ICT2012 Birkbeck — People inductively reason causality by calculating the probability of De Finetti’s biconditonal event – The pARIs rule, rarity assumption, and equiprobability — 2012 07Jul-06

1. People inductively reason causality
by calculating the probability of
DeFinetti’s biconditonal event –
The pARIs rule, rarity assumption,
and equiprobability
Junki Yokokawa and Tatsuji Takahashi
Jul 6th, 2012
Birkbeck, Univ. of London, UK
1

2. Summary
★ Our intuition for generative causality from co-
occurrence data is the probability of
biconditional event (or defective
biconditional).
★ In causal induction, biconditional event focuses on
rare events and neglects abundant events, in the
uncertain world.
★ pARIs: proportion of assumed-to-be rare instances
★ Biconditional event is turning out to have strong
normative nature and theoretical grounds, so
possibly will be proven to be normative as well.
2

3. Overview
★ DeFinetti's biconditional event and new paradigm
psychology of reasoning.
★ biconditional event in causal induction:
★ pARIs (proportion of assumed-to-be rare instances)
★ Meta-analysis to confirm the validity of pARIs
★ Three experiments to give candidate rationales to
pARIs
★ Theoretical background and connections to other
areas, such as:
★ Developmental study of conditionals by Gauffroy and
Barouillet (2009),
★ Amos Tversky's study of similarity (1977), and so on
3

4. toc
★ conditional and biconditional event
★ biconditional event in causal induction: pARIs
(proportion of assumed-to-be rare instances)
★ Meta-analysis
★ Three experiments
★ Theoretical background
4

5. ★ conditional and biconditional event
★ biconditional event in causal induction: pARIs
(proportion of assumed-to-be rare instances)
★ Meta-analysis
★ Three experiments
★ Theoretical background
5

6. de Finetti's conditional
event
★ The probability of "if p then q" is the conditional
probability P(q|p).
★ It neglects not-p cases.
★ "q|p" is itself a (conditional) event.
material conditional conditional biconditional
conditional event event event
p q p⊃q q|p p|q p⟛q
T T T T T T
T F F F V F V: void case
F T T V F F
F F T V V V
6

7. ★ conditional and biconditional event
★ biconditional event in causal induction: pARIs
(proportion of assumed-to-be rare instances)
★ Meta-analysis
★ Three experiments
★ Theoretical background
7

8. Causal induction
★ Diagnostic:
★ Example: we often want to know the cause of a health
problem.
★ I sometimes have stiff shoulders and a headache.
What's the cause? How about coffee?
★ How frequently I got a headache after having a cup of
coffee? ...
8

9. Causal induction experiment
Stimulus presentation: a
pair of two kinds of pictures
illustrating the presence and
absence of cause and effect, at
left and right, respectively
Response: participants
evaluate the causal intensity they
felt from 0 to 100, using a slider
E ¬E
C a b
¬C c d
9

10. Causal (intensity) induction
★ Here, not structure but intensity
★ Two phases of causal induction (Hattori & Oaksford 2007)
★ Phase 1: observational (statistical)
★ Phase 2: interventional (experimental)
★ We focus on causal induction of the phase 1 for
generative cause because preventive causes
are confusing and hard to treat especially in the
observation phase (Hattori & Oaksford, 2007).
10

11. Causal Induction
★ Here we study about the intensity.
★ Recent studies emphasize the structure (Bayes
network topology) rather than the intensity (node
weight)
★ But not about "structure vs. intensity" or one's
ascendancy.
★ Many problems about intensity remain untouched.
★ Why ∆P doesn't fit the data?
★ Structure and intensity, mutual relationship.
★ In an unknown situation, intensity is what matters
since structure is not known.
11

12. ∆P = P (E|C) − P (E|¬C) = (a + b)(c + d)
Framework and models of
(a + b)(c + d)
causal induction
∆P
∆P = P (E|C) − P (E|¬C) =
PowerPC =
ad − bc
(a + b)(c + d)
1 − P (E|¬C)
★ The ∆P =(input) is co-occurrence of the target
data P (E|C) − P (E|¬C)
effect (E) and= candidate cause (C).
a ∆P
∆P
PowerPC =
PowerPC
1 − P (E|¬C)
1 − P (E|¬C)
★ Normative: Power PC (Cheng, 1997)
∆P = P (E|C) − P (E|¬C)
★ Descriptive: H ∆P(Dual Factor bc
ad − Heuristics)
PowerPC = Oaksford 2007)
(Hattori & ∆P=
PowerPC − P (E|¬C) (a + b)d
1 =
1 − P = ad − bc
∆P = P (E|C) − P (E|¬C)
(E|¬C)
(a + b)(c + d)
∆P ad − bc E ¬E
PowerPC = ∆P = ad − bc
PowerPC = 1 − P (E|¬C) = (a + b)d
∆P = P (E|C) − PP (E|¬C)
ad − bc C a b
1 − (E|¬C) = (a + b)d
(a + b)(c + d)
a ¬C c d
H= P (E|C)P (C|E) =
∆P ∆P (a +ad −+ c)
b)(a bc
PowerPC =
PowerPC = = 12

