a bayesian perspective on info- gap decision theory ullrika sahlin, centre of environmental and...

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A Bayesian perspective on Info-Gap Decision Theory

Ullrika Sahlin, Centre of Environmental and Climate ResearchRasmus Bååth, Cognitive Science Lund University, Sweden

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Decision space D

Uncertainty space“uncertainty in parameters in an assessment model that

describes the consequences of the decisions on the system” A way to quantify uncertainty

or or + A sense of the strength of knowledge behind the assessment

An idea of what is a good decisione.g. rational, cautious or robust

Decision making

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Types of policy problems

Hage et al (2010). Futures

Decision making

Certainty about knowledge

HighLow

High

Low

moderately structured (scientific) problem

moderately structured (policy-ethical) problem

unstructured problem

structured problem

Norms/values consensus

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When do we have severe uncertainty?

Aven (2011). Risk Analysis.

Decision making

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When do we have severe uncertainty?

Not when– Large uncertainties in outcomes relative to the expected

values– A poor knowledge basis for the assigned probabilities – Large uncertainties about relative frequency-interpreted

probabilities (chances) p

Yes when– It is difficult to specify a set of possible consequences (state

space) (since it implies the next)– It is difficult to establish an accurate prediction model

Aven (2011). Risk Analysis.

Decision making

scientific

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Under risk

Under uncertainty

Under severe uncertainty

Decision making

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Info-Gap Decision Theory

Prof Yakov Ben-Haim Book and web page• Info-Gap Decision Theory:

Decisions under Severe Uncertainty

• http://info-gap.com/

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Info-Gap Decision Theory

IGDT is robust satisficing meant to evaluate decisions on the basis of robustness to loss when

uncertainty is unstructured

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Info-Gap Decision Theory

IGDT a method that is suitable when the information base is so depauperate that the analyst cannot parameterise a probability

distribution, decide on an appropriate distribution or even identify the lower or upper

bounds on a parameter

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Info-Gap Decision Theory

IGDT selects the decision which meets a given performance criterion under the largest possible

range of parameters with deep uncertainty

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Info-Gap Decision Theory

Instead of maximizing the expected net benefits of emissions control

we maximize the range of uncertainty under which the welfare loss from error in the

estimates the benefits and costs of emissions control can be limited

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Info-Gap Decision Theory

Publications per year Citations per year

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The steps of IGDT

1. Build and calibrate the assessment model by informing parameters

2. Expand uncertainty in parameters with severe uncertainty

where is a level of uncertainty and are the initial estimate of

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3. Define model to evaluate performance that includes a level of acceptability and cautiousness Reward function: Criterion on reward: Principle of cautiousness: consider worst case and look for the minimum reward over for a given level of uncertainty

The steps of IGDT

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4. Evaluate robustness for different requirements on what is acceptable performanceRobustness is the largest level of uncertainty that gives a satisficing reward

Decision A is more robust than B if

The steps of IGDT

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4. Evaluate robustness for different requirements on what is acceptable performanceRobustness is the largest level of uncertainty that gives a satisficing reward

Decision A is more robust than B if

The steps of IGDT

Opportunity is the smallest level of uncertainty that give a satisficing reward

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A robustness curve is the robustness for different acceptability on performance,

The steps of IGDT

Decision A is more robust than B if

𝑟𝑐

�̂� (𝑑 ,𝑟 𝑐)

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A robustness curve is the robustness for different acceptability on performance,

The steps of IGDT

𝑟𝑐

�̂� (𝑑 ,𝑟 𝑐)

Decision A is more robust than B if

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The wall against the sea

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The wall against the sea

is the height of the wall is the rise of the sea level is the loss per liter water that comes over the wallReward

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The wall against the sea

Unc in the rise of the sea level is severe

where

Unc in loss due to water is mild

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The wall against the sea

Acceptable reward

Robu

stne

ss

high wall

low wall

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The wall against the sea

Unc in the rise of the sea level is severe

Unc in loss due to water is mild severe

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Expanded uncertaintyThe wall against the sea

𝛼

𝛼

Loss

Sea

leve

l ris

e

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The wall against the sea

Acceptable reward

Robu

stne

ss

high wall

low wall

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Robustness curves

Stranlund and Ben-Haim (2008). J of Env Manag.

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Robustness curves

Korteling et al (2013). Water Resour Manage.

