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Causal Attribution 1
A Connectionist Approach to Causal Attribution
Frank Van Overwalle and Dirk Van Rooy
Vrije Universiteit Brussel, Belgium
=== FINAL VERSION ===
Chapt er pr epar ed f or : S. J. Read & L. C Mi l l er ( Eds. )
Connect i oni st and PDP model s of Soci al Reasoni ng and Soci al
Behavior . Lawrence Erlbaum.
We ar e ver y gr at ef ul t o t he edi t or s f or t hei r hel pf ul
comment s on ear l i er ver si ons of t he manuscr i pt . The r esear ch
r epor t ed i n t hi s chapt er was i n par t suppor t ed by t he Bel gi an
Nat i onal Foundat i on of Sci ent i f i c Resear ch ( N. F. W. O. ) under gr ant
8. 0192. 95. Addr ess f or cor r espondence: Fr ank Van Over wal l e,
Depar t ment of Psychol ogy, Vr i j e Uni ver si t ei t Br ussel , Pl ei nl aan 2,
B - 1050 Brussel, Belgium; or by e - mail: [email protected].
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Causal Attribution 2
Introduction
At t r i but i ng a cause t o an event i s an i ndi spensabl e ment al
capaci t y t hat enabl es humans t o i dent i f y t he f act or s i n t hei r
envi r onment r esponsi bl e f or t hei r har dshi p or wel l - bei ng, t o
pr edi ct s i mi l ar event s i n t he f ut ur e and t o i ncr ease t hei r cont r ol
over t hei r occur r ences. I n f act , not onl y humans but al so ani mal s
use t he capaci t y t o det ect cause- ef f ect r el at i onshi ps i n or der t o
pr osper and saf eguar d t hei r ever yday adapt at i on and l ong- term
sur vi val . Gi ven t hi s l ong evol ut i onar y hi st or y f r om si mpl e
i nver t ebr at es l i ke t he mol l usk ( Hawki ns, 1989) on, i t seems
r easonabl e t o assume t hat much, i f not al l causal l ear ni ng i s
gover ned by ver y el ement ar y and si mpl e cogni t i ve pr ocesses,
oper at i ng i n ani mal s as wel l as humans. I n t hi s paper , we wi l l
ar gue t hat t he causal l ear ni ng pr ocess i nvol ves t he devel opment of
ment al associations or connections bet ween pot ent i al causes and
t he ef f ect , and t hat t hi s l ear ni ng pr ocess can be pr of i t abl y
analyzed from a connectionist perspective.
The associ at i ve appr oach t o causal l ear ni ng gr ew f r om
resea r ch on ani mal condi t i oni ng ( e. g. , Rescor l a & Wagner , 1972) ,
and has gai ned i ncr easi ng suppor t i n cur r ent cogni t i ve r esear ch on
human causal i t y and cat egor i zat i on ( f or r evi ews see Al l en, 1993;
Shanks, 1993, 1995) and i n connect i oni st or adapt i ve net wor k
model s of human memor y and t hi nki ng ( e. g. , Gl uck & Bower , 1988a;
McCl el l and & Rumel har t , 1988) . The var i ous t heor et i cal pr oposal s
put f or war d i n t hese di ver se ar eas of r esear ch seem t o conver ge t o
a f ew f undament al pr i nci pl es. For i nst ance, t he popul ar
asso ci at i ve model of ani mal condi t i oni ng pr oposed by Rescor l a and
Wagner i n 1972 i s, i n f act , f or mal l y equi val ent t o a speci f i c
c l ass of t wo- l ayer connect i oni st model s based on t he delta
l ear ni ng al gor i t hm ( i . e. , pat t er n associ at or s; McCl el l and &
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Causal Attribution 3
Rumelhart, 1988).
I n cont r ast t o cogni t i ve psychol ogi st s ' r i s i ng i nt er est i n
associ at i ve or connect i oni st l ear ni ng pr i nci pl es, soci al
psychol ogi st s have been sl ow i n i ncor por at i ng t hese i deas i n t hei r
t heor i es of human causal i t y j udgment s. Rat her t han t he
descendant s of a l ong evol ut i onar y past , t hey v i ew humans as nai ve
sci ent i st s ( Kel l ey, 1967) , l ogi c i ans ( Hewst one & Jaspar s, 1987) or
st at i st i c i ans ( Cheng & Novi ck, 1990) , who comput e causal i t y i n
anal ogy t o l ogi cal and st at i st i cal pr ocedur es and nor ms. Al t hough
r esear ch has r epeat edl y demonst r at ed t hat humans somet i mes make
bi ased i nf er ences t hat devi at e f r om nor mat i ve pr obabi l i t i es ( cf . ,
Kahneman, Sl ovi c & Tver sky, 1982) , such f i ndi ngs have been
r out i nel y i nt er pr et ed i n t er ms of unf avor abl e condi t i ons or l ack
of cogni t i ve r esour ces t o cal cul at e t he cor r ect i nf er ences r at her
t han as evi dence of an al t er nat i ve l ear ni ng pr ocess. Even
r esear ch i n soci al cogni t i on whi ch adopt ed not i ons f r om ear l i er
net wor k st r uct ur es ( e. g. , i n per son i mpr essi on, Hami l t on, Dr i scol l
& Wor t h, 1989) or f r om r ecent connect i oni st net wor k model s ( e. g. ,
i n causal knowl edge st r uct ur es, Read & Mar cus- Newhal l , 1993) has
been mai nl y concer ned wi t h t he r et r i eval of exi st i ng memor i es,
r at her t han wi t h t he l ear ni ng i t sel f of new concept s and causal
rel ationships.
How mi ght we under st and t he causal l ear ni ng pr ocess ? What
i nsi ght s do connect i oni st appr oaches of f er t hat go beyond t hose
of f er ed by al t er nat i ve r ul e- based ( e. g. , st at i st i cal ) model s ? To
addr ess t hese f undament al quest i ons, we begi n t hi s chapt er wi t h a
r evi ew of t he nor mat i ve pr obabi l i s t i c t heor y of causal at t r i but i on
as exempl i f i ed by t he cont r ast model devel oped by Cheng and Novi ck
( 1990) and ext ended by Van Over wal l e ( 1996a; Van Over wal l e &
Heyl i ghen, 1995) ; and compar e i t wi t h a connectionist
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Causal Attribution 4
i mpl ement at i on of Rescor l a and Wagner ' s ( 1972) model of l ear ni ng.
Next , we wi l l pr esent evi dence i ndi cat i ng t hat t he connect i oni st
appr oach pr ovi des a bet t er expl anat i on f or some phenomena of
causal compet i t i on l i ke di scount i ng and augment at i on ( cf . Kel l ey,
1971) . Fi nal l y, we wi l l pr esent t he configural model by Pear ce
( 1994) whi ch r epr esent s a maj or advance i n connect i oni st model i ng
t hat over comes sever al l i mi t at i ons of ear l i er connect i oni st model s
( Rescor l a & Wagner , 1972; McCl el l and & Rumel har t , 1988) , and we
wi l l r evi ew some dat a suggest i ng t hat t hi s model i s super i or i n
deal i ng wi t h gener al i zat i on of causal i t y t o f act or s t hat ar e
similar to the true cause.
Probabilistic Approach
Kel l ey ( 1967) , one of t he f ounder s of at t r i but i on t heor y i n
soci al psychol ogy, pr oposed t hat per cei ver s i dent i f y t he causes of
an ef f ect by usi ng a pr i nci pl e of covariation whi ch speci f i es t hat
an " ef f ect i s at t r i but ed t o t hat condi t i on whi ch i s pr esent when
t he ef f ect i s pr esent and whi ch i s absent when t he ef f ect i s
absent " ( p. 194) . He speci f i ed t hr ee maj or compar i sons t hat ar e
i mpor t ant i n det ect i ng covar i at i on and causal i t y i n t he soci al
domai n, and l at er r esear ch ( e. g. , Hewst one & Jaspar s, 1987; Hi l t on
& Sl ugoski , 1986; Cheng & Novi ck, 1990) demonst r at ed t hat each of
t hese compar i sons gener at es an at t r i but i on : Low consensus ( t he
ef f ect occur s when t hi s per son i s pr esent but not when ot her s ar e)
pr oduces at t r i but i ons t o t he person ; hi gh distinctiveness ( t he
ef f ect occur s when t hi s st i mul us i s pr esent but not when other
st i mul i ar e) gener at es at t r i but i ons t o t he stimulus ; and l ow
consistency ( t he ef f ect i s pr esent at t hi s occasi on but not at
ot her occasi ons) pr oduces at t r i but i ons t o t he occasion . Al t hough
Kel l ey ' s ( 1967) covar i at i on i dea i s now wi del y accept ed i n t he
at t r i but i on l i t er at ur e, t her e i s di sagr eement about t he under l y i ng
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Causal Attribution 5
pr ocesses by whi ch covar i at i on and causal i t y i s det ect ed. Some
at t r i but i on r esear cher s have t aken a pr obabi l i s t i c appr oach t o
describe this process in more detail.
Probabilistic Contras ts
Accor di ng t o r esear cher s t aki ng a pr obabi l i s t i c per spect i ve,
per cei ver s st or e i n memor y t he f r equenci es about cause- effect
occur r ences, and t hen per f or m some quasi - st at i st i cal comput at i on
on t hese f r equenci es i n or der t o pr oduce a causal j udgment ( f or an
over vi ew see Al l en, 1993; Cheng & Novi ck, 1990; Shanks, 1993) . I n
Fi gur e 1, t he r el evant f r equenci es ( denot ed by a - d) ar e
r epr esent ed i n a 2 x 2 cont i ngency t abl e, wher e one axi s r ef l ect s
t he pr esence or absence of t he pot ent i al cause C, and t he second
axis the presence or absence of the effect.
--------------------------
Insert Figure 1 about here
--------------------------
For r easons t hat wi l l become cl ear l at er , t he f i r st axi s al so
di spl ays a second f act or X, r epr esent i ng t he backgr ound or cont ext
agai nst whi ch t he cause C occur s. Thi s cont ext i s i nvar i ant l y
pr esent , i r r espect i ve of t he pr esence or absence of t he t ar get
f act or C. To di st i ngui sh bet ween t he t wo t ypes of f act or s, t he
f act or C i s t er med a contrast f act or ( because i t i s pr esent i n one
case but not t he ot her ) , and t he f act or X i s t er med a context
f act or ( because i t r ef l ect s t he cont ext pr esent i n al l cases under
obser vat i on) . I t shoul d be not ed t hat we def i ne cont r ast or
cont ext f act or s wi t h r espect t o a f ocal set of obser vat i ons
sel ect ed by t he per cei ver or pr ovi ded by t he exper i ment er ; t hey
ar e not necessar i l y cont r ast i ve or cont ext ual under al l possi bl e
observations of interest.
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Causal Attribution 6
Accor di ng t o pr obabi l i s t i c t heor y, causal j udgment s i nvol ve
t he cal cul at i on of t he cont r ast bet ween t he pr obabi l i t y of t he
ef f ect gi ven t he pr esence of t he cause mi nus t he same pr obabi l i t y
gi ven t he absence of t he cause ( see Al l en, 1993; Cheng & Novi ck,
1990; Shanks, 1993) . St at ed mor e si mpl y, causal i t y i s at t r i but ed
t o a cause t hat di f f er s f r om an i nvar i ant backgr ound or cont ext
wher e t hat cause i s absent . Thi s i s mat hemat i cal l y expr essed by a
probabilistic contrast 1 :
∆PC = P[Effect|C] - P[Effect|~C] [1]
= a - c a + b c + d
i n whi ch P r epr esent s t he pr opor t i on i n whi ch t he ef f ect
occur s when t he cause i s pr esent ( C) or absent ( ~C; a t i l de
denot es t he absence of a f act or ) ; and t he smal l l et t er s a- d denot e
t he f r equenci es i n Fi gur e 1. Thi s pr obabi l i s t i c cont r ast i s
c l osel y r el at ed t o common st at i st i cal measur es of cor r el at i on
bet ween t wo st i mul i such as t he χ2 st at i st i c, and has t her ef or e
received a normative status.