13. The pARIs rule
★ The frequency information of rare instances
conveys more information than abundant instances
(rational analysis and rarity assumption,
see esp. McKenzie 2007).
★ Because of the frame problem-like aspect, the d-
cell information can be unreliable (depends
strongly on how we frame and count).
★ Hence we calculate the causal intensity only by the
proportion of assumed-to-be rare instances
(pARIs)
★ named after pCI: proportion of confirmatory
instances, White 2003.
13

14. H= Rarity(C|E)
P (E|C)P assumption
★ We assume the effect in focus and the candidate
a
cause to be rare: P(C) and P(E) to be small.
H= P (E|C)P (C|E) =
★ Originally in Oaksford + b)(a + c)
(a & Chater, 1994,
★ then in Hattori & Oaksford, 2007, McKenzie 2007, in the
H= study (C|E) =induction a
P (E|C)P of causal
(a + b)(a + c)
★ C and E to take small proportion in U.
U
lim φ = P (E|C)P (C|E) = H C E
d→∞
ba c
lim φ = rarity
extreme P (E|C)P (C|E) = H d
d→∞
14

15. The pARIs rule
★ C and E are both assumed to be rare (P(C) and
P(E) low)
★ pARIs = proportion of assumed-to-be rare
instances (a, b, and c).
pARIs = P(C iff E) = P(C and E | C or E)
P(C and E) a
= =
P(C or E) a+b+c
E -E U
C E
C a b
ba c d
-C c d
15

16. ★ conditional and biconditional event
★ biconditional event in causal induction: pARIs
(proportion of assumed-to-be rare instances)
★ Meta-analysis
★ Three experiments
★ Theoretical background
16

17. Data-fit of pARIs and PowerPC
AS95 BCC03exp1generative BCC03exp3 H03
100 100 100 100
80 80 80 80
Human rating
Human rating
Human rating
Human rating
60 60 60 60
40 40 40 40
20 20 20 20
0 0 0 0
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
Model prediction Model prediction Model prediction Model prediction
H06 LS00exp123 W03JEPexp2 W03JEPexp6
100 100 100 100
80 80 80 80
Human rating
Human rating
Human rating
Human rating
60 60 60 60
40 40 40 40
20 20 20 20
0 0 0 0
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
Model prediction Model prediction Model prediction Model prediction
17

18. Meta-analysis
★ Fit with experiments (the same as Hattori & Oaksford, 2007)
★ pARIs fits the data set with the lowest correlation r < 0.89,
the highest average correlation in almost all the data, and the
smallest average error.
best next best bad otherwise
experiment model pARIs DFH PowerPC ∆P Phi P(E|C) P(C|E) pCI
AS95 0.94 0.95 0.95 0.88 0.89 0.91 0.76 0.87
BCC03: exp1 0.98 0.97 0.89 0.92 0.91 0.82 0.51 0.92
BCC03: exp3 0.99 0.99 0.98 0.93 0.93 0.95 0.88 0.93
H03 0.99 0.98 -0.09 0.01 0.70 -0.01 0.98 0.40
H06 0.97 0.96 0.74 0.71 0.71 0.89 0.58 0.70
LS00 0.93 0.95 0.86 0.83 0.84 0.58 0.34 0.83
W03.2 0.90 0.85 0.44 0.29 0.55 0.47 0.18 0.77
W03.6 0.93 0.90 0.46 0.46 0.46 0.77 0.56 0.54
average r 0.95 0.94 0.65 0.63 0.75 0.67 0.60 0.75
average error 11.97 18.48 33.39 24.30 27.18 27.78 24.75 29.93
Values other than in error row are correlation coefficient r.
18

