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
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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