How can t he pr obabi l i s t i c cont r ast equat i on be appl i ed t o
Kel l ey ' s ( 1967) covar i at i on pr i nci pl e ? An i nnovat i ve at t empt t o
t ackl e t hi s quest i on was devel oped by Cheng and Novi ck ( 1990) i n
t hei r pr obabi l i s t i c cont r ast model . As t hei r appr oach i s al so
i mpor t ant i n under st andi ng connect i oni st appl i cat i ons, we wi l l
di scuss one exampl e i n some mor e det ai l . Let us t ake t he t ar get
event " Sar ah l aughed" , and l et us f ocus on Sarah as t he pot ent i al
cause of t he ef f ect laughed . Because t he ef f ect ( l aughed) i s
al ways pr esent when t he t ar get per son ( Sar ah) i s pr esent , t hi s can
be expr essed as P[ Laughed| Sar ah] = 1. Now, t o est i mat e Sar ah' s
causal r ol e, we need t o cont r ast t hi s pr obabi l i t y wi t h a r el evant
causal backgr ound wher e t he t ar get per son i s absent , f or exampl e,
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Causal Attribution 7
wi t h ot her per sons such as Sar ah' s f r i ends. The l ow consensus
i nf or mat i on t hat her f r i ends di d not l augh at t he same event can
be expr essed as P[ Laughed| ~Sar ah] = 0. Then, accor di ng t o
Equat i on 1, t he cont r ast bet ween t ar get and compar i son per sons
wi l l be hi gh, or ∆PSarah = 1, and Sar ah' s l aught er wi l l
consequent l y be at t r i but ed t o her sel f . I f , on t he ot her hand, t he
hi gh consensus i nf or mat i on i s gi ven t hat Sar ah' s f r i ends al so
l aughed dur i ng t he same event or P[ Laughed| ~Sar ah] = 1, t hen
∆PSarah = 0 and causal i t y wi l l not be at t r i but ed t o her . The same
cont r ast l ogi c appl i es f or hi gh di st i nct i veness and l ow
consi st ency i nf or mat i on. Gener al l y, a f act or wi l l be desi gnat ed
t he cause when i t s out come i s di f f er ent f r om t hat of t he
compar i son cases. Empi r i cal r esear ch has conf i r med t hat peopl e
make at t r i but i ons t o t he per son, st i mul us or occasi on i n l i ne wi t h
the probabilistic contrast model (Cheng & Novick, 1990) .
However , an i mpor t ant shor t comi ng of pr obabi l i s t i c t heor y i s
t hat i t does not speci f y how at t r i but i ons ar e made about t he
causal backgr ound gi ven a f ocal set of obser vat i ons. For exampl e,
when j udgi ng t he causal r ol e of Sar ah, her behavi or was cont r ast ed
wi t h t hat of ot her per sons who const i t ut ed a r el evant causal
backgr ound or cont ext ; but t hi s causal backgr ound i t sel f coul d not
be est i mat ed. Cheng and Novi ck gave t he causal cont ext onl y a
r at her shal l ow i nt er pr et at i on as enabl i ng condi t i on or i r r el evant
f act or , and speci f i ed t hat compar i sons wi t h an al t er nat i ve set of
obser vat i ons wer e necessar y t o j udge t hei r st r engt h. However , as
we wi l l see, i n connect i oni st appr oaches, causal cont ext s can be
est i mat ed and do pl ay a cr uci al r ol e i n det er mi ni ng t he causal
st r engt h of a cont r ast f act or , even wi t hi n a gi ven set of
obser vat i ons. Gi ven t hat t hi s i s t he case, we mi ght s i mpl y st op
her e and not e t hat t he pr obabi l i s t i c model i s ser i ousl y def i c i ent
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Causal Attribution 8
and much mor e l i mi t ed t han connect i oni st appr oaches. However , an
al t er nat i ve appr oach whi ch seems much mor e i nf or mat i ve, i s t o
ext ent pr obabi l i s t i c t heor y wi t h addi t i onal pr edi ct i ons f or causal
cont ext s, so t hat we st i l l can make sensi bl e compar i sons bet ween
the two approaches.
Probabilistic Contexts
What i s a cont ext f act or pr eci sel y, and how can i t s causal
st r engt h be i nf er r ed f r om a f ocal set of obser vat i ons ?
For t unat el y, t hi s i ssue was al r eady addr essed i n our ear l i er wor k
on causal at t r i but i on ( Van Over wal l e, 1996a; Van Over wal l e &
Heyl i ghen, 1995) . A cont ext i s def i ned as a r el at i vel y constant
background condi t i on consi st i ng, f or i nst ance, of " s i t uat i onal
st i mul i ar i s i ng f r om t he . . . envi r onment " ( Rescor l a & Wagner ,
1972, p. 88) . Because r esear cher s on t he f undament al di mensi ons
of human causal i t y ( e. g. , Abr amson, Sel i gman & Teasdal e, 1978;
Wei ner , 1986) al r eady i nt r oduced a t er mi nol ogy f or f act or s t hat
cor r espond ver y much t o our not i on of const ant backgr ound
condi t i ons, we bor r owed t hei r t er ms. Hence, t he cont ext of a
per son i s denot ed as external c i r cumst ances ( whi ch r emai n const ant
acr oss di f f er ent per sons) ; t he cont ext of a st i mul us as a global
cause ( whi ch r emai ns const ant acr oss di f f er ent st i mul i ) ; and t he
cont ext of an occasi on as a stable cause ( whi ch r emai ns per manent
over t i me) . As can be seen, each cont ext r ef l ect s a const ant
condi t i on agai nst whi ch a cont r ast cause may be di st i ngui shed.
Each pai r r ef l ect s t he t wo ext r emes of st andar d di mensi ons of
causal i t y, i ncl udi ng locus ( per sonal vs. ext er nal ) , globality
(stimulus - speci f i c vs. gener al ) and stability ( occasi onal vs.
st abl e) . Al t hough t hi s t er mi nol ogy st i l l l eaves open t he quest i on
whi ch speci f i c el ement s i n t he cont ext ar e r esponsi bl e f or t he
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Causal Attribution 9
ef f ect , t hi s same pr obl em i s i n f act al so t r ue f or cont r ast
causes. For i nst ance, an at t r i but i on t o t he per son does not
i ndi cat e whi ch el ement i nsi de t he per son i s causal i t y r el evant --
hi s or her abi l i t y , mot i vat i on, and so on ( see f or exampl es of
contrast and context causes, Weiner, 1986).
To comput e t he causal st r engt h of a cont ext f act or , Van
Over wal l e ( 1996a) devel oped a anal ogous pr obabi l i s t i c equat i on
which is mathematically expressed by a probabilistic context :
∆PX = P[Effect|~C] [2]
= c c + d
The l ogi c behi nd t hi s equat i on i s st r ai ght f or war d, because i t
has t he same gener al f or mat as t he cont r ast equat i on ( see Equat i on
1) , but r et ai ns onl y t he second t er m whi ch r epr esent s t he causal
cont ext X. I t speci f i es t hat t he st r engt h of a causal cont ext can
be est i mat ed f r om t he pr obabi l i t y of t he ef f ect gi ven t he r el evant
comparison cases where the target factor C is absent.
The pr obabi l i s t i c cont ext equat i on can al so easi l y be appl i ed
t o Kel l ey ' s ( 1967) covar i at i on var i abl es. I n our pr evi ous
exampl e, t he l ow consensus i nf or mat i on t hat most of Sar ah' s
f r i ends di d not l augh can be expr essed as P[ Laughed| ~Sar ah] = 0,
and t hi s l ow pr obabi l i t y i ndi cat es t hat t he causal cont ext ( e. g. ,
ext er nal c i r cumst ances) had l i t t l e causal ef f ect on Sar ah' s l augh.
I n cont r ast , t he hi gh consensus i nf or mat i on t hat most of her
f r i ends l aughed can be expr essed as P[ Laughed| ~Sar ah] = 1 so t hat
i n t hi s case causal i t y wi l l be st r ongl y at t r i but ed t o t he ext er nal
cont ext . The same l ogi c appl i es t o l ow di st i nct i veness and t o
hi gh consi st ency. Thus, i n gener al , a causal cont ext wi l l acqui r e
a subst ant i al amount of causal wei ght when bot h t he t ar get and
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Causal Attribution 10
comparison cases obtain the same effect.
The pr obabi l i s t i c pr edi ct i ons f or bot h cont r ast and cont ext
causes wer e combi ned i n what we have t er med t he j oi nt model ( Van
Over wal l e, 1996a; Van Over wal l e & Heyl i ghen, 1995) t o r ef l ect t he
j oi nt oper at i on of t he t wo pr obabi l i s t i c equat i ons. The j oi nt
model bor r ows Equat i on 1 f r om Cheng and Novi ck ' s ( 1990) cont r ast
model f or t he comput at i on of cont r ast f act or s, and adds Equat i on 2
f or t he comput at i on of cont ext f act or s. Hence, t he t wo equat i ons
ar e used separ at el y f or est i mat i ng cont r ast and cont ext f act or s.
Resear ch on t he j oi nt model has conf i r med t hat peopl e make
at t r i but i ons t o cont r ast and cont ext causes i n l i ne wi t h t he
pr edi ct i ons of t he j oi nt model ( Van Over wal l e & Heyl i ghen, 1995;
Van Overwalle, 1996a).
Connectionist Approach
Al t hough t he empi r i cal r esul t s ar e suppor t i ve of a
pr obabi l i s t i c anal ysi s of causal at t r i but i on, i t seems qui t e
i mpl ausi bl e t hat ani mal s and humans possess t he capaci t y t o t al l y
and memor i ze f r equenci es i n t he pr esence and absence of al l
pot ent i al causes and expl i c i t l y comput e t he r el evant cont r ast and
cont ext pr obabi l i t i es. The ear l y associ at i ve model s and t he mor e
r ecent connect i oni st appr oaches addr essed t hese cogni t i ve
l i mi t at i ons by assumi ng t hat t he per cei ved st r engt h of causes i s
di r ect l y st or ed i n memor y under t he f or m of ment al connect i ons
bet ween t he pot ent i al cause and t he ef f ect . These cause- effect
connect i ons ar e gr adual l y adj ust ed gi ven i nf or mat i on on t he co-
occur ence of t he cause and t he ef f ect . Var i ous l ear ni ng
mechani sms descr i bi ng t hese adj ust ment s have been pr oposed i n t he
ani mal and human l ear ni ng l i t er at ur e, but t he l ear ni ng al gor i t hm
devel oped by Rescor l a and Wagner i n 1972 gai ned most popul ar i t y.
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Causal Attribution 11
Because t hi s al gor i t hm pr eceded r ecent devel opment s i n
connect i oni st model i ng and st i l l i s a maj or sour ce of i nspi r at i on
i n t he f i el d of associ at i ve l ear ni ng ( cf . , Pear ce, 1994) , we wi l l
f i r st di scuss t he Rescor l a- Wagner model and t hen t ur n t o a
connectionist implementation of it.
