1. EXPECTED VALUE OF INFORMATION Omkar Apahle

2. INFORMATION A B C

3. EXPECTED VALUE OF INFORMATION (EVI) EVI is required to negate the effects of, -overconfidence, -underestimation of risk and -surprise EVI is required often in, -Risk Analysis -Sensitivity Analysis - Decision problem

4. DEFINITION Expected Value of Information (EVI) is the integral over all possible posterior distributions of the opportunity loss prevented by improved information, weighted by probability of that information.

5. CLASSIFICATION Expected Value of Information (EVI) Expected Value of Imperfect Information (EVII) Expected Value of Including Uncertainty (EVIU) Expected Value of Ignoring Uncertainty (EVEU) Expected Value of Perfect Information (EVPI)

6. Example, Weather condition and Camping activity EVPI = Highest price the decision maker is willing to pay for being able to know “Weather Condition” before making camping decision. EVII = Highest price the decision maker is willing to pay for being able to know “Weather Forecast” before making camping decision.

7. CHARECTERISTICS OF EVI Expected Value of Information (EVI) can never be less than zero. No other information gathering / sharing activities can be more valuable than that quantified by value of perfect information.

8. SOURCES Sample data Expert judgments

9. EVI & BAYES RULE Bayesian analysis relies on both sample information and prior information about uncertain prospects. Bayesian analysis provides a formal representation of human learning. An individual would update his / her “subjective beliefs” after receiving new information.

10. EVI & Bayes rule continued …. Investment in stock market Expert will provide perfect information Perfect information = always correct P ( Expert says “Market Up” Market Really Goes Up ) = 1

11. Applying Bayes’ theorem P ( Market Up Exp Says “Up” ) = P ( Exp “Up” Market Up ) P (Market Up ) P ( Exp “Up” Market Up ) P (Market Up ) + P ( Exp “Up” Market Down) P (Market Down) EVI & Bayes rule continued ….

12. EVI & PRIOR DISTRIBUTION EVI depends on the prior distribution used to represent current information. Subject experts and lay people often produce distributions that are far too tight. New or more precise measurements are often found to be outside the reported error bars of old measurements.

13. Expected posterior probability before the results are known is exactly the prior probability content of that region. Expected value of posterior mean is equal to the prior mean. Prior variance = Posterior variance + Variance of posterior mean

14. UNCERTAINTY A random variable y is more uncertain than another random variable z if -y = z + random noise -Every risk averter prefers a gamble with payoffs equal to z to one with payoffs equal to y -The density of y can be obtained from density of z by shifting weight to the tails through a series of mean-preserving spreads.

15. Uncertainty continued….. The decision as to whether to include uncertainty or not purely depends on the decision maker Expected Value of Including Uncertainty (EVIU) Expected Value of Ignoring (Excluding) Uncertainty (EVEU)

16. NOTATION d ∈ Dis a decision chosen from space D x ∈ Xis an uncertain variable in space X L ( d, x )is the loss function of d and x f ( x )is prior subjective probability density on x x iu = E ( x ) is the value of x when ignored uncertainty E [ L ( d, x ) ] = ∫x L ( d, x ) f ( x ) dx is the prior expectation over x loss for d

17. Notation continued … Bayes’ decision d y = Min -1 d E [ L ( d, x )] Deterministic optimum decision ignoring uncertainty d iu = Min -1 d L ( d, x iu )

18. EVIU Expected Value of Including Uncertainty (EVIU) is the expectation of the difference in loss between an optimal decision ignoring uncertainty and Bayes’ decision. EVIU = E [ L ( d iu, x ) ] - E [ L ( d y, x ) ]

19. EVPI Expected Value of Perfect information (EVIU) is the expectation of the difference in loss between an Bayes’ decision and the decision made after the uncertainty is removed by obtaining perfect information on x. EVPI = E [ L ( d y, x ) ] - E [ L ( d pi (x), x ) ]