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Robustness curves

Parameter estimation error (%)

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Critique of IGDT

• Sensitivity to initial estimates• Localised nature of the analysis• Arbitrary parameterisation

– Combine multiple parameters• The ad hoc introduction of notions of plausibility when

applied in practice– Reactions when the curves cross– The meaning of α

• Paradox: while focus is on a few parameters with severe uncertainty it disregard parameters with mild uncertainty over-estimate robustness

Hayes et al (2013). Methods in Ecology and Evolution.

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Uncertainty the Bayesian way

Approaches to quantify uncertainty

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Uncertainty the Bayesian way

Approaches to quantify uncertainty

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Uncertainty the Bayesian way

Approaches to quantify uncertainty

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Uncertainty the Bayesian way

Approaches to quantify uncertainty

is someone’s degree of belief

is the degree of belief of an agent thinking like a Bayesian

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A Bayesian perspective on IGDT

• Can IGDT be integrated in a Bayesian framework?

• IGDT ≈ IMP Analysis with worst-case optimization

• IMprecise Prob ≈ Robust Bayesian AnalysisTroffaes and Gosling (2012). International Journal of Approximate Reasoning

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A Bayesian perspective on IGDT Step IGDT Bayesian IGDT

1 Build and calibrate the assessment model

No specific statistical framework for parameterisation

Priors from experts, Bayesian updating, Bayesian calibration

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3

4

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A Bayesian perspective on IGDT Step IGDT Bayesian IGDT

1 Build and calibrate the assessment model

No specific statistical framework for parameterisation

Priors from experts, Bayesian updating, Bayesian calibration

2 Define model to expand parameters with severe uncertainty, u

Nest non-probabilistic models for using α as the “highest level of uncertainty”

Assign an hierarchical probability model to each parameter and use nested priors that become less informative with increasing α

3

4

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A Bayesian perspective on IGDT Step IGDT Bayesian IGDT

1 Build and calibrate the assessment model

No specific statistical framework for parameterisation

Priors from experts, Bayesian updating, Bayesian calibration

2 Define model to expand parameters with severe uncertainty, u

Nest non-probabilistic models for using α as the “highest level of uncertainty”

Assign an hierarchical probability model to each parameter and use nested priors that become less informative with increasing α

3 Define model to evaluate performance including cautiousness

Consider worst case such as minimum utilityor maximum loss

Bayesian decision theory with cautiousness: e.g. minimise possible worst case rewards over the predictive posterior.

4

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A Bayesian perspective on IGDT Step IGDT Bayesian IGDT

1 Build and calibrate the assessment model

No specific statistical framework for parameterisation

Priors from experts, Bayesian updating, Bayesian calibration

2 Define model to expand parameters with severe uncertainty, u

Nest non-probabilistic models for using α as the “highest level of uncertainty”

Assign an hierarchical probability model to each parameter and use nested priors that become less informative with increasing α

3 Define model to evaluate performance including cautiousness

Consider worst case such as minimum utilityor maximum loss

Bayesian decision theory with cautiousness: e.g. minimise possible worst case rewards over the predictive posterior.

4 Evaluate robustness for different requirements of performance

Explore how much α that is needed to make sure worst case reward is at acceptable level

Apply Robust Bayesian Analysis to explore how α influence the worst case reward and evaluate robustness

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The wall against the sea

Unc in sea level

with hyper parameter

Unc in loss due to water

with hyper parameter

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The wall against the sea

Unc in sea level

with hyper parameter

Unc in loss due to water

with hyper parameter

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Expanded uncertaintyThe wall against the sea

𝛼

𝛼𝛼

𝛼

Loss

Sea

leve

l ris

e

Robustness curves

Acceptable reward

Robu

stne

ss

high wall

low wall

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A Bayesian perspective on IGDT

• Consider both mild and severe uncertainty when evaluating robustness

• Robustness is influenced by – mild uncertainty– cautiousness in relation to mild uncertainty

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A Bayesian perspective on IGDT

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A Bayesian perspective on IGDT

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A Bayesian perspective on IGDT

Priors for severe unc introduce noise in the Bayesian model to mimic a decreasing confidence in the assessment model

Acceptable reward

Robu

stne

ss

High confidence

Low confidence

A way to quantify uncertainty or or

+ A sense of the strength of knowledge behind the

assessment

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Conclusions

• Most would agree that, given data and models, the optimal way to quantify uncertainty is the Bayesian approach

• There is a need to moving back and forth into the Bayesian approach

• IGDT is useful for decision making under severe uncertainty• IGDT can be integrated in the Bayesian framework• Challenges remain how to combine multiple parameters and

to interpret robustness • We suggest to associate robustness to lack of confidence in

the assessment model

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Financial support from the Swedish research council FORMAS is highly appreciated