Rescorla - Wagner Associative Model
A cent r al not i on of t he Rescor l a- Wagner l ear ni ng al gor i t hm i s
t hat " or gani sms onl y l ear n when event s v i ol at e t hei r expect at i ons"
( p. 75) . Thus, changes i n associ at i ve wei ght s of a causal f act or
are dr i ven by r educi ng t he di f f er ence bet ween t he act ual ef f ect
and t he ef f ect expect ed by t he or gani sm. The r educt i on of t hi s
di f f er ence or er r or t akes pl ace af t er each t r i al i n whi ch t he
f act or i s pr esent , and t he r esul t ant adj ust ment i n wei ght or ∆w of
the factor, is expressed in the following learning formula :
∆w = α βw ( λ - Σw). [3]
wher e α r epr esent s t he sal i ence or t he pr obabi l i t y of
at t endi ng t o t he f act or , and i s nor mal l y set t o 1 i f t he f act or i s
pr esent and t o 0 i f absent ; and βw i s a l ear ni ng r at e par amet er ,
r angi ng bet ween 0 and 1, whi ch r ef l ect s t he speed of l ear ni ng.
The λ var i abl e denot es t he magni t ude of t he ef f ect , and i s
t ypi cal l y set t o 1 when t he ef f ect i s pr esent and t o 0 when
absent ; and Σw r epr esent s t he expect ed ef f ect based on t he summed
association weights of all factors present on the trial.
The cour se of l ear ni ng i n t he s i mpl e case wi t h one causal
f act or ( Σw = wC) i s st r ai ght f or war d. The wei ght wC st ar t s at
zer o, and successi ve t r i al s cause an i ncr ease or decr ease until
t he poi nt i s r eached when t he er r or or ( λ - wC) i s zer o, i mpl y i ng
t hat t he ef f ect i s per f ect l y pr edi ct ed by t he cause C, and no
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Causal Attribution 12
f ur t her changes ar e necessar y. I n t hat case, we say t hat l ear ni ng
has r eached asymptote . The l ear ni ng r at e par amet er βw det er mi nes
t he pr opor t i on or speed by whi ch t he di scr epancy or er r or bet ween
expect ed and act ual ef f ect ar e t aken i n t o account f or adj ust i ng
associ at i ve wei ght s. Thus, causal l ear ni ng i s concei ved as a
gr adual pr ocess t hat i s cont i nuousl y updat ed r at her t han a f i nal
j udgment at t he end of a ser i es of obser vat i ons as assumed by t he
probabilistic approach.
Connectionist Implementation
An i mpor t ant f eat ur e of Rescor l a and Wagner ' s associ at i ve
model i s t hat t hei r l ear ni ng al gor i t hm ( Equat i on 3) i s i dent i cal
t o t he delta or Wi dr ow- Hof f updat i ng al gor i t hm whi ch has pl ayed a
maj or r ol e i n some f eedf or war d connect i oni st model s ( McCl el l and &
Rumel har t , 1988) . Consequent l y, t he Rescor l a- Wagner model can be
easi l y i mpl ement ed by a t wo- l ayer f eedf or war d ar chi t ect ur e, as
i l l ust r at ed i n Fi gur e 2 ( see al so Gl uck & Bower , 1988a; McCl el l and
& Rumel har t , 1988) . The f i r st l ayer compr i ses i nput nodes t hat
encode t he pr esence of causal f act or s, and t he second l ayer
compr i ses t he out put node r epr esent i ng t he pr edi ct ed ef f ect . The
i nput nodes ar e connect ed t o t he out put node vi a adj ust abl e
connections or weights, denoted by dashed lines in Figure 2.
--------------------------
Insert Figure 2 about here
--------------------------
When a f act or i s pr esent at a t r i al , t he act i vat i on of i t s
i nput node i s t ur ned on at val ue 1; and when a f act or i s absent ,
t he act i vat i on i s t ur ned of f t o 0. Ther e ar e t wo f eat ur es i n t hi s
i nput codi ng t hat di f f er somewhat f r om a t ypi cal f eedf or war d
net wor k. Fi r st , t he absence of a f act or i s coded as an act i vat i on
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Causal Attribution 13
of 0 r at her - 1 ( McCl el l and & Rumel har t , 1988) . Thi s codi ng scheme
r ef l ect s t he common obser vat i on i n associ at i ve r esear ch t hat
ani mal s and humans l ear n l i t t l e about obj ect s t hat ar e absent .
Second, t her e i s al ways a const ant cont ext f act or X pr esent on
ever y t r i al , i r r espect i ve of t he pr esence of t he t ar get cause C,
( see some i l l ust r at i ve codi ng i n Fi gur e 2) . As not ed bef or e, t hi s
cont ext assumpt i on was al so i nt r oduced i n t he pr obabi l i s t i c model
by Van Over wal l e and Heyl i ghen ( 1995) . Al t hough a ver y s i mi l ar
i dea can be i mpl ement ed i n connect i oni st model s by t he use of bias
t er ms ( see McCl el l and & Rumel har t , 1988, p. 121) , associ at i ve
appr oaches assume t hat t hese cont ext f act or s r ef l ect i mpor t ant and
meani ngf ul el ement s of t he envi r onment i n whi ch l ear ni ng t akes
place (e.g., the animal's cage).
Next , act i vat i on f r om t he i nput nodes spr eads aut omat i cal l y
t o t he out put node i n pr opor t i on t o t hei r connect i on wei ght s, and
ar e summed t o det er mi ne t he act i vat i on of t he out put node. Thi s
out put act i vat i on i s t hus a l i near f unct i on of t he i nput
act i vat i on, and r epr esent s t he st r engt h of t he ef f ect pr edi ct ed by
t he net wor k. The act ual l y obt ai ned ef f ect i s r epr esent ed by a
t eachi ng val ue ( not shown i n t he f i gur e) , usi ng t he same +1 and 0
codi ng scheme as t he i nput nodes. Thi s t eachi ng val ue ser ves as
i nput t o t he out put node; and t he er r or bet ween t he out put
( expect ed ef f ect ) and t eachi ng act i vat i ons ( act ual ef f ect )
det er mi nes t he adj ust ment s t o t he st r engt hs of t he connect i ons i n
t he net wor k, as i n t he Rescor l a- Wagner model . The change i n t he
connect i on wei ght of f act or j ( ∆wj) i s pr opor t i onal t o t hi s er r or ,
and t hi s i s f or mal l y expr essed by t he del t a l ear ni ng al gor i t hm
(McClelland & Rumelhart, 1988, p. 87) :
∆wj = ε (t - o) a j . [4]
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Causal Attribution 14
wher e ε i s t he l ear ni ng r at e, and wher e t he ot her symbol s t ,
o, and aj denot e, r espect i vel y, t he t eachi ng, out put and i nput
act i vat i ons. Compar i ng t hi s equat i on wi t h Equat i on 3 of Rescor l a-
Wagner , shows t hat t hey ar e compl et el y i dent i cal , except f or some
not at i onal var i at i ons. The l ear ni ng r at e βw i n t he Rescor l a-
Wagner equat i on i s r epr esent ed her e by ε, t he magni t ude of t he
act ual ef f ect λ i s r epl aced by t he t eachi ng act i vat i on t , the
expect ed ef f ect based on t he summed associ at i on wei ght s Σw i s
denot ed her e by t he out put act i vat i on o, and t he sal i ence α of a
f act or i s r ef l ect ed i n t he act i vat i on of t he cor r espondi ng i nput
node a j of all factors present on the trial
I n r esear ch on causal l ear ni ng, af t er t he subj ect s have gone
t hr ough a t r i al - by - t r i al acqui s i t i on phase, t hey ar e pr esent ed
wi t h quest i ons concer ni ng t he causal i nf l uence of some f act or s.
An appr opr i at e measur e i n t he net wor k f or subj ect s ' causal
j udgment s i s s i mpl y t he act i vat i on of t he out put node, gi ven t hat
t he appr opr i at e i nput nodes ar e t ur ned on. Thus, t o t est t he
pr edi ct i ons of t he net wor k, t he f act or s t o- be- t est ed ar e t ur ned on
at t he i nput l ayer as bef or e, except t hat t he st r engt h of a
cont r ast f act or i s now t est ed separ at el y wi t hout i t s accompanyi ng
cont ext ( see some i l l ust r at i ve codi ng i n Fi gur e 2) . I n t he
pr esent t wo- l ayer net wor k, t he out put act i vat i on i s i dent i cal t o
t he connect i on wei ght of each f act or t est ed, because t hi s i s t he
sol e f act or wi t h i t s i nput act i vat i on t ur ned on. I n t he r emai nder
of t hi s chapt er , t hi s connect i oni st i mpl ement at i on of associ at i ve
learning will be referred to as the Rescorla - Wagner network.
Fi gur e 3 i l l ust r at es t wo l ear ni ng hi st or i es wi t h 6 t r i al s f or
f act or s C and X, s i mul at ed by a connect i oni st net wor k j ust
descr i bed. I n t he f i r st exampl e wher e onl y t he cont r ast f act or
and i t s cont ext ar e f ol l owed by t he ef f ect ( CX → Ef f ect ; X → No
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Causal Attribution 15
Ef f ect ; l ef t panel ) , t he wei ght s t end t o asympt ot i c val ues of wC
= 1. 00 and wX = 0. 00. The cont r ast f act or acqui r es much posi t i ve
st r engt h because i t i s al ways f ol l owed by t he ef f ect . Al t hough
t he cont ext acqui r es some posi t i ve wei ght when pr esent ed t oget her
wi t h t he cont r ast f act or ( when CX → Ef f ect ) , t hi s wei ght i s
neut r al i zed ever y t i me the context is presented alone (when X → No
ef f ect ) , so t hat t he net r esul t i s l i t t l e causal st r engt h.
Conver sel y, i n t he second exampl e wher e onl y t he cont ext i s
f ol l owed by t he ef f ect ( X → Ef f ect , CX → No Ef f ect ; r i ght panel ) ,
t he wei ght s t end t o asympt ot i c val ues of wX = 1. 00 and wC = - 1.00.
The cont ext acqui r es st r ong posi t i ve st r engt h as i t i s al ways
f ol l owed by t he ef f ect . However , t he cont r ast f act or at t ai ns
st r ong negat i ve causal st r engt h, because when i t i s pr esent ed
t oget her wi t h t he cont ext t he ef f ect i s not pr oduced, so t hat i t
must compensate for the positive weight of the context.
--------------------------
Insert Figure 3 about here
--------------------------
An i mpor t ant char act er i st i c of t he Rescor l a- Wagner net wor k i s
t hat , gi ven suf f i c i ent l ear ni ng t r i al s, i t wi l l ar r i ve at t he same
causal pr edi ct i ons as t he pr obabi l i s t i c cont r ast and j oi nt model s
wi t hout st or i ng f r equenci es. Thi s somewhat sur pr i s i ng r esul t was
demonst r at ed mat hemat i cal l y by sever al aut hor s ( Chapman & Robbi ns,
1990; Van Over wal l e, 1996b) , and suggest s t hat t he nor mat i ve
pr edi ct i ons f r om pr obabi l i s t i c t heor y ar e, i n f act , al so emer gent
pr oper t i es of t he Rescor l a- Wagner l ear ni ng pr ocess. Fr om an
evol ut i onar y per spect i ve, i t seems i mper at i ve t hat associ at i ve
t hi nki ng i n ani mal s and humans has evol ved i n such a way t hat
pr edi ct i ve f act or s ar e i dent i f i ed wi t h r easonabl e accur acy
(Shanks, 1995).
