainty

Probability

Impact

Consequence

Risk


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WHAT IS RISK?

RiskInsurance industry: risk = consequence

High Risk

Low Risk

Co

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WHAT IS RISK?

RiskPublic health: risk = incidence (frequency or probability)

High Risk

Low Risk

Probability


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WHAT IS RISK?

RiskManagement & engineering:

risk = probability * consequence

High Risk

Low Risk

ProbabilityC

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Federal agency definitions of risk

Quantitative Flood Risk

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WHAT IS RISK?

High Risk

Low Risk

Probability

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High Risk

Low Risk

ProbabilityC

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WHAT IS RISK?


Federal agency response

Quantitative Flood Risk

What does “ yellow-and-a-half ”Mean?


WHAT IS RISK?

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


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NATURE OF UNCERTAINTY:

=≠


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NATURE OF UNCERTAINTY


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Aleatory vs. epistemic uncertaintyAleatory 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.


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Aleatory Epistemic




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WHAT IS PROBABILITY?

“The theory of probability is at the bottom nothing but common sense reduced to calculus.”

— Laplace 1812


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Mathematical probability (1654)Axioms of probability (Kolmogorov 1933)

Frequency measures Aleatory uncertainty

Degree-of-belief measures Epistemic uncertainty

0 P(A) 1

P(S) 1,P() 0

P(E1 E2 ...) E i for E i disjointi


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


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PARTIAL BELIEF

Axioms of probabilityAny ‘meaning’ that adheres to the axioms is, de facto, ‘probability’

Relative frequencyClassical frequency (DeMoivre)Limiting frequency (Venn)Infinite trials (Fisher, Gibbs)

Degree of beliefNecessary (rational) degree of belief (H.Jeffreys)Subjective degree of belief (Ramsey, Savage,

deFinetti)“Radical probabilism” (R.Jeffrey)


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“It’s an excellent proof, but it lacks warmth and feeling”


Frank P. Ramsey 1903 – 1930

PARTIAL BELIEF

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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

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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.


QUANTIFYING SUBJECTIVE PROBABILITY

IntrospectionAsk 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 approachHow many dollars would it take to make the decision

maker indifferent about which side to bet on?Can solve for the probability


Verbal DescriptionProbability Equivalent

Low High

virtually impossible 0.01 0.00 0.05

very unlikely 0.10 0.02 0.15

unlikely 0.15 0.04 0.45

fairly unlikely, rather unlikely 0.25 0.02 0.75

fair chance, toss-up 0.50 0.25 0.85

usually, good chance, probable, likely

0.75 0.25 0.95

quite likely 0.80 .030 0.99

very likely, very probably 0.90 0.75 0.99

virtually certain 0.99 0.90 1.00


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


Engineering Risk and Reliability

Subjective probability assessment


Difficulties …People don’t have well-formed subjective

probabilitiesMany 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-aversionSome people have difficulty grasping the

hypothetical game that is asked of them.Some may fail to take the measuring seriously.



50%

98%


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


PRESENTATION OVERVIEW

What is “risk”What is “uncertainty”What is “probability”How well do people assign probabilities?

Documents

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