Decision-Principles to Justify Carnap's Updating Method and to Suggest Corrections

of Probability JudgmentsPeter P. WakkerEconomics Dept.

Maastricht University

Nir Friedman (opening)

2

+

dimensionmap

densitylabelsplayer

ancestralgenerativedynamics

boundfiltering

iterationancestral

graph

Good words

agentBayesiannetworklearning

elicitationdiagramcausality

utilityreasoning

Bad words

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

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

5

(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

6

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.

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

8

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?

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) =

.

.

.

9

_ (C(35) C(25))

_ (C(35) C(25))

slowfast

t

t´

t1 ~

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

_ (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

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.

13

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?

Theorem. EC model holds

14

Descriptive application:EC model falsified ifftradeoff consistency violated.

tradeoff consistency holds.

15

Normative application: Can convince client to use ECiffcan convince client that tradeoff consistency is reasonable.

2. A New Foundation of (Static) Bayesianism

16

We examine:Rudolf Carnap’s (1952, 1980) ideas aboutthe Dirichlet family of probty distributions.

3. Carnap’s Updating Method

17

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.

For simplicity:Assumption. Results of treatment can be expressed in monetary terms.

18

Definition. Treatment (di:1) : if true disease is di, it saves $1, compared to common treatment; otherwise, it is equally expensive.

19

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.

20

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.

21

You are Bayesian.

So, expected uility.

Assumption.

22

Given info E, probs are to be taken as follows:

Imagine someone, say me, gives you advice:

23

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.

24

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!

25

Let us change subject.

Forget about advice, for the time being.

E

26

Positive relatedness of the observations.(di:1) ~E $x

(1) Wouldn’t you want to satisfy:

(di:1) $x . ( ,di)

27

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.

28

Ej. . . . . . Et

¬ni

di attime t+1

E1

ni ns. . . . . .n1

past-exchange-bility

disjoint causality

next, 29

31

31

29

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:

(di:1) $x ( ,dj)

30

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

31

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:

32

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

33

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):

34

W(AB) W(A) + W(B) if disjoint (superadditivity). (e.g., Dempster-Shafer belief functions).

Economists/AI: w is convex. Enhances:

p

w

1

1

0

35

Psychologists:

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

= 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

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

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

40

Support theory (Tversky & Koehler 1994). Typical finding:For disjoint Aj,

(A1) + ... + (An) – (A1 ... An)

increases as n increases.