19. correlation
Cor
AS95 BCC03exp1 BCC03exp3 H03 H06 LS00 W03.2
7.00
0.90 0.85
0.93 0.95 0.55
5.25 0.77
0.44 0.29 0.84
0.97 0.96 0.47 0.83
0.86 0.18
0.83 0.71 0.58 0.34
3.50 0.99 0.98 0.58 0.70
0.74 0.71 0.70 0.89
0.01 0.40
0.99 0.99 0.98
0.98 0.93 0.93 0.95 0.93
1.75 0.88
0.98 0.97 0.89 0.92 0.91 0.82 0.92
0.51
0.94 0.95 0.95 0.88 0.89 0.91 0.76 0.87
0 -0.09 -0.01
-1.75
pARIs DFH PowerPC ΔP Phi P(E|C) P(C|E) pCI
300
225
150
75
0
error 19

20. ★ conditional and biconditional event
★ biconditional event in causal induction: pARIs
(proportion of assumed-to-be rare instances)
★ Meta-analysis
★ Three experiments
★ Theoretical background
20

21. Experiments
★ Experiment 1.1
★ To test the validity of rarity assumption in ordinary
2x2 causal induction
★ Experiment 1.2
★ To test the validity of rarity assumption in 3x2 causal
induction
★ Difference in the cognition between rare events (a, b, and
c-type) and non-rare d-type event, people just vaguely
recognize and memorize the occurrence of d-type events.
★ Experiment 2
★ Rarity and presence-absence (yes-no)
21

22. Experiment 1.1:
c and d in 2x2 table
★ 27 undergraduates, 9 stimuli. stim. a b c d
1 1 9 1 9
★ p: to give artificial diet to your 2 1 9 5 5
horse, q: your horse gets ill. 3 1 9 9 1
★ After the presentation of 4 5 5 1 9
(a,b,c,d), participants are asked 5 5 5 5 5
the causal intensity and then 6 5 5 9 1
the frequency of c- and d-type 7 9 1 1 9
event. 8 9 1 5 5
9 9 1 9 1
22

23. Result of exp. 1.1
stim. a b c d
1 1 9 1 9
c cell d cell
2 1 9 5 5 10 10
3 1 9 9 1 8 8
4 5 5 1 9 6
5
5 5 5 5 5 4
6 5 5 9 1 2 3
7 9 1 1 9 0
1 2 3 4 5 6 7 8 9
0
1 2 3 4 5 6 7 8 9
8 9 1 5 5 real c estimated c real d estimated d
9 9 1 9 1
★ Participants' estimation of c and d occurrence
was basically faithful.
★ d is estimated moderately than the real stimuli.
23

24. Experiment 1.2:
c and d in 3x2 table
★ 54 undergraduates, 2
stimuli.
stimulus A q not-q
★ As a medical scientist, p: to
give a medicine (three p1 6 4
types, p1, p2 and p3) to a p2 9 1
patient q: the patient p3 2 8
develops antibodies against
a virus.
stimulus B q not-q
★ After the presentation of six p1 5 5
kinds of events, participants
are asked the causal p2 8 2
intensity of p1 to q and p2 to p3 1 9
q, and then the frequency of
c- and d-type event.
24

25. Experiment 1.2:
c and d in 3x2 table
★ Each participant estimates
the intensity of causal stimulus A q not-q
relationship from p1 to q.
p1 6 a 4 b
★ Then asked the value of focus
p2 9 c 1 d
c, as "How often q + +
happened in the absence p3 2 8
of p1?." The given value of
c is 9+2=11.
25

26. Exp. 1.2: Result
c cell d cell
13 14
10 11
7 7
3 4
0 0
1 2 3 4 1 2 3 4
real c estimated c real d estimated d
26

27. ★ conditional and biconditional event
★ biconditional event in causal induction: pARIs
(proportion of assumed-to-be rare instances)
★ Meta-analysis
★ Three experiments
★ Theoretical background
27

28. Theoretical background of
biconditional event and pARIs
★ Angelo Gilio is studying biconditional event as with
the notion of quasi-conjunction.
★ There are some equivalent indices.
★ Possibly it contributes to the maximization of
information acquisition as in Wason selection task by
Oaksford & Chater 1994.
★ Computer simulations shows that pARIs is very
efficient in inferring the correlation of the
population from a small sample set, with the highest
reliability and precision.
28

