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Nir Friedman (opening)

Decision-Principles to Justify Carnap's Updating Method and to Suggest Corrections of Probability Judgments Peter P. Wakker Economics Dept. Maastricht University. Nir Friedman (opening). Good words. Bad words. dimension map density labels player ancestral generative dynamics bound

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Nir Friedman (opening)

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  1. Decision-Principles to Justify Carnap's Updating Method and to Suggest Corrections of Probability Judgments Peter P. WakkerEconomics Dept.Maastricht University Nir Friedman (opening)

  2. Good words Bad words dimension map density labels player ancestral generative dynamics bound filtering iteration ancestral graph agent Bayesian network learning elicitation diagram causality utility reasoning 2 - - - - - - - - - - - - + 0 + + 0 + - + + -

  3. 3 “Decision theory =probability theory + utility theory.” Bayesian networkers care about prob. th. However, why care about utility theory? (1) Important for decisions. (2) Helps in studying probabilities: If you are interested in the processing of probabilities, then still the tools of utility theory can be useful.

  4. 4 Outline 1. Decision Theory: Empirical Work (on Utty); 2. A New Foundation of (Static) Bayesianism; 3. Carnap’s Updating Method; 4. Corrections of Probability Judgments Based on Empirical Findings.

  5. 5 1. Decision Theory; Empirical Work (Hypothetical) measurement of popularity of internet sites. For simplicity, Assumption. We compare internet sites that differ only regarding (randomness in) waiting time. Question: How does random waiting time affect popularity of internet sites? Through average?

  6. 6 Problem: Users’ subjectively perceived costof waiting time may be nonlinear. More refined procedure: Not average of waiting time, but average ofhow people feel about waiting time, (subjectively perceived) cost of waiting time.

  7. 1 Subj.perc. of costs 5/6 4/6 3/6 2/6 1/6 0 waiting time (seconds) 3 5 7 0 9 14 20 7 Graph

  8. 8 How measure subjectively perceived cost of waiting time? For simplicity,Assumption. Internet can be in two states only: fast or slow. P(fast) = 2/3; P(slow) = 1/3.

  9. t´ ~ t1  slow fast    _ _ 1/3 1/3 (C(35)-C(25)) (C(35)-C(25))  ~  35 35 25 25 2/3 2/3 ~ =   . . .   = 9 Tradeoff (TO) method C(25) + C(t1) = C(35) + C(t0) EC slow 35 25 _ 1/3 C(t1)-C(t0)= (C(35)-C(25))  fast  2/3 t 0 (=t0) = C(t2)-C(t1)= t1 t2 . . . C(t6)-C(t5)= t5 t6

  10. Subj.cost 5/6 4/6 3/6 t3 t4 t5 waiting time 10 Consequently: C(tj) = j/6. Normalize: C(t0) = 0; C(t6) = 1. 1 2/6 1/6 0 t1 t2 t6 0 = t0

  11. 35 25 _ d1 d1 d1 1/3 C(t1)-C(t0)= (C(35)-C(25)) ~  2/3 0 t1  (=t0) = ! ?    _ _ 1/3 1/3 (C(35)-C(25)) (C(35)-C(25))  C(t2)-C(t1)= ~  35 35 25 25 d2 d2 d2 2/3 2/3 ~ t1 t2 = ! ?   . . . . . .   = ! ? C(t6)-C(t5)= t5 t6 11 Tradeoff (TO) method revisited unknown probs misperceived probs EC

  12. If pslow pslow 35 25 ~ fast1-p fast1-p 0 t1 (=t0) C(t1)–C(t0) C(35)–C(25)+C(t1)–C(t0) 12 Measure subjective/unknown probs from elicited choices: then p(C(35) – C(25)) = (1-p)(C(t1) – C(t0)), so p = P(slow) = Abdellaoui (2000), Bleichrodt & Pinto (2000),Management Science.

  13. 13 What if inconsistent data? Say, some observations show: C(t2) -C(t1) = C(t1) -C(t0). Other observations show: C(t2’) -C(t1) = C(t1) -C(t0), for t2’ > t2. Then you have empirically falsified EC model! Definition. Tradeoff consistency holds if this never happens.

  14. 14 Theorem. EC model holds  tradeoff consistency holds. Descriptive application: EC model falsified iff tradeoff consistency violated.

  15. 15 2. A New Foundation of (Static) Bayesianism Normative application: Can convince client to use EC iff can convince client that tradeoff consistency is reasonable.

  16. 16 3. Carnap’s Updating Method We examine: Rudolf Carnap’s (1952, 1980) ideas about the Dirichlet family of probty distributions.

  17. 17 Example. Doctor, say YOU, has to choose the treatment of a patient standing before you. Patient has exactly one (“true”) diseasefrom set D = {d1,...,ds} of possible diseases. You are uncertain aboutwhich the true disease is.

  18. 18 For simplicity:Assumption. Results of treatment can beexpressed in monetary terms. Definition. Treatment (di:1) : if true disease is di, it saves $1, compared to common treatment; otherwise, it is equally expensive.