20. SOCRATIC RATIO Dimensionless index of relative severity of EVIU and EVPI EVIU S iu = EVPI

21. LOSS FUNCTIONS When ignoring uncertainty doesn’t matter ? Classes of common loss functions, 1.Linear 2.Quadratic 3.Cubic

22. LINEAR LOSS FUNCTION Assume, x iu = x d ∈ { d 1, d 2,…. d n } Loss function, a 1 + b 1 x if d = d 1 L ( d, x ) = a 2 + b 2 x if d = d 2 ……. a n + b n x if d = d n

23. Linear loss function continued…. E [ L ( d, x )] = L ( d, x ) = L ( d, x iu ) Bayes decision, d y = Min d -1 E [ L ( d, x )] = Min d -1 L ( d, x iu ) = d iu EVIU = E [ L ( d iu, x )] - E [ L ( d y, x )] = 0 Considering uncertainty makes no difference to decision and hence to the outcome.

24. QUADRATIC LOSS FUNCTION Let the loss function be, L ( d, x ) = k ( d – x ) 2 E [L ( d, x ) ] = k ( d 2 – 2 d E( x ) + x 2 ) On derivation we get, 2 d – 2E( x ) = 0 d y = E( x ) = d iu Uncertainty makes no difference in decision

25. CUBIC ERROR LOSS FUNCTION Decisions involving uncertain future demand L ( d, x ) = r ( d – x ) 2 + s ( d – x ) 3 r, s > 0 Henrion (1989) showed that, EVIU1 S iu = < EVPI3 Obtaining better information about x is better than including uncertainty ( in all cases ).

26. BILINEAR LOSS FUNCTION Newsboy problem How many newspapers to order ? d = newspaper to be ordered x = uncertain demand a = $ loses if ordered too many b = $ forgoes if ordered too few

27. Newsboy problem continued …. Loss function, a ( d – x )if d > x L ( d, x ) = where a, b > 0 b ( x – d )if d < x

28. Newsboy problem continued …. Uniform prior on x, with mean x and width w 1 / w if x - w/ 2 < x < x + w/ 2 f(x) = w > 0 0 otherwise d iu = x

29. Newsboy problem continued …. Paper Demanded (x) Probability density: f (x) xd w

30. Newsboy problem continued …. b ( x – d ) a ( d – x ) Loss Too fewToo many 0 Error (excess newspapers) = ( d – x )

31. Results EVPI = EVIU = S iu = EVIU / EVPI = ( a – b ) 2 / 4 a b Newsboy problem continued …. w a b / 2 ( a + b ) w ( a – b ) / 8 ( a + b )

32. Newsboy problem continued …. Socratic ratio is independent of the uncertainty EVIU does not increase with uncertainty value relative to the EVPI Considering uncertainty is more important than getting better information

33. CATASTROPHIC LOSS PROBLEM Plane-catching problem How long to allow for the trip to the airport ? d = decision x = uncertainty ( actual travel time ) k = marginal cost per minute of leaving earlier M = loss due to missing the plane

34. Plane-catching problem continued …. Loss function 0if d > x L ( d, x ) = k ( d – x ) + M if d < x k and M are positive M > k ( d – x ) for all d, x

35. Plane-catching problem continued …. - 60- 50- 40- 30- 20- 100 300 0 150 L ( d, x = 35 ): Loss as a function of d M : Loss due to missing plane k ( d – x ) = Wasted time d : departure time ( minutes before plane ) Loss (in min )

36. Plane-catching problem continued …. x is uncertain and the decision is subjective f ( x ) = subjective probability density function Baye’s decision ( d y ) will allow x such that, k f (d y ) = M

37. Plane-catching problem continued …. In case we ignore uncertainty d iu = x 0.5 where, x 0.5 = median value d iu will lead us to miss the plane half the time EVIU = E [ L ( d iu, x ) ] - E [ L ( d y, x ) ] EVPI = E [ L ( d y, x ) ]