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Causal Attribution 16
I n sum, t he t wo most i mpor t ant char act er i st i cs of an
connect i oni st appr oach t o causal l ear ni ng t hat di f f er ent i at es i t
f r om a pr obabi l i s t i c appr oach ar e t hat ( a) t he connect i oni st
appr oach does not r equi r e t hat f r equency i nf or mat i on on al l
pr evi ous t r i al s must be t al l i ed and memor i zed dur i ng t he l ear ni ng
per i od, but onl y t he cause- ef f ect connect i ons; and ( b) t he
connect i oni st appr oach does not r equi r e an expl i c i t and l abor i ous
comput at i onal pr ocess, but i mmedi at el y i nt egr at es i ncomi ng
i nf or mat i on of t he l ast t r i al by t he aut omat i c spr eadi ng of
act i vat i on f r om i nput t o out put and, i f i nconsi st enci es ar i se,
i mmedi at el y adj ust s t he connect i ons i n memor y so t hat causal
j udgment s ar e r eadi l y avai l abl e. I t i s evi dent t hat t he cogni t i ve
s i mpl i c i t y and evi dent gener al i t y of connect i oni st model s makes
t hem a ver y at t r act i ve al t er nat i ve t o pr obabi l i s t i c model s.
However , t he val i di t y of t he t wo appr oaches t o causal i t y depends
ul t i mat el y on t hei r empi r i cal conf i r mat i on. I t i s t o such
empirical findings that we turn now.
Discounting and Augmentation
To di st i ngui sh bet ween t he pr obabi l i s t i c and connect i oni st
account s, we expl or e t hei r pr edi ct i ons i n t wo si t uat i ons wher e a
per cei ver must l ear n whi ch one among mul t i pl e f act or s caused an
event . Despi t e t he cent r al pl ace accor ded i n at t r i but i on t heor y
t o t he pr i nci pl e of covar i at i on, Kel l ey ( 1971) r eal i zed t hat t hi s
pr i nci pl e al one was not suf f i c i ent t o expl ai n how compet i ng causal
expl anat i ons ar e sel ect ed, and he t her ef or e suggest ed t wo
auxiliary principles of discounting and augmentation .
The di scount i ng pr i nci pl e speci f i es t hat when t he i nf l uence
of one or mor e causes i s al r eady est abl i shed, per cei ver s wi l l
di sr egar d ot her possi bl e causes as l ess r el evant . A common
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Causal Attribution 17
exampl e i s when at t r i but i ons t o t he per son ar e di scount ed gi ven
evi dence on t he pot ent i nf l uence of ext er nal pr essur e. The
r ever se t endency i s descr i bed i n t he augment at i on pr i nci pl e, whi ch
suggest s t hat gi ven t wo opposi ng ( f aci l i t at or y vs. i nhi bi t or y)
causes, per cei ver s wi l l val ue t he st r engt h of t he cause t hat
pr oduces t he ef f ect hi gher t o compensat e f or t he i nhi bi t or y
i nf l uence of t he ot her cause. For i nst ance, success wi l l be mor e
st r ongl y at t r i but ed t o a per son' s capaci t i es when t he t ask was
har d r at her t han easy. Numer ous i nvest i gat i ons have shown t hat
t hese t wo compet i t i ve pr i nci pl es oper at e i n causal j udgment s of
adul t s ( e. g. , Hansen & Hal l , 1985; Kr ugl anski , Schwar t z, Mai des &
Hamel , 1978) and chi l dr en of 3- 4 year s of age ( e. g. , Kassi n &
Lowe, 1979; Kassin, L owe & Gibbons, 1980; Newman & Ruble, 1992).
Di scount i ng and augment at i on show a r emar kabl e s i mi l ar i t y
wi t h ef f ect s known i n t he associ at i ve l i t er at ur e as blocking and
condi t i oned i nhi bi t i on r espect i vel y ( cf . Val l ée- Tour angeau, Baker
& Mer ci er , 1994) . These t er ms r ef er t o speci f i c pr ocedur es by
whi ch compet i t i ve ef f ect s have been di scover ed, f i r st i n ani mal
condi t i oni ng ( e. g. , Kami n, 1968) , and subsequent l y i n causal
j udgment t asks wi t h humans ( e. g. , Baker , Mer ci er , Val ée-
Tour angeau, Fr ank & Pan, 1993; Chapman & Robbi ns, 1990; Chapman,
1991; Shanks, 1985, 1991) . As expl ai ned bef or e, a cent r al f eat ur e
of t he Rescor l a- Wagner and ot her connect i oni st model s i s t hat t he
adj ust ment s of cause- ef f ect connect i ons ar e det er mi ned by t he
di scr epancy bet ween t he act ual and expect ed ef f ect , gi ven not onl y
t he t ar get cause but all ot her s i mul t aneousl y pr esent ed causes, or
Σw. Hence, compet i t i on f or pr edi ct i ve st r engt h bet ween all causes
pr esent i s an i nher ent pr oper t y of associ at i ve or connect i oni st
learning.
We conduct ed an exper i ment ( Van Rooy & Van Over wal l e, 1996)
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Causal Attribution 18
i n whi ch, i n an i ni t i al phase, t he i nf l uence of t he cont ext was
st r engt hened i n or der t o expl or e whet her t hi s woul d l ead t o t he
di scount i ng or augment at i on of at t r i but i ons t o t he t ar get per son
or st i mul us l at er on. The causal st r engt h of t he cont ext was
enhanced by i ncr easi ng t he number of compar i son per sons or st i mul i
-- whi ch i mpl y t he cont ext -- f r om one ( smal l compar i son set ) t o
five ( l ar ge compar i son set ) . As di scussed bef or e, when sever al
compar i son per sons or st i mul i shar e t he same out come, st r ong
at t r i but i ons wi l l be made t o t he ext er nal or gl obal cont ext ,
r espect i vel y ( Van Over wal l e & Heyl i ghen, 1995; Van Over wal l e,
1996a) . For i nst ance, when many at hl et es r ecor d a ver y f ast t i me
i n a spr i nt r ace, i t i s mor e l i kel y t o at t r i but e t hi s out come t o
ext er nal c i r cumst ances such as a f avor abl e back wi nd r at her t han
t o per sonal t al ent . The cr uci al poi nt now i s t hat we expect
gr eat er di scount i ng or augment i ng ef f ect s when f i ve r at her t han
onl y one compar i son per son or st i mul us i s avai l abl e. For exampl e,
at t r i but i ons t o per sonal at hl et i c t al ent wi l l be mor e di scount ed
t he mor e ot her at hl et es al so r ecor ded f ast t i mes. Conver sel y,
per sonal t al ent wi l l be mor e augment ed t he mor e ot her at hl et es
r ecor ded sl ow r at her t han f ast r unni ng t i mes. To what ext ent can
probabilistic or connectionist models reproduce these predictions ?
Connectionist Predictions
Accor di ng t o t he Rescor l a- Wagner net wor k model , an i ncr easi ng
number of compar i son cases ( or t r i al s) wi t h a s i mi l ar out come wi l l
cause an i ncr ease i n t he per cei ved i nf l uence of t he cont ext . Thi s
i s i l l ust r at ed f or a per son f act or and an ext er nal cont ext i n
Fi gur e 4. As can be seen i n bot h t he t op and bot t om panel , t he
ext er nal cont ext acqui r es a much st r onger st r engt h af t er 5 t r i al s
i n t he l ar ge compar i son set ( wE = . 97) , t han af t er 1 t r i al i n t he
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Causal Attribution 19
smal l compar i son set ( wE = . 50) . Thi s cour se of causal l ear ni ng
was f ol l owed i n t he f i r st phase of our di scount i ng and
augmentation conditions.
--------------------------
Insert Figure 4 about here
--------------------------
The t wo condi t i ons di f f er ed i n t he second phase, af t er t he
causal st r engt h of t he cont ext had been est abl i shed. I n t he
di scount i ng condi t i on, a novel cont r ast per son or st i mul us was
i nt r oduced shar i ng t he same out come as t he pr evi ous compar i son
cases. Accor di ng t o t he Rescor l a- Wagner net wor k, i n t he l ar ge
compar i son set , t he r ol e of t hi s cont r ast f act or wi l l be st r ongl y
di scount ed or bl ocked because t he cont ext f act or al r eady f ul l y
pr edi ct s t hi s out come. Thi s can be seen i n t he t op panel of
Fi gur e 4, as mor e di scount i ng of t he cont r ast per son f act or i s
obser ved i n t he l ar ge compar i son set wher e t he cont ext has
pr evi ousl y acqui r ed st r ong causal st r engt h, t han i n t he smal l
comparison set where the context has acquired less strength.
On t he ot her hand, i n t he augment at i on pr ocedur e, a novel
per son or st i mul us cont r ast case i s i nt r oduced wi t h an out come
opposite t o t hat of t he pr ecedi ng compar i son cases. Accor di ng t o
t he Rescor l a- Wagner model , i n t he l ar ge compar i son set , t hi s
out come i s t ot al l y opposi t e t o t hat pr edi ct ed by t he cont ext
f act or so t hat t he cont r ast f act or wi l l acqui r e st r ong negat i ve
st r engt h t o compensat e f or t hi s di scr epancy. As can be seen f r om
t he bot t om panel i n Fi gur e 4, t he negat i ve st r engt h of t he
cont r ast f act or wi l l be mor e augment ed ( i . e. , mor e negat i ve) i n
t he l ar ge compar i son set wher e t he cont ext has acqui r ed a st r ong
causal r ol e, t han i n t he smal l set wher e t he cont ext has acqui r ed
less strength.
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Causal Attribution 20
Probabilistic Predictions
These pr edi ct i ons ar e, however , pr obl emat i c f or pr obabi l i s t i c
t heor y. The r eason i s t hat al l cur r ent pr obabi l i s t i c model s of
causal i t y ( Cheng & Novi ck, 1990; Cheng & Hol yoak, 1995; Van
Over wal l e, 1996a; Wal dmann & Hol yoak, 1992) base t hei r pr edi ct i ons
on a pr obabi l i s t i c cal cul at i on of t he ef f ect , t hat i s , t he
r el at i ve proportion of cause- ef f ect covar i at i on r at her t han i t s
raw f r equency. Consequent l y, al l model s pr edi ct t hat t he
pr obabi l i t y of t he ef f ect gi ven t he cont ext i s t he same whet her
t he number of compar i son cases i s one or f i ve ( i . e. , ∆PX = 1/ 1 and
5/ 5 r espect i vel y f or bot h di scount i ng and augment at i on) , so t hat
t he cont r ast f act or i s expect ed t o r ecei ve t he same causal
st r engt h i n each condi t i on ( see Equat i on 1) . Thus, i ncr easi ng t he
number of compar i son cases shoul d not make any di f f er ence t o
subjects' causal judgments.
Experiments and Results
Subj ect s wer e pr esent ed wi t h di f f er ent st or i es cont ai ni ng
i nf or mat i on about f i ve compar i son per sons or st i mul i i n t he l ar ge
compar i son set , or onl y one compar i son per son or st i mul us i n t he
smal l compar i son set . Thi s i nf or mat i on was f ol l owed by one t ar get
per son or st i mul us wi t h t he same out come ( di scount i ng) or a
opposi t e out come ( augment at i on) . The f ol l owi ng descr i pt i on
i l l ust r at es t he di scount i ng of a per son f act or i n t he l ar ge set
( wi t h t he smal l set bet ween br acket s) : " Fi ve ot her sal esl adi es
[ One ot her sal esl ady] and al so Anni e at t ai ned hi gh sal es f i gur es
f or per f umes. " The next descr i pt i on depi ct s t he augment at i on of a
per son f act or : " Fi ve ot her spor t swomen [ One ot her spor t swoman]
f el l dur i ng t he spr i nt r ace, but Sandr a di d NOT f al l dur i ng t he
spr i nt r ace. " Si mi l ar descr i pt i ons wer e gi ven t o mani pul at e t he
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Causal Attribution 21
discounting and augmentation of a stimulus factor.
