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Decision Making under Uncertainty and Bayesian Probability Hanan S. Bell PhD March 4, 2015 What is a decision? • Alternatives – something to choose among • Outcomes – what you get based on your choice • Uncertainty and structure – how the outcomes are computed • Values – what is the worth of the outcomes Jane’s Party Problem • Jane wants to hold a birthday party for her daughter Sally. • Jane is trying to decide whether to hold it indoors or outdoors. • Jane would much prefer an outdoor party if the weather is sunny, but an outdoor party would be a disaster if it rains. Jane’s Values Indoors Outdoors Sun 50 100 Rain 60 0 • If the probability of rain is 5/11 (.454), then Jane’s expected value of either option is 54.5 • If the probability of rain is .3, then Jane’s expected value of indoors is .3*60+. 7*50=53 and Jane’s expected value of outdoors is .7*100+.3*0=70 • If the probability of rain is .7, then Jane’s expected value of indoors is .7*60+. 3*50=57 and Jane’s expected value of outdoors is .7*0+.3*100=30 • Jane maximizes expected value by being outdoors if the probability of rain is less than .454 But wait – what do you mean by probability? What is this expected value? It will either rain or it won’t and Jane either gets the good outcome or not. Subjectivist Bayesian Probability • Probability encodes all our information about an uncertain event • It reflects a state of information, not a state of things • It is not necessarily the same thing as frequency although it often is • Different people can have different states of information—hence different probabilities and different decisions Probability in weather and climate • Information sources ▫ Statistical history ▫ Current climate conditions ▫ Model results ▫ Experience • Note that ensemble results are not necessarily a complete probability—they are a mean over a set of varying initial conditions that may or may not be equally likely—systemic model flaws not always included. What is expected value • Expected value computed by the sum of multiplying each potential outcome value by its probability • It represents the mean value of the outcomes that could occur if the decision could be repeated multiple times • Note that the expected value may not be one of the outcome values that could actually occur in a single decision • Expected value maximizing decision makers expect to do well over many decisions even though some turn out badly Farmer John’s planting problem • John is deciding whether to plant his field with his regular crop or a drought resistant crop • The drought resistant seed is more expensive and has a lower yield in normal rainfall conditions • The drought resistant crop has a lower yield in a drought than under normal rain, but the regular seed has no net return in a drought Climate John’s Values Seed Type Resistant Regular Normal 50 100 Drought 60 0 • Note that this is exactly the same problem as Jane’s except that it depends on intra-seasonal or seasonal climate probabilities as opposed to daily weather probabilities—same solution • But the problem is too simple. John could plant part of his field with each type of seed (Jane can’t hold half of the party outside and half inside in the same way. • Let x=% of field planed with resistant seed; 1-x= % with regular seed • Let p = probability of drought • Expected value of planting = p*(60x)+(1-p)*(100(1-x)+50x) • To max expected value take derivative of above and set to zero and find relevant value of x • Oops—there is no x in the derivative—the expected value is linear in x Why doesn’t John hedge his bets? • We haven’t really talked about value of outcomes • Because we use expected value decision making, we are saying we are not risk averse • Most people are risk averse • Von-Neumann-Morgenstern utility transforms outcome values to a form such that maximizing expected utility will maximize utility • Other things that may impact values ▫ Time preferences ▫ Social preferences (public policy questions) Other issues • Is drought vs normal really the right issue? • Perhaps soil moisture, temperature, humidity, radiation are the right issues • In this case we have multiple uncertain variables with potential probabilities that are continuous, not binary. • These variables are not necessarily independent. Sequential decisions • Frequently, a decision problem involves multiple sequential decisions that arise as uncertainty evolves • For example, climate change=>there are decisions now and down the line • Formal structures such as decision trees or influence diagrams can be used to deal with these situations Summary • Probabilities encode all our information about an uncertain event. • Decisions can be set up in a way that maximizing expected utility maximizes the expected value of the outcomes. • Decision problems can have very complex structures. • Weather and climate are important uncertain variables in many questions facing individuals and societies.