  19. 19 d1 . . . di . . . ds treatment (di:1) 0 . . . 1 . . . 0 Uncertain which disease dj is true  uncertain what the outcome (money saved) of the treatment will be.

  20. 20 Assumption. When deciding on your patient, you have observed t similar patientsin the past, and found out their true disease. Notation. E = (E1,...,Et), Ei describes disease of ith patient.

  21. 21 Assumption. You are Bayesian. So, expected uility.

  22. 22 Imagine someone, say me, gives you advice: Given info E, probs are to be taken as follows:

  23. : obsvd relative frequency of di in E1,…,Et ni ni ni i i i p p p - statistical information i p 0 0 0 E - subject-matter info t t t ) ( ) ( (as are the ‘s) 23 t +  = + t >0: subjective parameter Appealing! Natural way to integrate Subjective parameters disappear as t . Alternative interpretation: combining evidence.

  24. 24 Appealing advice, but, a hoc! Why not weight t2 iso t? Why not take geometric mean? Why not have  depend on t and ni, and on other nj’s? Decision theory can make things less ad hoc. An aside. The main mathematical problem: to formulate everything in terms of the“naïve space,” as Grünwald & Halpern (2002) call it.

  25. 25 Forget about advice, for the time being. Let us change subject.

  26. 26 (1)Wouldn’t you want to satisfy: Positive relatedness of the observations. (di:1) ~E $x  ( ,di) E  (di:1) $x .

  27. 27 (2) Wouldn’t you want to satisfy: Past-exchangeability: (di:1) ~E $x  (di:1) ~E' $x whenever: E = (E1,...,Em-1,dj,dk,Em+2,...,Et) and E' = (E1,...,Em-1, , ,Em+2,...,Et) dk dj for some m < t, j,k.

  28. disjoint causality di attime t+1 31 next, 29 31 28 Et E1 Ej . . . . . . past-exchange-bility ni n1 ns . . . . . . ¬ni

  29. 29 (3) Wouldn’t you want to satisfy: Future-exchangeability Assume$x ~E (dj:y) and $y ~(E,dj) (dk:z). Interpretation:$x ~E(dj and then dk:z). Assume $x‘~E (dk:y’) and $y' ~(E,dk) (dj:z’). Interpretation:$x’ ~E(dk and then dj:z’). Now: x = x‘ z = z’. Interpretation: [dj then dk] is as likely as [dk then dj], given E.

  30. Othercause Badnutrition d3 d2 d1 Fig, 28 Fig, 28 30 (4) Wouldn’t you want to satisfy: Disjoint causality: for all E & distinct i,j,k, ( ,dj) E ( ,dk) E ~ ~  (di:1) $x (di:1) $x A violation:

  31. t +  = + t ni i p i p 0 E t 31 Decision-theoretic surprise: Theorem. Assume s3. Equivalent are: (i)(a)Tradeoff consistency; (b)Positive relatedness of obsns; (c)Exchangeability (past and future); (d) Disjoint causality. (ii) EU holds for each E with fixed U, and Carnap’s inductive method:

  32. 32 4. Corrections of Probability Judgments Based on Empirical Findings Abdellaoui (2000), Bleichrodt & Pinto (2000) (and many others): Subj.Probs nonadditive. Assume simple model: (A:x)  W(A)U(x) U(0) = 0; W nonadditive; may be Dempster-Shafer belief function. Only nonnegative outcomes.

  33. 33 Tversky & Fox (1995): two-stage model, W = w ; : direct psychological judgment of probability w: turns judgments of probability into decision weights. w can be measured from case where obj. probs are known.

  34. 34 Economists/AI: w is convex. Enhances: W(AB)  W(A) + W(B) if disjoint (superadditivity). (e.g., Dempster-Shafer belief functions).

  35. 35 Psychologists: 1 w 1 0 p

  36. 36 p, q moderate: w(p + q)  w(p) + w(q) (subadditivity) . The w component of W enhances subadditivity of W, W(A  B)  W(A) + W(B) for disjoint events A,B, contrary to the common assumptions about belief functions as above.

  37. 37 = winvW: behavioral derivation of judgment of expert. Tversky & Fox 1995: more nonlinearity in  than in w 's and W's deviations from linearity are of the same nature as Figure 3. Tversky & Wakker (1995): formal definitions

  38. 38 Non-Bayesians: Alternatives to the Dempster-Shafer belief functions. No degeneracy after multiple updating. Figure 3 for  and W: lack of sensitivity towards varying degrees of uncertainty Fig. 3 better reflects absence of information than convexity

  39. 39 Fig. 3: from data Suggests new concepts. e.g., info-sensitivity iso conservativeness/pessimism. Bayesians: Fig. 3 suggests how to correct expert judgments.

  40. 40 Support theory (Tversky & Koehler 1994). Typical finding: For disjoint Aj, (A1) + ... + (An) – (A1 ...  An) increases as n increases.

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