29. Simulation
mean value through MC sim.
0.8"
0.7"
0.6"
0.5"
0.4"
pARIs"
DFH" ★ pARIs enables
the reliable and
Delta"P"
Phi"
0.3"
PowerPC"
precise grasp of
0.2"
0.1"
0"
1" 2" 3" 4" 5" 6" 7" 8" 9" 10" 11" 12" 13" 14" 15" 16" 17" 18" 19" 20" 21" 22" 23" 24" 25" 26" 27" 28" 29"
population
1"
correlation with
a very small
0.9"
0.8"
sample
0.7"
pARIs"
0.6"
DFH"
0.5"
Delta"P"
0.4" Phi"
0.3" PowerPC"
0.2"
0.1"
0"
1" 2" 3" 4" 5" 6" 7" 8" 9" 10" 11" 12" 13" 14" 15" 16" 17" 18" 19" 20" 21" 22" 23" 24" 25" 26" 27" 28" 29" 30" 31"
sd value Correlation of the population is 0.2
29

30. Indices equivalent to the
probability of biconditional event
★ Psychology
★ Tversky index of similarity, Tversky (1977)
★ Asymmetric similarity measure comparing a variant to a
prototype. Also in: Gregson (1975) and Sjöberg (1972)
★ Mathematics, machine learning and statistics:
★ Probable equivalence, or the probabilistic
indentity of two sets A and B, P(A=B) by Kosko
(2004)
★ Tanimoto similarity coefficient
★ Jaccard similarity measure
30

31. Tversky index
Psychological Review
J 330
Copyright © 1977 C_? by the American Psychological Association, Inc.
AMOS TVERSKY
V O L U M E 84 NUMBER 4 JULY 1977
not to
compon
A-B
2. Mo
Features of Similarity APIB
wheneve
Amos Tversky
Hebrew University
FEATURES OF- A
B
SIMILARI
Jerusalem, Israel and
The metric and dimensional assumptions that underlie the geometric Figure 1. A graphical illustration of the relation between
represen-
matching function of interest is the ratio model,
tation of similarity are questioned on both theoretical and empirical two feature sets.
grounds.
A new set-theoretical approach to similarity is developed in which objects are
Hence, sMoreo
represented as collections of features, and similarity is described as a feature-
_
matching process. Specifically, a set of qualitative assumptions is shown to of features is viewed as a product of a
lection f(A - Thatin
B
either
imply the contrast model, which expresses the similarity between objects as process of extraction and compilation.
prior a
linear combination of the measures of their common and distinctive features.
. , - ( nB)+af(A-B)+^f(B-A)' f(B),ofprov
Second, the term, feature usually denotes the
Several predictions of the contrast model are tested in studies of similarity with
comm
tive fea
f A model is used to uncover,value of a binary variable (e.g., voiced vs.
both semantic and perceptual stimuli. The analyze,
symmetry
object b
«,/3>0,
and explain a variety of empirical phenomena such as the role of common and consonants) or the value of a nominal
voiceless
distinctive features, the relations between judgments of similarity and differ-
ence, the presence of asymmetric similarities, and the effects of context on (e.g., eye color). Feature representa-
variable
axiom c
in measui
letters
tions, however, are not restricted to binary or
judgments of similarity. The contrast model generalizes standard representa- compon
31

32. Conjunctive MP
Def Bicond Other
Biconditional
Def Cond Weak
90%
event
80%
70%
60%
50%
40%
30%
20%
★ Developmental 10%
0%
3 6 9 adults
★ Merely transient in the Grades
process of narrowing the Conjunctive MP
scope, between
Def Bicond Other
Def Cond Strong
conjunctive and 90%
80%
conditional? (Gauffroy and 70%
Barouillet, 2009) 60%
50%
★ Probably there are 40%
30%
theoretical reasons for the 20%
dominance of defective 10%
0%
biconditional 3 6
Grades
9 adults
(biconditional event).
Gauffroy and Barouillet, 2009
Fig. 3. Percent of response patterns categorized as conjunctive, defective biconditional (Def Bicond), defective conditi
Cond), matching (MP), and others as a function of grades for strong and weak causal conditionals in Experiment 2.
32

33. Conclusion
★ Our intuition for generative causality from co-
occurrence data is the probability of
biconditional event (or defective
biconditional).
★ In causal induction, biconditional event focuses on
rare events and neglects abundant events, in the
uncertain world.
★ pARIs: proportion of assumed-to-be rare instances
★ Biconditional event is turning out to have strong
normative nature and theoretical grounds, so
possibly will be proven to be normative as well.
33