nir friedman (opening)
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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 - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: Nir Friedman (opening)](https://reader035.vdocuments.mx/reader035/viewer/2022062721/5681379a550346895d9f3f78/html5/thumbnails/1.jpg)
Decision-Principles to Justify Carnap's Updating Method and to Suggest Corrections
of Probability JudgmentsPeter P. WakkerEconomics Dept.
Maastricht University
Nir Friedman (opening)
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+
dimensionmap
densitylabelsplayer
ancestralgenerativedynamics
boundfiltering
iterationancestral
graph
Good words
agentBayesiannetworklearning
elicitationdiagramcausality
utilityreasoning
Bad words
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“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.
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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.
Outline
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(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?
1. Decision Theory; Empirical Work
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More refined procedure:Not average of waiting time, but average ofhow people feel about waiting time,(subjectively perceived) cost of waiting time.
Problem: Users’ subjectively perceived costof waiting time may be nonlinear.
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Subj.perc.of costs
waiting time (seconds)
1
00 3 20
1/6
5/6
14
4/6
9
3/6
7
2/6
5
7
Graph
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For simplicity,Assumption.Internet can be in two states only:fast or slow.P(fast) = 2/3;P(slow) = 1/3.
How measure subjectively perceived cost of waiting time?
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C(25) + C(t1) = C(35) + C(t0)
_ (C(35) C(25))
Tradeoff (TO) method
t2
25 35
t1
~
t6
25 35
t5
~
25 35
0
slowfast (= t0)
EC
=C(t2) C(t1) ==
.
.
.
=
C(t6) C(t5) =
C(t1) C(t0) =
.
.
.
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_ (C(35) C(25))
_ (C(35) C(25))
slowfast
t
t´
t1 ~
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1
0
Subj.cost
waiting time
Normalize: C(t0) = 0; C(t6) = 1.
0=t0
t1 t6
1/6
5/6
t5
4/6
t4
3/6
t3
2/6
t2
Consequently: C(tj) = j/6.10
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_ (C(35) C(25))
t2
25 35
t1
~
t6
25 35
t5
~
~ 25
t1
35
0
(= t0)
=C(t2) C(t1) ==
.
.
.
=
C(t6) C(t5) =
C(t1) C(t0) =
.
.
.
_ (C(35) C(25))
_ (C(35) C(25))
Tradeoff (TO) method revisited11
misperceived probs
1
2
1
2
1
2
?
?
?
!
!
!
ECunknown probs
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12Measure subjective/unknown probs from elicited choices:
then
p(C(35) – C(25)) = (1p)(C(t1) – C(t0)),
sop =
C(35) – C(25) + C(t1) – C(t0)C(t1) – C(t0)
~25
t1
35
0
pslowfast1-p (= t0)
pslowfast1-p
If
P(slow) =
Abdellaoui (2000), Bleichrodt & Pinto (2000),Management Science.
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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.
What if inconsistent data?
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Theorem. EC model holds
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Descriptive application:EC model falsified ifftradeoff consistency violated.
tradeoff consistency holds.
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Normative application: Can convince client to use ECiffcan convince client that tradeoff consistency is reasonable.
2. A New Foundation of (Static) Bayesianism
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We examine:Rudolf Carnap’s (1952, 1980) ideas aboutthe Dirichlet family of probty distributions.
3. Carnap’s Updating Method
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Example. Doctor, say YOU, has to choose the treatment of a patient standing before you.
Patient has exactly one (“true”) disease from set D = {d1,...,ds} of possible diseases.
You are uncertain aboutwhich the true disease is.
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For simplicity:Assumption. Results of treatment can be expressed in monetary terms.
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Definition. Treatment (di:1) : if true disease is di, it saves $1, compared to common treatment; otherwise, it is equally expensive.
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treatment (di:1)d1 . . . di . . . ds
0 . . . 1 . . . 0
Uncertain which disease dj is true uncertain what the outcome (money saved) of the treatment will be.
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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.
Assumption.
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You are Bayesian.
So, expected uility.
Assumption.
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Given info E, probs are to be taken as follows:
Imagine someone, say me, gives you advice:
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pEi =ip0 +
ni
tt
+ t
(as are the ‘s)ip0
Appealing! Natural way to integrate- subject-matter info
ip0( )- statistical informationni
t( )
: obsvd relative frequency of di in E1,…,Etni
t > 0: subjective parameter
Subjective parameters disappear as t .
Alternative interpretation: combining evidence.
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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.
Appealing advice, but, a hoc!
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Let us change subject.
Forget about advice, for the time being.
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E
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Positive relatedness of the observations.(di:1) ~E $x
(1) Wouldn’t you want to satisfy:
(di:1) $x . ( ,di)
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Past-exchangeability:(di:1) ~E $x (di:1) ~E' $x
whenever:E = (E1,...,Em1,dj,dk,E
m+2,...,Et)
andE' = (E1,...,Em1, , ,Em+2,...,Et)
(2) Wouldn’t you want to satisfy:
dk dj
for some m < t, j,k.
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Ej. . . . . . Et
¬ni
di attime t+1
E1
ni ns. . . . . .n1
past-exchange-bility
disjoint causality
next, 29
31
31
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Future-exchangeabilityAssume $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.
(3) Wouldn’t you want to satisfy:
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(di:1) $x ( ,dj)
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Disjoint causality: for all E & distinct i,j,k,
(4) Wouldn’t you want to satisfy:
E~
E(di:1) $x ~( ,dk)
Badnutrition
Othercause
d2d1 d3
A violation:
Fig, 28
Fig, 28
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Theorem. Assume s3. Equivalent are: (i) (a) Tradeoff consistency;
Decision-theoretic surprise:
pEi =ip0 +
ni
tt
+ t
(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:
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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.
4. Corrections of Probability Judgments Based on Empirical Findings
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two-stage model, W = w ;: direct psychological judgment of probabilityw: turns judgments of probability into decision weights. w can be measured from case where obj. probs are known.
Tversky & Fox (1995):
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W(AB) W(A) + W(B) if disjoint (superadditivity). (e.g., Dempster-Shafer belief functions).
Economists/AI: w is convex. Enhances:
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p
w
1
1
0
35
Psychologists:
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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.
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= 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
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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
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Fig. 3: from dataSuggests new concepts. e.g., info-sensitivity iso conservativeness/pessimism.Bayesians: Fig. 3 suggests how to correct expert judgments.
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Support theory (Tversky & Koehler 1994). Typical finding:For disjoint Aj,
(A1) + ... + (An) – (A1 ... An)
increases as n increases.