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)
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Good words
dimension
map
density
labels
player
ancestral
generative
dynamics
bound
filtering
iteration
ancestral
graph
+
Bad words
agent
Bayesian
network
learning
elicitation
diagram
causality
utility
reasoning
0
+
+
0
+
+
+
-
“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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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.
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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?
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Problem:
Users’ subjectively perceived cost
of waiting time may be nonlinear.
More refined procedure:
Not average of waiting time, but
average of
how people feel about waiting time,
(subjectively perceived) cost of waiting time.
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Graph
Subj.
perc.
of
costs
1
5/6
4/6
3/6
2/6
1/6
0
0
3 5 7 9
14
20
waiting time
(seconds)
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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.
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Tradeoff (TO) method
=
C(25) + C(t1) = C(35) + C(t0)
EC


slow 25
slow 35
1/3
_
~
C(t
)
C(t
)
=
(
C(35) - C(25))


1
0
fast tt´1
fast 0
2/3

 (= t0)
 35
 25

t1
.
.
.
.
.
.

t6
~

t5
=
 35
 25
=

t2
~
1/3
_
C(t2) - C(t1) = (C(35) - C(25))
2/3
1/3
_
C(t6) - C(t5) = (C(35) - C(25))
2/3
Normalize: C(t0) = 0;
C(t6) = 1.
Subj.
cost
Consequently: C(tj) = j/6.
1
5/6
4/6
3/6
2/6
1/6
0
0
=
t0
t1 t2 t3 t4
t5
t6
waiting time
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Tradeoff (TO) method revisited
EC
 35
 25
t1
 35
 25
~

t1

t6

t5
=
 35
~
d_1
1/3
C(t2) - C(t1) = (C(35) - C(25))
d2
2/3
?!
.
.
.
.
.
.
 25
?!
=

t2
0
 (= t )
0
d1
1/3
_
C(t1) - C(t0) = (C(35) - C(25))
d2
2/3
=

~
unknown probs
misperceived probs
?!
d_1
1/3
C(t6) - C(t5) = (C(35) - C(25))
d2
2/3
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Measure subjective/unknown
from elicited choices:
If
then
p
slow 25
fast t
1
1-p
~
probs
p
35
slow
fast 0
1-p (= t )
0
p(C(35) – C(25)) = (1-p)(C(t1) – C(t0)),
so
C(t1) – C(t0)
P(slow) = p =
C(35) – C(25) + C(t1) – C(t0)
Abdellaoui (2000), Bleichrodt & Pinto (2000),
Management Science.
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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.
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Theorem. EC model holds 
tradeoff consistency holds.
Descriptive application:
EC model falsified iff
tradeoff consistency violated.
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2. A New Foundation of (Static) Bayesianism
Normative application:
Can convince client to use EC
iff
can convince client that tradeoff consistency is
reasonable.
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3. Carnap’s Updating Method
We examine:
Rudolf Carnap’s (1952, 1980) ideas about
the Dirichlet family of probty distributions.
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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 about
which the true disease is.
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For simplicity:
Assumption. Results of treatment can be
expressed in monetary terms.
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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Assumption.
When deciding on your patient,
you have observed t similar patients
in the past,
and found out their true disease.
Notation.
E = (E1,...,Et),
Ei describes disease of ith patient.
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Assumption.
You are Bayesian.
So, expected uility.
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Imagine someone, say me, gives you advice:
Given info E, probs are to be taken as follows:
p Ei =
ni
+ t
t
+t
0
p
 i
ni
: obsvd relative frequency of di in E1,…,Et
t
 > 0: subjective parameter
(as are the p 0‘s)
i
Appealing! Natural way to integrate
- subject-matter info (p 0i )
ni
- statistical information ( )
t
Subjective parameters disappear as t  .
Alternative interpretation: combining evidence.
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Appealing advice, but, a hoc!
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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) cal
it.
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Forget about advice, for the time being.
Let us change subject.
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(1) Wouldn’t you want to satisfy:
Positive relatedness of the observations.
(di:1) ~E $x

( E,di)
(di:1)  $x .
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(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, ,dk,E
djm+2,...,Et)
for some m < t, j,k.
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E1
n1
...
...
Ej
ni
...
...
¬ni
di at
time t+1
Et
pastexchangebility
ns
disjoint
causality
next,
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31
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(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.
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(4) Wouldn’t you want to satisfy:
Disjoint causality: for all E & distinct i,j,k,
( E,dj)
(di:1) ~
$x
( E,dk)
 (di:1) ~
$x
A violation:
Bad
nutrition
d1
Other
cause
d2
d3
Fig,
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Fig,
28
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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:
ni
0
pi + t
t
p Ei =
+t
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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.
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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.
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Economists/AI: w is convex. Enhances:
W(AB)  W(A) + W(B) if disjoint
(superadditivity).
(e.g., Dempster-Shafer belief functions).
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Psychologists:
w
1
0
1
p
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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 data
Suggests 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.