I n one condi t i on, t he i nf or mat i on was pr esent ed i n a pr e-
packed summar y f or mat ( as i n t he exampl es above) whi ch capt ur es
some aspect s of peopl e' s ver bal i nt er act i ons wi t h one anot her , and
i n anot her condi t i on t he i nf or mat i on was pr esent ed i n a sequent i al
f or mat ( i . e. , case- after - case) whi ch r ef l ect s peopl e' s i nci dent al
l ear ni ng dur i ng ever yday l i f e. Af t er r eadi ng each st or y, subj ect s
r at ed t he causal i nf l uence of f our f act or s i ncl udi ng t he per son
( i . e. , somet hi ng about Anni e) , t he ext er nal cont ext ( i . e. ,
somet hi ng ext er nal out s i de Anni e) , t he st i mul us ( i . e. , something
about per f umes) and t he gl obal cont ext ( somet hi ng gener al about
toiletry ) , usi ng a 11- poi nt r at i ng scal e r angi ng f r om 0
( absolutely no influence ) to 10 ( very strong influence ).
The most r el evant at t r i but i on r at i ngs ar e l i s t ed i n Tabl e 1.
I n l i ne wi t h t he connect i oni st pr edi ct i ons, i n t he di scount i ng
condi t i on, t he mean at t r i but i ons t o t he per son and t he st i mul us
wer e mor e di scount ed i n t he l ar ge t han t he smal l compar i son set ,
and i n t he augment at i on condi t i on, t hey wer e mor e augment ed i n t he
l ar ge t han smal l compar i son set . Thus, i ncr easi ng t he number of
compar i son cases l ead t o mor e di scount i ng and mor e augment at i on.
Thi s ef f ect was si gni f i cant i n s i x out of ei ght compar i sons, and
most consi st ent i n t he summar y pr esent at i on condi t i on. Over al l ,
t hese r esul t s ar e consi st ent wi t h t he Rescor l a- Wagner net wor k
model, but clearly contradict probabilistic models.
-------------------------
Insert Table 1 about here
-------------------------
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Causal Attribution 22
Model Simulations
I n or der t o assess t he over al l per f or mance of t he t wo model s,
we comput ed si mul at i ons of Van Over wal l e' s ( 1996a) pr obabi l i s t i c
j oi nt model usi ng Equat i ons 1 and 2, and of t he Rescor l a- Wagner
net wor k usi ng Equat i on 3 ( or 4) wi t h wei ght updat es af t er each
t r i al . The i nf or mat i on f r om t he st or i es was encoded i n t he model s
i n exact l y t he same or der and number as pr ovi ded t o our subj ect s.
Gi ven t hat t he Rescor l a- Wagner net wor k has one f r ee par amet er , t he
l ear ni ng r at e βw, we sought t he best f i t f or t hi s model by r unni ng
si mul at i ons f or t he whol e r ange of admi ssi bl e par amet er val ues
( bet ween 0 and 1) , and t hen sel ect ed t he si mul at i on wi t h t he
hi ghest cor r el at i on bet ween si mul at ed and obser ved r esponses ( see
bel ow) . Al t hough t hi s pr ocedur e may r ef l ect some capi t al i zat i on
on chance, t he mer e exi st ence of a f r ee par amet er mi ght be
consi der ed as yet anot her way i n whi ch connect i oni st model s ar e
super i or . To our knowl edge, t her e i s no publ i shed r esear ch i n
whi ch t he Rescor l a- Wagner l ear ni ng r at e par amet er was est i mat ed on
soci al dat a, so t hat an appr opr i at e val ue can onl y be est abl i shed
post hoc. Not e, however , t hat t he r epor t ed best - f i t par amet er
val ues ar e gener al l y qui t e r obust , and t hat devi at i ons of . 10 i n
the parameter values decrease the fit only minimally.
The sequent i al and summar y f or mat s wer e s i mul at ed separ at el y
wi t h i ndependent l ear ni ng r at es βw, because we sur mi sed t hat t he
f or mat i n whi ch i nf or mat i on was pr esent ed dur i ng t he exper i ment
mi ght have af f ect ed t he speed of l ear ni ng of our subj ect s. Tabl e
2 di spl ays t he cor r el at i on R bet ween t he si mul at ed and obser ved
r esponses f or t he cont r ast and cont ext f act or s ( aver aged over al l
subj ect s) . Thi s measur e pr ovi des an i ndex of t he summar y f i t of
t he model s ( Gl uck & Bower , 1988a) . As can be seen, al t hough t he
pr obabi l i s t i c model capt ur ed some var i ance i n t he dat a ( mean R =
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Causal Attribution 23
. 314) , t he Rescor l a- Wagner net wor k model f i t t ed t he dat a much
bet t er ( mean R = . 802) . Cont r ar y t o our suspi c i on, t he l ear ni ng
r at e at t ai ned a hi gh val ue of . 80 i n bot h pr esent at i on f or mat s.
Thi s may suggest t hat t he det ect i on of covar i at i on among soci al
st i mul i occur r ed r el at i vel y f ast r egar dl ess of pr esent at i on mode,
per haps because t he soci al st or i es used i n our r esear ch wer e qui t e
simple and perhaps familiar to our subjects.
-------------------------
Insert Table 2 about here
-------------------------
Configural Attributions
So f ar , we have deal t wi t h s i ngl e cont r ast f act or s and t hei r
cont ext s whi ch make up si ngl e di mensi ons of causal i t y. However ,
i n r eal l i f e, humans ar e most of t en conf r ont ed wi t h a mor e compl ex
s i t uat i on wher e mul t i pl e di mensi ons ar e per cei ved si mul t aneousl y.
As Kel l ey ( 1967) not ed, we do not onl y obser ve and compar e
r egul ar l y wi t h ot her peopl e, but al so wi t h ot her s i t uat i ons or
st i mul i , and wi t h ot her t i me occasi ons. When anal yzi ng
covar i at i on wi t h mul t i pl e di mensi ons, causal i t y can not onl y be
at t r i but ed t o s i ngl e causes, but al so t o t hei r interactions or
configurations . For exampl e, a car acci dent i s of t en due t o a
concur r ence of c i r cumst ances wher e var i ous causal f act or s must be
pr esent ( e. g. , speedi ng, bad weat her , et c. ) t o pr oduce t he ef f ect .
Such i nt er act i ve causes ar e pr edi ct ed by t he pr obabi l i s t i c model
( Cheng & Novi ck, 1990; Van Over wal l e & Heyl i ghen, 1995; Van
Over wal l e, 1996a) , but ar e pr obl emat i c f or t he or i gi nal Rescor l a-
Wagner model . Sever al connect i oni st amendment s t o t he Rescor l a-
Wagner model have been pr oposed t o deal wi t h conf i gur at i ons, but
none wer e par t i cul ar l y el egant or pl ausi bl e ( e. g. , Gl uck & Bower ,
1988b, Van Over wal l e, 1996b) . Recent l y, however , Pear ce ( 1994)
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Causal Attribution 24
devel oped an connect i oni st ext ensi on t o Rescor l a and Wagner ' s
model whi ch deal s qui t e ni cel y wi t h conf i gur at i ons. We wi l l f i r st
di scuss Pear ce' s model and t hen expl or e some r el evant empi r i cal
data.
Pearce's Configural Network Model
Pear ce was i nspi r ed by t he common obser vat i on i n condi t i oni ng
r esear ch t hat ani mal s r espond not onl y t o t he t r ai ni ng cue but
al so t o ot her cues t hat ar e s i mi l ar t o i t , a phenomenon t hat i s
t er med generalization . The mor e t he ot her cues r esembl e t he
t r ai ni ng cue, t he mor e t he or gani sm expect s t he same ef f ect and
r esponds i n a s i mi l ar manner ( Pear ce, 1987, 1994) . Si mi l ar l y,
humans may not onl y at t r i but e causal i t y t o t he t r ue cause t hat
covar i es wi t h t he ef f ect , but al so t o ot her f act or s t hat shar e
s i mi l ar f eat ur es wi t h i t . However , such si mi l ar f act or s, i n
t hemsel ves, do not necessar i l y covar y wi t h t he ef f ect . Al t hough
gener al i zat i on may t hus be qui t e subopt i mal f r om a st at i st i cal
poi nt of v i ew, i t does have adapt i ve val ue i n r eal l i f e. Ther e
may al ways be some doubt about t he cr i t i cal f eat ur es i n t he t r ue
cause t hat pr oduced t he ef f ect . Because t hese cr i t i cal f eat ur es
may be pr esent i n ot her , s i mi l ar f act or s, i t may be advant ageous
t o gener al i ze causal i t y t o t hese f act or s. Hence, gener al i zat i on
pr ovi des t he basi s f or bui l di ng gener i c knowl edge t hat can be
appl i ed i n many mor e si t uat i ons t han t he or i gi nal l ear ni ng
situation.
To addr ess t he phenomenon of gener al i zat i on, Pear ce ( 1994)
pr oposed a connect i oni st net wor k whi ch assumes t hat per cei ver s
st or e i n memor y r epr esent at i ons of conf i gur al exempl ar s.
Exempl ar s r ef l ect t he whol e st i mul us conf i gur at i on as i t i s
encount er ed i n t he envi r onment ( e. g. , a per son i n a par t i cul ar
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Causal Attribution 25
s i t uat i on at a par t i cul ar t i me) r at her t han i sol at ed f act or s
( e. g. , a per son) . Pear ce' s ( 1994) connect i oni st net wor k consi sts
of t hr ee l ayer s, i n whi ch t he exempl ar s ar e r epr esent ed as
conf i gur al nodes si t uat ed at an i nt er medi at e l ayer i n bet ween t he
i nput and out put l ayer . These conf i gur al nodes di f f er f r om
st andar d hi dden nodes i n ot her wel l - known connect i oni st model s
( McCl el l and & Rumel har t , 1988) . An i l l ust r at i on of t he net wor k i s
gi ven i n Fi gur e 5 f or f our possi bl e combi nat i ons of consensus and
di st i nct i veness i nf or mat i on. Assumi ng t hat t he ext er nal ( E) and
gl obal ( G) cont ext f act or s ar e al ways pr esent , t he P* E* S* G
con f i gur al node r ef l ect s t he pr esence of bot h t he t ar get per son
( P) and st i mul us ( S) , P* E* G r ef l ect s t he pr esence of t he t ar get
per son, E* S* G i ndi cat es t he pr esence of t he t ar get st i mul us, and
E*G denotes that both the target person and stimulus are absent.
--------------------------
Insert Figure 5 about here
--------------------------
Pear ce' s ( 1994) net wor k c l osel y f ol l ows t he Rescor l a- Wagner
speci f i cat i ons wi t h r espect t o t he i nput and t ar get act i vat i ons,
t hat i s , act i vat i on i s set t o 1 f or a f act or or ef f ect t hat i s
pr esent , ot her wi se t he act i vat i on i s 0. The i nput act i vat i on t hen
spr eads t o t he conf i gur al nodes i n pr opor t i on t o t hei r s i mi l ar i t y
( see bel ow) . The mor e a conf i gur al node i s s i mi l ar t o t he pat t er n
of i nput nodes, t he mor e i t wi l l be act i vat ed. Hence, act i vat i on
f r om i nput can spr ead t o var i ous ot her conf i gur at i ons and so
i ndi r ect l y t o ot her f act or s due t o t hei r mut ual s i mi l ar i t y r at her
t han act ual covar i at i on wi t h t he ef f ect . Thi s pr oduces t he ef f ect
of generalization.
