Causal Attribution 1 A Connectionist Approach to Causal ...ritter.ist.psu.edu/misc/dirk-files/Papers/VanOverwalle/pubread.pdf · Causal Attribution 2 Introduction Attributing a cause

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].

No license: PDF produced by PStill (c) F. Siegert - http://www.this.net/~frank/pstill.html


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 &



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



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



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.



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,



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



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



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



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.



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



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.

--------------------------


--------------------------

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



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]



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



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.

--------------------------


--------------------------

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).



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



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)



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



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.

--------------------------


--------------------------

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.



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



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

-------------------------



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 =



. 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.

-------------------------


-------------------------

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)



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



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.

--------------------------


--------------------------

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



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



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



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



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



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.

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-------------------------

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,



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



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.

-------------------------


-------------------------

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



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



. 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.

-------------------------


---------------- ---------

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,



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



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



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 .



References

Abr amson, L. V. , Sel i gman, M. E. P. , & Teasdal e, J. D. ( 1978) .

Lear ned hel pl essness i n humans : Cr i t i que and r ef or mul at i on.

Journal of Abnormal Psychology, 87 , 49 - 74.

Al l en, L. G. ( 1993) . Human cont i ngency Judgment s : Rul e based or

associative ? Psychological Bulletin, 114 , 435 - 448.

Baker , A. G. , Mer ci er , P. , Val l ée- Tour angeau, F. , Fr ank, R. &

Pan, M. ( 1993) . Sel ect i ve associ at i ons and causal i t y

j udgment s : Pr esence of a st r ong causal f act or may r educe

j udgment s of a weaker one. Jour nal of Exper i ment al

Psychology : Learning, Memory and Cognition, 19 , 414 - 432.

Beecher , M. D. ( 1988) . Some comment s on t he adapt at i oni st

appr oach t o l ear ni ng. I n R. C. Bol l es & M. D. Beecher ( Eds. )

Evolution and learning . Hillsdale, NJ : Erlbaum.

Chapman, G. B. & Robbi ns, S. J. ( 1990) . Cue i nt er act i on i n human

contingency judgment. Memory and Cognition, 18 , 537 - 545.

Chapman, G. B. ( 1991) . Tr i al or der af f ect s cue i nt er act i on i n

cont i ngency j udgment Jour nal of Exper i ment al Psychol ogy :

Learning, Memory and Cognition, 17 , 837 - 854.

Cheng, P. W. , & Hol yoak, K. J. ( 1995) . Compl ex adapt i ve syst ems as

intuitive st at i st i c i ans : Causal i t y, cont i ngency, and

pr edi ct i on. I n H. L. Roi t bl at , & J. - A. Meyer ( Eds. )

Compar at i ve appr oaches t o cogni t i ve sci ence. Cambr i dge, MA :

MIT Pre ss.

Cheng, P. W. , & Novi ck, L. R. ( 1990) . A pr obabi l i s t i c cont r ast

model of causal i nduct i on. Jour nal of Per sonal i t y and Soci al

Psychology, 58 , 545 - 567.



Gi l ber t , D. T. ( 1989) Thi nki ng l i ght l y about ot her s : Aut omat i c

component s of t he soci al i nf er ence pr ocess. I n J. S. Ul eman

& J. A. Bar gh ( Eds. ) Uni nt ended t hought . New Yor k, NY :

Guilford.

Gl uck, M. A. & Bower , G. H. ( 1988a) . Fr om condi t i oni ng t o cat egor y

l ear ni ng : An adapt i ve net wor k model . Jour nal of

Experimental Psychology: General, 117 , 227 - 247.

Gl uck, M. A. & Bower , G. H. ( 1988b) . Eval uat i ng an adapt i ve

net wor k model of human l ear ni ng. Jour nal of Memor y and

Language, 27 , 166 - 195.

Hami l t on, D. L. & Gi f f or d, R. K. ( 1976) . I l l usor y cor r el at i on i n

i nt er per sonal per cept i on : A cogni t i ve basi s of st er eot ypi c

j udgment . Jour nal of Exper i ment al Soci al Psychol ogy, 12,

392 - 407.

Hami l t on, D. L. , Dr i scol l , D. M. & Wor t h, L. T. ( 1989) . Cogni t i ve

or gani zat i on of i mpr essi ons : Ef f ect s of i ncongr uency i n

compl ex r epr esent at i ons. Jour nal of Per sonal i t y and Soci al

Psychology, 57 , 925 - 939.

Hansen, D. H. & Hal l , C. A. ( 1985) . Di scount i ng and augment i ng

f aci l i t at i ve and i nhi bi t or y f or ces : t he wi nner t akes al most

al l . Jour nal of Per sonal i t y and Soci al Psychol ogy, 49, 1482-

1493.

Hawki ns, R. D. ( 1989) . A bi ol ogi cal l y r eal i st i c neur al net wor k

model f or hi gher - or der f eat ur es of c l assi cal condi t i oni ng.

I n R. G. M. Mor r i s ( Ed. ) Par al l el di st r i but ed pr ocessi ng :

I mpl i cat i ons f or psychol ogy and neur obi ol ogy ( pp. 214- 247).

