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A Bayesian perspective on Info- Gap Decision Theory Ullrika Sahlin, Centre of Environmental and Climate Research Rasmus Bååth, Cognitive Science Lund University, Sweden 1

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Types of policy problems Hage et al (2010). Futures 3 Certainty about knowledge High Low 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? 4 Aven (2011). Risk Analysis.

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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 5 Aven (2011). Risk Analysis. scientific

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Under risk Under uncertainty Under severe uncertainty 6

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Prof Yakov Ben-HaimBook and web page Info-Gap Decision Theory: Decisions under Severe Uncertainty http://info-gap.com/ 7

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IGDT is robust satisficing meant to evaluate decisions on the basis of robustness to loss when uncertainty is unstructured 8

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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 9

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IGDT selects the decision which meets a given performance criterion under the largest possible range of parameters with deep uncertainty 10

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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 11

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Publications per yearCitations per year 12

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22 Acceptable reward Robustness high wall low wall

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Expanded uncertainty 24 Loss Sea level rise

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25 Acceptable reward Robustness high wall low wall

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26 Stranlund and Ben-Haim (2008). J of Env Manag. Korteling et al (2013). Water Resour Manage.

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27 Korteling et al (2013). Water Resour Manage.

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28 Hayes et al (2013). Methods in Ecology and Evolution. Regan et al (2005). Ecol Appl. Wintle et al. Nature Climate Change Letters. Parameter estimation error (%)

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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. 29

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Can IGDT be integrated in a Bayesian framework? IGDT ≈ IMP Analysis with worst-case optimization IMprecise Prob ≈ Robust Bayesian Analysis Troffaes and Gosling (2012). International Journal of Approximate Reasoning 34

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StepIGDTBayesian IGDT 1 Build and calibrate the assessment model No specific statistical framework for parameterisation Priors from experts, Bayesian updating, Bayesian calibration 2 3 4 35

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StepIGDTBayesian 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 3 4 36

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StepIGDTBayesian 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 3 Define model to evaluate performance including cautiousness Consider worst case such as minimum utility or maximum loss Bayesian decision theory with cautiousness: e.g. minimise possible worst case rewards over the predictive posterior. 4 37

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StepIGDTBayesian 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 3 Define model to evaluate performance including cautiousness Consider worst case such as minimum utility or 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 38

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Expanded uncertainty 41 LossSea level rise

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Acceptable reward Robustness high wall low wall

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Consider both mild and severe uncertainty when evaluating robustness Robustness is influenced by – mild uncertainty – cautiousness in relation to mild uncertainty 43

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Priors for severe unc introduce noise in the Bayesian model to mimic a decreasing confidence in the assessment model 46

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Acceptable reward Robustness High confidence Low confidence

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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 48

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Financial support from the Swedish research council FORMAS is highly appreciated 49

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