The si mi l ar i t y bet ween t he i nput and conf i gur al nodes i s
f i xed ( i ndi cat ed i n t he f i gur e by st r ai ght l i nes) and det er mi ned
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Causal Attribution 26
by the number of common factors, as formalized below :
i
Sh
= n
cn
i
nc
nh
∗ [5]
I n t hi s expr essi on, whi ch i s a s i mpl i f i ed ver si on of t he
or i gi nal f or mul a by Pear ce ( 1987, p. 65; 1994, p. 599) , nc i s t he
number of f act or s common t o t he i nput node and t he conf i gur al
node, ni i s t he number of f act or s pr esent i n t he i nput node, and
nh i s t he number of f act or s pr esent i n t he conf i gur al ( hi dden)
node.
Af t er t he conf i gur al nodes have been act i vat ed, t hei r
act i vat i on spr eads t hr ough adj ust abl e connect i ons ( i ndi cat ed by
br oken l i nes) t o t he out put node. As not ed bef or e, due t o
gener al i zat i on, al l conf i gur al nodes wi l l r ecei ve some degr ee of
act i vat i on, and al l of t hem cooper at e i n act i vat i ng t he out put
node. However , onl y t he conf i gur al node whi ch encodes exact l y t he
i nput i nf or mat i on wi l l r ecei ve maxi mum act i vat i on ( because i Sh =
1) . I t i s onl y t he connect i on bet ween t hi s maximally act i vat ed
conf i gur at i on and t he out put t hat i s adj ust ed af t er each t r i al ,
whi l e t he connect i ons bet ween t he ot her conf i gur at i ons r emai n
unchanged. The adj ust ment f ol l ows t he Rescor l a- Wagner l ear ni ng
al gor i t hm ( Equat i on 3) except , of cour se, t hat t he l i near
summat i on of t he i nput nodes ( Σw) i s now r epl aced by t he l i near
summat i on of t he act i vat i on r ecei ved f r om of t he conf i gur al nodes.
Pear ce' s ( 1994) assumpt i on t hat onl y t he connect i on of t he
maxi mal l y act i vat ed conf i gur at i on i s adj ust ed, st ems f r om t he
or i gi nal Rescor l a- Wagner model wher e onl y t he f act or s pr esent wer e
updated.
Li ke i n t he Rescor l a- Wagner model , an appr opr i at e measur e f or
subj ect s ' causal j udgment s i s s i mpl y t he act i vat i on of t he out put
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Causal Attribution 27
node gi ven t hat t he appr opr i at e i nput nodes ar e t ur ned on. Thi s
pr ocedur e wor ks f i ne when onl y one cont r ast - cont ext pai r ( or
causal di mensi on) i s i nvol ved. Usi ng a s i mi l ar l ogi c as Chapman
and Robbi ns ( 1990) , i t can be shown mat hemat i cal l y t hat Pear ce' s
model -- l i ke t he Rescor l a- Wagner model -- wi l l ar r i ve at
probabilistic norms given sufficient learning trials.
However , when mul t i pl e cont r ast - cont ext pai r s ar e i nvol ved,
t hi s pr ocedur e mi ght not be ent i r el y sat i sf act or y f or Pear ce' s
( 1994) net wor k, because t he act i vat i on of one f act or gener al i zes
t o ot her f act or s f r om ot her di mensi ons vi a t he conf i gur al nodes,
so t hat t he st r engt h of a f act or i s conf ounded and cannot be
t est ed i n i sol at i on. One possi bi l i t y t o r emedy t hi s shor t comi ng,
i s by assumi ng t hat when t he i nput nodes ar e t ur ned on f or one
contrast - cont ext di mensi on, t he at t ent i on or sal i ence f or t he
ot her di mensi ons i s dr ast i cal l y at t enuat ed. For exampl e, when t he
per cei ver t est s t he st r engt h of t he P or E f act or ( i . e. , l ocus
di mensi on) , t hen he or she may st r ongl y r educe t he at t ent i on t o
f act or s f r om ot her di mensi ons. Thi s assumpt i on i s consi st ent wi t h
t he common obser vat i on t hat many di mensi ons of causal i t y ar e
r el at i vel y i ndependent ( cf . Wei ner , 1986) , and t hus can be
at t ended t o separ at el y. We r ef er t o t hi s sel ect i ve at t ent i on
mechanism as the attention strength for other dimensions or αo.
The αo mechani sm was i mpl ement ed i n Pear ce' s s i mi l ar i t y
Equat i on 5 as f ol l ows : I f at l east one f act or of a di mensi on i s
pr esent at i nput , t hen t he def aul t at t ent i on ( = 1) i s used t o
est i mat e t he number of t hese f act or s i n Equat i on 5; conver sel y, i f
no f act or of a di mensi on i s pr esent at i nput , t hen t hei r number i s
est i mat ed by t he ( much r educed) αο at t ent i on val ue. A l ow αo
at t ent i on val ue i mpl i es t hat t he s i mi l ar i t y and st r engt h of a
f act or ( e. g. , P) i s i nf l uenced l ess by conf i gur al nodes cont ai ni ng
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Causal Attribution 28
many di f f er ent f act or s ( e. g. , P* E* S* G) and mor e by conf i gur al
nodes cont ai ni ng l i t t l e el se but t hi s f act or ( e. g. , P* E* G) . Thi s
i s t r ue f or bot h cont r ast and cont ext f act or s. I t i s i mpor t ant t o
not e t hat t hi s sel ect i ve at t ent i on mechani sm af f ect s onl y t he
t est i ng phase of t he model , because dur i ng l ear ni ng al l t he
cont ext f act or s ar e assumed t o be pr esent at i nput , so t hat al l
f act or s ar e equal l y act i ve and cont r i but e equal l y t o t he l ear ni ng
of conf i gur at i ons. Gi ven t hat Pear ce ( 1994) al l owed f or var yi ng
i nt ensi t y or at t ent i on i n t he t er ms of hi s s i mi l ar i t y equat i on, i n
t he net wor k s i mul at i ons descr i bed l at er , αo was f r eel y est i mat ed
bet ween 0 and 1. We expect t hat l ow val ues cl ose t o 0 wi l l r esul t
in a better fit of the model.
Other Networks with Hidden Nodes
Kr uschke ( 1992) r ecent l y pr oposed an exempl ar net wor k wi t h a
s i mi l ar i t y gr adi ent ver y s i mi l ar t o Pear ce' s conf i gur al net wor k,
but t hi s net wor k has f al l en out of f avor because i t s pr edi ct i ons
depend t oo much on l ear ni ng or der ( Lewandowsky, 1995) . Comput er
s i mul at i ons wi t h our dat a pr esent ed i n t he next sect i on, al so
showed t hat t hi s net wor k expl ai ned l ess t han hal f of t he var i ance.
Therefore, we will not further discuss this model.
Anot her c l ass of ver y popul ar connect i oni st net wor ks t hat
make use of an i nt er medi at e l ayer ar e back- pr opagat i on net wor ks
( McCl el l and & Rumel har t , 1988) . These net wor ks di f f er , however ,
f r om t he conf i gur al net wor k i n sever al r espect s. Fi r st , t he
i nt er medi at e nodes i n t he conf i gur al net wor k ar e t r anspar ent as
t hey exact l y copy t he i nput pat t er ns, wher eas back- propagation
net wor ks consi st s of nodes t hat ar e hi dden i n t he sense t hat t he
net wor k i t sel f sear ches f or an opt i mal r epr esent at i on wi t hout
di r ect i nt er vent i on f r om t he i nput . Second, t he l i nks bet ween
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Causal Attribution 29
i nput and conf i gur al nodes ar e f i xed and det er mi ned by t hei r
s i mi l ar i t y, wher eas t he connect i ons bet ween i nput and hi dden nodes
i n back- pr opagat i on net wor ks ar e adapt i ve and det er mi ned by t he
di scr epancy bet ween expect ed and act ual ef f ect t hat i s pr opagat ed
f r om out put l ayer t o t he hi dden l ayer , and t hen t o t he i nput
l ayer . Thi r d, and per haps mor e i mpor t ant l y, t he t wo net wor ks wer e
st i mul at ed by a di f f er ent t heor et i cal concer n. Wher eas t he
conf i gur al net wor k i s based on empi r i cal f i ndi ngs of
gener al i zat i on ef f ect s wi t h ani mal s ( Pear ce, 1987) , back-
pr opagat i on net wor ks have been devel oped mai nl y f r om mat hemat i cal
consi der at i ons on how t o pr opagat e t he er r or di scr epancy
ef f i c i ent l y t hr ough t he net wor k ( Rumel har t , Dur bi n, Gol den &
Chauvi n, 1995) . We sur mi se t hat t he st r onger empi r i cal r oot s of
t he conf i gur al net wor k make i t a mor e appr opr i at e candi dat e f or
model i ng causal l ear ni ng t han convent i onal back- propagation
net wor ks. Because t he conf i gur al net wor k has al r eady been t est ed
ext ensi vel y wi t h ani mal s ( see Pear ce, 1994) , we now t ur n t o some
empi r i cal evi dence wi t h human subj ect s t o compar e t he behavi or of
the configural network with the predictions of other networks.
Experiments and Results
We conduct ed t wo exper i ment s ( Van Over wal l e & Van Rooy, 1996)
i n whi ch subj ect s wer e pr esent ed di f f er ent scenar i os r ef l ect i ng
al l possi bl e t wo- by - t wo combi nat i ons of Kel l ey ' s t hr ee covar i at i on
var i abl es. Af t er r eadi ng each scenar i o, t he subj ect s r at ed t he
causal i nf l uence of s i ngl e cont r ast f act or s ( e. g. , somet hi ng about
t he per son, stimulus , or occasion ) , s i ngl e cont ext f act or s ( e. g. ,
somet hi ng ext er nal , global , or stable ) as wel l as combi nat i ons of
t hese f act or s, usi ng a scal e r angi ng f r om 0 ( no cause) t o 10 ( most
compl et e cause). Li ke i n t he pr evi ous exper i ment , t he i nf or mat i on
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Causal Attribution 30
was pr esent ed under a summar y f or mat or a sequent i al t r i al - by -
t r i al f or mat . The pr edi ct i ons of t he t heor et i cal model s under
consi der at i on ar e i l l ust r at ed i n Tabl e 3 f or consensus and
di st i nct i veness, and can be r eadi l y ext ended t o ot her combi nat i ons
of covar i at i on var i abl es. To f aci l i t at e t he di scussi on, under
each i nf or mat i on pat t er n we l i s t ed t he conf i gur al nodes i mpl i ed by
Pear ce' s ( 1994) model t hat wi l l acqui r e t he st r ongest connect i on
weights.
------ -------------------
Insert Table 3 about here
-------------------------
Si nce t he pr obabi l i s t i c model f ai l ed t o pr edi ct t he i mpor t ant
phenomenon of di scount i ng and augment at i on, we t ur n i mmedi at el y t o
t he pr edi ct i ons of Pear ce' s pr edecessor , t he Rescor l a- Wagner
net wor k. As suggest ed bef or e, gi ven suf f i c i ent l ear ni ng t r i al s,
t hi s net wor k wi l l conver ge t o pr obabi l i s t i c pr edi ct i ons, pr ovi ded
t hat t he i nput nodes ar e coded separ at el y f or each cont r ast -
cont ext di mensi on and t hei r i nt er act i ons ( Van Over wal l e, 1996b).
Hence, t he Rescor l a- Wagner net wor k pr edi ct s t hat when t her e i s a
cont r ast i n t he obser vat i ons, t hen at t r i but i ons ar e made t o
cont r ast causes; conver sel y when t her e i s no such cont r ast ,
at t r i but i ons ar e made t o t he cont ext . Appl i ed t o Tabl e 3, t hi s
i mpl i es t hat at t r i but i ons wi l l be made t o t he per son ( P) gi ven l ow
consensus, and t o ext er nal ( E) causes gi ven hi gh consensus; and
si mi l ar l y, t hat at t r i but i ons wi l l be made t o t he st i mul us ( S)
gi ven hi gh di st i nct i veness, and t o gener al ( G) causes gi ven l ow
di st i nct i veness. These pr edi ct i ons can be combi ned t o pr oduce
at t r i but i ons t o i nt er act i ons ( see t op panel ) . I n sum, t he
Rescorla - Wagner model assumes t hat t hese si ngl e f act or s and t hei r
i nt er act i ons wi l l r ecei ve s i gni f i cant positive causal st r engt h,
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Causal Attribution 31
wher eas al l ot her f act or s ( i . e. , not t abl ed i n t he t op panel ) ar e
assumed to have approximately null causal strength.