38. HEURISTIC FACTORS Four heuristic factors contribute to understand the EVI 1.Uncertainty (about parameter value) 2. Informativeness (the extent to which the current uncertainty may be reduced)

39. Heuristic factors continued … 3. Promise (the probability that improved information will result in a different decision and the magnitude of the resulting gain) 4. Relevance (the extent to which uncertainty about the parameter to overall uncertainty)

40. Heuristic factors continued … Example, Whether to permit or prohibit use of a food additive Expected social cost = Number of life years lost θ = Risk from consuming additive = Excess cancer risk K = Expected social cost with use of substitute in case additive is prohibited

41. Heuristic factors continued … f 0 = probability distribution representing current information about θ f + = probability distribution representing θ is hazardous f- = probability distribution representing θ is safe L 0 = expected social cost if additive is permitted L 1 = expected social cost if additive is permitted / prohibited after research

42. Heuristic factors continued … EVI = L 0 – L 1 Condition: Additive is permitted if and only if L 0 > K Substantial chance that additive is riskier than alternative and should be prohibited

43. Prohibit additive use Permit additive use f - f + f 0 K L1 L1 L0 L0 θ Expected Social Loss

44. Effect of greater prior uncertainty Prohibit additive use Permit additive use f - f + f 0 K L1 L1 L0 L0 θ L - Expected Social Loss

45. Effect of greater informativeness Prohibit additive use Permit additive use f - f + f 0 K L1 L1 L0 L0 θ Expected Social Loss

46. Effect of greater promise Prohibit additive use Permit additive use f - f + f 0 K L1 L1 L0 L0 θ Expected Social Loss

47. Effect of relevance θ1 θ1 θ2 θ2 Permit additive useProhibit additive use

48. RISK PREMIUM How to measure the monetary value of risk ? Let, a = uncertain monetary reward w = initial wealth a + w = terminal wealth U ( w + a ) = Utility function

49. Selling price of risk R s = selling price of risk = sure amount of money a decision maker would be willing to receive to sell the risk a { w + R s } ~* { w + a } Under expected utility function, U {w + R s } = EU {w + a}

50. Bid price of risk R b = bid price of risk = sure amount of money a decision maker would be willing to pay to buy the risk a { w } ~* { w + a - R b } Under expected utility function, U {w} = EU {w + a - R b }

51. Risk premium R = risk premium = sure amount of money one would be willing to receive to become indifferent between receiving the risky return a versus receiving the sure amount [ E ( a ) – R ] { w + a } ~* { w + E ( a ) - R } EU {w + a } = U { w + E ( a ) - R }

52. Risk premium continued … v = { w + E ( a ) - R } = certainty equivalent R = E [ v ] – vE [ v ] = expected loss Willingness to insure

53. COMBINING PRIORS A single expert report an idiosyncratic perception of a consensus Useful if use combined judgments Aggregation procedure – Weighted average – Bayes’ rule – Copula method

54. Example Climate sensitivity ( Morgan and Keith, 1995) ∆ T 2x = equilibrium increase in global annual mean surface temperature as a result of doubling of atmospheric CO 2 from its pre- industrial concentration

55. Example continued … Estimates gathered from different experts. All experts are treated equal. Range given by IPCC: 1.5 to 4.5 0 C Most likely value : 2.5 0 C Tails extended to account underestimation Sensitivity analysis : include / exclude expert 5

56. Example continued … ∆ T 2x Experts

57. Example continued … PDF All Experts Excluding 5 All experts with exponential tails

58. BEST ESTIMATE x iu = Mean of the SPD What to choose – mean, median or mode ? If Mean >> Median Make decision based on median and ignore uncertainty If Mean ~ Median Make decision considering possibility of extreme scenarios

59. IN CONCLUSION “As for me, all I know is I know nothing.” Socrates Expected Value of Information depends upon the expected benefits of Socratic wisdom (i.e. admitting one’s limits of knowledge) relative to the expected benefits of perfect wisdom (i.e. knowing the truth).

60. THNAK YOU !