Pear ce' s conf i gur al net wor k model makes si mi l ar pr edi ct i ons
wi t h r espect t o t he causal f act or s depi ct ed i n t he t op panel of
Tabl e 3, but f or ent i r el y di f f er ent r easons. As expl ai ned bef or e,
t he conf i gur al net wor k pr edi ct s t hat most causal st r engt h wi l l be
acqui r ed by t he conf i gur al node t hat i s al ways f ol l owed by t he
ef f ect ( see t abl e header ) . For i nst ance, hi gh consensus and hi gh
di st i nct i veness i ndi cat es t hat t he cont r ast st i mul us ( S) , t oget her
wi t h t he ext er nal ( E) and gl obal ( G) cont ext s, ar e f ol l owed by t he
ef f ect . Thi s r ei nf or ces most st r ongl y t he connect i on wei ght of
t he E* S* G conf i gur al node. Thi s wei ght wi l l t hen gener al i ze t o
other causes t hat f or m a par t of t he conf i gur at i on ( e. g. , E* S* G
gener al i zes t o E, S & E* S) . Ther ef or e, t hese causes wi l l r ecei ve
subst ant i al positive causal st r engt h. At hi s poi nt , Pear ce' s
conf i gur al net wor k makes ver y s i mi l ar pr edi ct i ons as t he Rescor l a-
Wagner model . I n addi t i on, Pear ce' s net wor k makes t he uni que
pr edi ct i on t hat some amount of t he causal st r engt h of t he
maxi mal l y connect ed conf i gur al node wi l l al so gener al i ze t o ot her
i nt er act i ons t hat ar e not par t of i t , but shar e a f act or ( e. g. ,
E*S* G gener al i zes t o P* G & E* G) . These i nt er act i ons ar e denot ed
generalized causes because t hey do not covar y at al l wi t h t he
ef f ect , but wi l l r ecei ve some causal st r engt h ( see mi ddl e panel ) .
The st r engt h of t hese gener al i zed causes shoul d be st r onger t han
t hat of t he r emai ni ng causes whi ch ar e compl et el y di ssi mi l ar , and
therefore receive null causal strength (see bottom panel).
A summar y of t he r esul t s i s gi ven i n Tabl e 4. As can be
seen, t he r at i ngs pr ovi ded by our subj ect s wer e most consi st ent
wi t h Pear ce' s ( 1994) conf i gur al net wor k. Ther e was st r ong suppor t
f or t he pr edi ct i on, shar ed wi t h t he Rescor l a- Wagner model , t hat
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Causal Attribution 32
t he at t r i but i on r at i ngs f or posi t i ve f act or s ar e hi gher t han f or
gener al i zed and nul l f act or s. Conf or m t o t he uni que pr edi ct i on of
t he conf i gur al net wor k, however , most gener al i zed causes r ecei ved
at t r i but i on r at i ngs t hat wer e subst ant i al l y hi gher t han nul l
causes.
-------------------------
Insert Table 4 about here
-------------------------
Model Simulations
To pr ovi de addi t i onal conf i r mat i on f or t hese r esul t s and t o
t est our pr oposed amendment wi t h r espect t o t he αo par amet er , we
comput ed si mul at i ons of Pear ce' s conf i gur al net wor k and compar ed
t hem wi t h t he pr edi ct i ons of t he Rescor l a- Wagner model . I n
addi t i on, we wi l l al so pr esent s i mul at i ons wi t h t he wi del y used
back - propagation network.
We used t he same pr ocedur e as i n t he ear l i er s i mul at i ons.
The i nf or mat i on pr ovi ded t o t he subj ect s was encoded i n each
net wor k, and connect i on wei ght s wer e updat ed af t er each t r i al .
For t he Rescor l a- Wagner net wor k, t her e wer e separ at e out put nodes
f or each cont r ast - cont ext di mensi on and f or t hei r i nt er act i ons,
wi t h a common t eachi ng val ue and a common βw l ear ni ng r at e
par amet er . Thi s ar chi t ect ur e guar ant ees t hat t he net wor k wi l l
conver ge t owar ds pr obabi l i s t i c nor ms ( Van Over wal l e, 1996b) . For
Pear ce' s conf i gur al net wor k, t he t r i al i nf or mat i on i mpl i ed f our
conf i gur al nodes ( see Tabl e 3) , wi t h a βw l ear ni ng r at e and a αo
sel ect i ve at t ent i on par amet er . For r easons of compar abi l i t y , we
t ook t he same number of hi dden nodes i n t he back- propagation
net wor k, and al so t he same number of par amet er s, i ncl udi ng a βw
l ear ni ng r at e and a αm moment um par amet er . The moment um par amet er
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Causal Attribution 33
r ef l ect s t he ef f ect of t he past wei ght updat e on t he cur r ent
updat e, and so ef f ect i vel y f i l t er s out st r ong osci l l at i ons i n t he
updat es ( McCl el l and & Rumel har t , 1988, p. 136) . Because, as not ed
ear l i er , t he cont ext f act or s act as a k i nd of bi as t er ms, we di d
not i ncl ude addi t i onal bi as wei ght s ( McCl el l and & Rumel har t ,
1988) . As bef or e, f or al l model s, we cal cul at ed al l admi ssi bl e
par amet er s val ues bet ween 0 and 1, and t hen sel ect ed t he val ues
whi ch at t ai ned t he hi ghest cor r el at i on bet ween si mul at ed and
observed data.
Because t he i nf or mat i on i n t he exper i ment s was ei t her
pr esent ed r andoml y ( i n t he sequent i al f or mat ) or wi t hout any
par t i cul ar or der ( i n t he summar y f or mat ) , f or each st or y, we r an
100 si mul at i ons wi t h di f f er ent r andom t r i al or der s2. Tabl e 5
pr esent s t he summar y f i t R f or each model . The r esul t s show t hat
al t hough t he Rescor l a- Wagner net wor k obt ai ned an adequat e f i t
( mean R = . 759) , i t was t he conf i gur al net wor k whi ch r eached a
s l i ght l y bet t er f i t over al l ( mean R = . 793) . The l ear ni ng r at e of
bot h model s was hi ghest f or t he sequent i al pr esent at i on f or mat ,
suggest i ng ( per haps cont r ar y t o i nt ui t i on) t hat t he somewhat mor e
compl ex st i mul us mat er i al i n t he pr esent exper i ment s was l ear ned
most qui ckl y when pr esent ed t r i al - by - t r i al . I n cont r ast , t he
back - pr opagat i on s i mul at i on f ai l ed t o r epr oduce our dat a t o any
reasonable degree ( mean R = .171).
Per haps, t he hi gh f i t of t he conf i gur al model was par t l y due
t o t he i nt r oduct i on of t he novel αo par amet er . As expect ed, t he
est i mat es of t hi s par amet er wer e ver y l ow, conf i r mi ng our i dea
t hat t o t est causal st r engt hs i ndependent l y, t he i nf l uence of
ot her di mensi ons needs t o be cancel ed out . Omi t t i ng t hi s
par amet er so t hat al l di mensi ons wer e equal l y act i vat ed dur i ng
t est i ng, r educed t he R f i t measur e f or t he conf i gur al net wor k by
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Causal Attribution 34
. 086 and . 146 i n t he sequent i al and summar y f or mat r espectively.
However , on t he basi s of t he pr esent r esul t s al one, i t i s
di f f i cul t t o assess t he gener al i t y of our αo sol ut i on f or ot her
material.
Why di d t he back- pr opagat i on model per f or m so poor l y ?
Addi t i onal s i mul at i ons may pr ovi de some hi nt s. When 20 i nst ead of
4 hi dden nodes wer e used, t he f i t di d not i mpr ove. However ,
r epeat i ng t he or i gi nal t r i al i nf or mat i on of each st or y i mpr oved
t he f i t subst ant i al l y , but not t o t he same degr ee as t he ot her
model s ( mean R = . 277 wi t h 10 r epet i t i ons and mean R = . 430 wi t h
100 r epet i t i ons, al l wi t h t he same par amet er val ues of Tabl e 5) .
Al t hough per haps mor e r epet i t i ons or s l i ght l y di f f er ent par amet er
val ues mi ght have accommodat ed t he dat a bet t er , t he model seems
unabl e t o l ear n t he conf i gur at i ons ( and t hei r hi dden
r epr esent at i on) at a r easonabl e speed. Thi s suggest s t hat
Pear ce' s ( 1994) exempl ar r epr esent at i on of hi dden conf i gur at i ons
is crucial in the superior and faster performance of his model.
-------------------------
Insert Table 5 about here
---------------- ---------
Conclusions
The dat a pr esent ed i n t hi s chapt er c l ear l y demonst r at e t hat a
connect i oni st appr oach has much pr omi se f or our under st andi ng of
t he pr ocesses under l y i ng causal r easoni ng. We have shown how t he
Rescorla - Wagner net wor k can easi l y deal wi t h di scount i ng and
augment at i on ef f ect s and how Pear ce' s conf i gur al net wor k pr edi ct s
gener al i zat i on of causal i t y, t wo f i ndi ngs whi ch ar e pr obl emat i c
f or t he or i gi nal covar i at i on pr i nci pl e of Kel l ey ( 1967, 1971) as
wel l as f or a pr obabi l i s t i c concept i on of i t ( Cheng & Novi ck,
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Causal Attribution 35
1990) . Ot her cogni t i ve r esear ch wi t h humans has document ed t hat
t he Rescor l a- Wagner model i s super i or t o t he pr obabi l i s t i c
appr oach i n expl ai ni ng compet i t i on ef f ect s l i ke di scount i ng and
augment at i on ( Baker et al . , 1993; Chapman & Robbi ns, 1990;
Chapman, 1991; Gl uck, & Bower , 1988a, 1988b; Shanks, 1985, 1993,
1995; Val l ée- Tour angeau, Baker & Mer ci er , 1994) , and t her e i s al so
i ncr easi ng evi dence t o suggest t hat ani mal s pr ocess covar i at i on
i nf or mat i on i n conf i gur al uni t s r at her t han el ement al f eat ur es
( f or a r evi ew see Pear ce, 1989, 1994) . On a t heor et i cal l evel ,
t he connect i oni st appr oach may expl ai n how humans ar e capabl e t o
det ect causal r el at i onshi ps whi l e usi ng l i t t l e cogni t i ve sour ces
and ef f or t , so t hat i t pr ovi des a mor e pl ausi bl e account of causal
reasoning during the hurry of everyday social life.
The pr esent connect i oni st appr oach t o at t r i but i on l eaves a
number of unr esol ved i ssues. The pi ct ur e of l ear ni ng t hat emer ges
f r om connect i oni st net wor ks i s of a r at her passi ve pr ocess, i n
whi ch act i vat i ons spr ead aut omat i cal l y and wei ght s ar e adj ust ed
i mmedi at el y. However , humans may al so t ake a mor e act i ve r ol e i n
whi ch t hey consi der var i ous causal hypot heses t hat may expl ai n an
out come. Recent l y, Read ( Read & Mar cus- Newhal l , 1993) pr oposed a
connect i oni st net wor k t o account f or t hi s pr ocess of hypot hesi s
sel ect i on. He suggest ed t hat humans' causal knowl edge i n a
par t i cul ar domai n can be r epr esent ed by a l ar ge net wor k st r uct ur e,
wi t h each node r epr esent i ng a domai n- r el evant causal f act or . Al l
t hese f act or s compet e t o acqui r e some degr ee of act i vat i on, but
onl y t he node wi t h t he hi ghest act i vat i on i s chosen as t he most
pl ausi bl e hypot hesi s. However , because t he connect i ons i n Read' s
net wor k ar e not adapt i ve, mor e wor k needs t o be done t o i nt egr at e
it with the present approach.
Anot her i nt r i gui ng quest i on i s how pr eci sel y connect i ons ar e
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Causal Attribution 36
adj ust ed gi ven summar i zed i nf or mat i on. The connect i oni st appr oach
has no obvi ous means of expl ai ni ng t hi s because t her e ar e no
separ at e t r i al s dur i ng whi ch i nput act i vat i on spr eads t hr ough t he
net wor k and wei ght s ar e adj ust ed. Our s i mul at i ons of t he summar y
dat a wer e, i n f act , car r i ed out by i mposi ng phant om " t r i al s" t o
t he net wor ks. I t i s possi bl e t hat t he pr i mi t i ve mechani sm of
associ at i ve l ear ni ng has been adapt ed dur i ng human evol ut i on f or
t he novel t ask of i nt er pr et i ng ver bal summar y st at ement s, because
nat ur e t ypi cal l y r e- uses subsyst ems t hat ar e al r eady capabl e of
f unct i oni ng on t hei r own ( Beecher , 1988) . One possi bi l i t y i s t hat
humans por t r ay t he summar y i nf or mat i on i n t he f or m of dummy
ent i t i es or ment al model s ( cf . , Johnson- Lai r d, 1983) , whi ch ar e
t hen sequent i al l y anal yzed by an associ at i ve pr ocessor . However ,
t hi s i s mer e specul at i on and we know of no di r ect evi dence t hat
may support this hypothesis.
Leavi ng t hese i nt er est i ng quest i ons asi de, we suspect t hat
t her e i s pot ent i al f or f eedf or war d connect i oni st model s t o expl ai n
even a br oader r ange of soci al phenomena i n whi ch t he det ect i on of
covar i at i on pl ays a r ol e. For i nst ance, t he connectionist
appr oach may pr ovi de an al t er nat i ve account f or some i nt r i gui ng
f i ndi ngs i n gr oup st er eot ypi ng, such as i l l usor y cor r el at i ons
( Hami l t on & Gi f f or d, 1976) . I l l usor y cor r el at i on i s t he r obust
phenomenon t hat per cei ver s j udge mi nor i t y gr oups mor e negat i vel y
t han maj or i t y gr oups even when t he pr opor t i on of t hei r posi t i ve
and negat i ve behavi or s i s i dent i cal ( e. g. , t wi ce as much posi t i ve
t o negat i ve behavi or s) . As not ed ear l i er , connect i oni st model s
concei ve of l ear ni ng as a gr adual pr ocess by whi ch t he connect i on
wei ght s i ncr ement al l y i ncor por at e new i ncomi ng i nf or mat i on. Let
us assume t hat t he posi t i ve and negat i ve behavi or s ar e encoded i n
t wo i nput nodes, connect ed vi a modi f i abl e wei ght s wi t h t wo out put
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Causal Attribution 37
nodes r epr esent i ng t he maj or i t y and mi nor i t y gr oup. Because t her e
i s much more behavi or al i nf or mat i on about t he majority gr oup t han
about t he min or i t y gr oup, and because l ear ni ng occur s
i ncr ement al l y, t he connect i ons bet ween t he behavi or al i nput nodes
and t he maj or i t y out put node wi l l gr ow much strong er t han t hose of
t he min or i t y out put node. I n f act , t he maj or i t y ' s connect i ons
wi l l al most r each asympt ot e, t hat i s , t hey wi l l r each wei ght s t hat
r ef l ect t he t r ue pr opor t i on of pos i t i ve and negat i ve behaviors .
I n cont r ast , t he mi nor i t y ' s connect i ons wi l l r each onl y weak pr e-
asymptotic wei ght s, so t hat any di f f er ence bet ween t he per cei ved
strength of posi t i ve and negat i ve behavi or s i s mi ni mal and
insignificant . Thi s r esul t , whi ch can be easi l y s i mul at ed wi t h a
s i mpl e two - la yer f eedf or war d network ( McCl el l and & Rumel hart,
1988) , may be r esponsi bl e f or per cei ver s ' bi ased per cept i on of the
minority group as being l ess positive.
Gi ven t hat connect i oni st model s ar e an i deal i zed r ef l ect i on
of t he neur al wor ki ngs of t he human br ai n, we suspect t hat t hey
wi l l per haps i ncr ease our under st andi ng of ot her causal phenomena,
such as how per cei ver s i nf er di sposi t i onal at t r i but i ons about
ot her per sons usi ng covar i at i on i nf or mat i on ( cf . , Hi l t on, Smi t h &
Ki m, 1995) , and how peopl e f al l pr ey t o ot her causal i l l usi ons,
such as t he cor r espondence bi as ( Gi l ber t , 1989) . We suspect t hat
t hese and many ot her phenomena i n soci al r easoni ng ar e not so much
a t r i cky r esul t of our soci al per cept i ons, societal r ul es, or of
t he demandi ng ci r cumst ances of ever yday soci al l i f e, but s i mpl y
the outcome of a connectionist processing mechanism .
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Causal Attribution 38
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Causal Attribution 43
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Causal Attribution 44
Table 1.
Mean Attribution ratings per Factor and Condition
Discounting Augmentatio n
Large Set Small Set Large Set Small Set
Sequential Presentation
Person 3.07 < 5.33 8.87 > 7.30
Stimulus 3.13 3.57 7.67 7.23
Summary Presentation
Person 4.03 < 7.67 8.63 > 5.93
Stimulus 2.67 < 5.90 7.90 > 6.03
Note. Significant differences ( p<0.5) are indicated by > or <.
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Causal Attribution 45 Table 2. Fits of the Models to the Discounting and Augmentation Data Model R Model Parameter
Sequential Presentation
Probabilistic Joint .358
Rescorla - Wagner .854 βw = .80
Summarized Presentation
Probabilistic Joint .270
Rescorla - Wagner .751 βw = .80
Note . R = correlation, βw = learning rate parameter.
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Causal Attribution 46
Table 3.
Causal Strength predictions of Rescorla - Wagner's and Pearce's
Network illustrated for Consensus and Distinctiveness.
Consensus High Low Distinctiveness Low High Low High Configural nodes a E*G E*S*G P*E*G P*E*S*G
Causal Type Positive E*G E*S P*G P*S E E P P G S G S Generalized b P*G P*S P*S P*G E*S E*G E*G E*S Null P*S P*G E*S E*G P P E E S G S G
Note . P = person, E = external, S = stimulus, G = general. a Conf i gur al nodes i n Pear ce' s model that will acquire the
st r ongest wei ght s. b Nul l st r engt h predicted by Rescorla &
Wagner, generalized strength predicted by Pearce.
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Causal Attribution 47
Table 4.
Mean Causal Ratings in function of Causal Strength Type.
Positive Generalized Null
Presentation Sequential 5.46 3.87 2.90 Summary 6.84 4.17 2.82
Note . Means di f f er s i gni f i cant l y bet ween al l t hr ee causal
types ( p<.002)
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Causal Attribution 48
Table 5. Fits of the Models to the Attribution Data Model R Model Parameters
Sequential Presentation
Rescorla - Wagner .706 βw = .62
Configural .742 βw = 1.00; αo = .13
Back - Propagation .152 βw = .81; αm = .52
Summarized Presentation
Rescorla - Wagner .812 βw = .41
Configural .844 βw = .42; αo = .03
Back - Propagation .191 βw = .88; αm = .99
Note . R = maxi mum cor r el at i on; Model par amet er s ar e : βw =
l ear ni ng r at e; αo = at t ent i on f or ot her di mensi ons not pr esent
at i nput ; αm = moment um. Ther e wer e 3 compar i son t r i al s i n t he
sequent i al f or mat ; t hei r number i n t he summar y f or mat was
simulated as 1/5 (see footnote 2).
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Causal Attribution 49
Figure Captions
Figure 1. A cont i ngency t abl e i l l ust r at i ng a cont r ast f act or ( C)
and i t s cont ext f act or ( X) when t he ef f ect i s pr esent and
absent. The letters a - d reflect frequencies.
Figure 2 . A net wor k r epr esent at i on of t he Rescor l a- Wagner model
gi ven a cont r ast f act or ( C) and i t s cont ext ( X) , t oget her
wi t h some i l l ust r at i ve codi ng f or i nf or mat i on pr esent ed
during learning and questions presented during testing.
Figure 3 . A s i mul at i on of causal l ear ni ng wi t h par amet er βw = . 50.
The l ef t panel r ef l ect s a l ear ni ng hi st or y of CX → Effect
and X → No Ef f ect t r i al s; wher eas t he r i ght panel r ef l ect s
X → Effect and CX → No Effect trials.
Figure 4 . A s i mul at i on of di scount i ng and augment at i on wi t h
par amet er βw = . 50, i l l ust r at ed f or a per son ( P) cause and
i t s ext er nal ( E) cont ext . The condi t i ons i nvol ved f i ve
( l ar ge set ) or one ( smal l set ) E → Ef f ect t r i al s, f ol l owed
by one PE → Ef f ect t r i al i n t he di scount i ng condi t i on, or
one PE → No Effect trial in the augmentation condition.
Figure 5 . The conf i gur al net wor k pr oposed by Pear ce ( 1994) ,
illustrated for four factors P, E, S and G.
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Figure 1
C X
X
Effect
No Effect
a b
c d
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Figure 2
Out put Layer
I nput Layer C X
Information coding : C =X =
Question coding : C =X =
10
10
11
01
Associative links
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Figure 3
-1
0
1
0 1 2 3 4 5 6
Trials
Wei
gh
t
X
C
-1
0
1
0 1 2 3 4 5 6
Trials
C
X
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Figure 4
Discounting
Large Set
0
1
0 1 2 3 4 5 6
Trials
Str
eng
ht
P
E
Small Set
0
1
0 1 2
Trials
P
E
Augmentation
Large Set
-1
0
1
0 1 2 3 4 5 6
Trials
Str
eng
ht
P
E
Small Set
-1
0
1
0 1 2
Trials
P
E
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Figure 5
Out put Layer
Conf i gur al Layer
Information coding : PS =ES =
10
11
11
11
Question coding :
10
01
11
P =E =
10
01
00
00
00
PS =ES =
PESG
P E S G
PEG EGESG
I nput Layer
Associative links
Similarity links
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Causal Attribution
Footnotes
1 Thi s pr obabi l i s t i c r ul e i s al so t er med the delta - P rule. To
avoi d any conf usi on wi t h t he delta l ear ni ng al gor i t hm f r om
connectionist learning models, we will not use this term.
2 Gi ven t hat t he amount of compar i son cases was not speci f i ed
i n t he summar y f or mat , t hei r number was est i mat ed by r unni ng
si mul at i ons of t he pr obabi l i s t i c j oi nt model wi t h di f f erent
wei ght s f or t he f r equenci es of t he compar i son cases. The hi ghest
f i t was obt ai ned when t he compar i son cases r ecei ved one f i f t h of
t he wei ght of t he t ar get cases. The si mul at i ons wi t h t he
connect i oni st model s wer e t hen car r i ed out wi t h t he number of
t r i al s of compar i son and t ar get cases adj ust ed t o t hat same
proportion, that is, five target trials for each comparison trial.
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