Oxford : Clarendon Press



Hewst one, M. & Jaspar s, J. ( 1987) . Covar i at i on and causal

at t r i but i on : A l ogi cal model of t he i nt ui t i ve anal ysi s of

var i ance. Jour nal of Per sonal i t y and Soci al Psychol ogy, 53,

663 - 673.

Hi l t on, D. J. , & Sl ugoski , B. R. ( 1986) . Knowl edge- based causal

at t r i but i on : The abnor mal condi t i ons f ocus model .

Psychological Review, 93 , 75 - 88.

Hi l t on, D. J. , Smi t h, R. H. , & Ki m S. H. ( 1995) . The pr ocess of

causal expl anat i on and di sposi t i onal at t r i but i on. Jour nal of

Personality and Social Psychology, 68 , 377 - 387.

Johnson - Lai r d, P. N. ( 1983) . Ment al Model s : Towar ds a cogni t i ve

sci ence of l anguage, i nf er ence, and consci ousness. Cambr i dge

: Cambridge University Press.

Kahneman, D. , Sl ovi c, P. & Tver sky, A. ( 1982) Judgment s under

uncer t ai nt y : Heur i st i cs and bi ases. Cambr i dge : Cambr i dge

University Press.

Kami n, L. J. ( 1968) . At t ent i on- l i ke pr ocesses i n c l assi cal

condi t i oni ng. I n M. R. Jones ( Ed. ) , Mi ami Symposi um on t he

pr edi ct i on of behavi or : Aver si ve st i mul at i on, Mi ami :

University of Miami Press.

Kassi n, S. M. & Lowe, C. A. ( 1979) . On t he devel opment of t he

augment at i on pr i nci pl e : A per cept ual appr oach. Child

Development , 50 , 728 - 734.

Kassi n, S. M. , Lowe, C. A. & Gi bbons, F. X. ( 1980) . Chi l dren’s

use of t he di scount i ng pr i nci pl e : A per cept ual appr oach.

Journal of Personality and Social Psychology, 39 , 719 - 728.

Kel l ey, H. H. ( 1967) . At t r i but i on i n soci al psychol ogy. Nebraska



Symposium on Motivation, 15 , 192 - 238.

Kel l ey, H. H. ( 1971) . At t r i but i on i n soci al i nt er act i on. I n E.

E. Jones, D. E. Kanouse, H. H. Kel l ey, R. E. Ni sbet t , S.

Val i ns & B. Wei ner ( Eds. ) At t r i but i on : Per cei v i ng t he causes

of behavior. Morristown, NJ : General Learning Press.

Kr ugl anski , A. W. , Schwar t z, S. M. & Hamel , I . Z. ( 1978) .

Covar i at i on, di scount i ng, and augment at i on : Towar ds a

c l ar i f i cat i on of at t r i but i onal pr i nci pl es. Jour nal of

Personality, 76, 176 - 189.

Kr uschke, J. K. ( 1992) . ALCOVE : An exempl ar - based connect i oni st

model of category learning. Psychological Review, 99 , 22 - 44.

Lewandowsky, S. ( 1995) . Base- r at e negl ect i n ALCOVE : A cr i t i cal

reevaluation. Psychological Review, 102 , 185 - 191.

McCl el l and, J. M. & Rumel har t , D. E. ( 1988) . Expl or at i ons i n

par al l el di st r i but ed pr ocessi ng : A handbook of model s,

programs and exercises . Cambridge, MA : Bradford.

Newman, L. S. & Rubl e, D. N. ( 1992) . Do young chi l dr en use t he

di scount i ng pr i nci pl e ? Jour nal of Exper i ment al Soci al

Psychology, 28 , 572 - 593.

Pear ce, J. M. ( 1987) . A model f or st i mul us gener al i zat i on i n

Pavlovian Conditioning. Psychological Review, 94 , 61 - 73.

Pear ce, J. M. ( 1994) . Si mi l ar i t y and di scr i mi nat i on : A sel ect i ve

r evi ew and a connect i oni st model . Psychol ogi cal Revi ew, 101,

587 - 607.

Read, S. J. & Mar cus- Newhal l , A. ( 1993) Expl anat or y coher ence i n

soci al expl anat i ons : A par al l el di st r i but ed pr ocessi ng



account . Jour nal of Per sonal i t y and Soci al Psychol ogy, 65,

429 - 447.

Rescor l a, R. A. & Wagner , A. R. ( 1972) . A t heor y of Pavl ovi an

condi t i oni ng : Var i at i ons i n t he ef f ect i veness of

r ei nf or cement and nonr ei nf or cement ( pp. 64- 98) . I n A. H.

Bl ack & W. F. Pr okasy ( Eds. ) Cl assi cal condi t i oni ng I I :

Current research and theory.

Rumel har t , D. E. , Dur bi n, R. , Gol den, R. & Chauvi n, Y. ( 1995) .

Back - pr opagat i on : The basi c t heor y. I n Y. Chauvi n & D. E.

Rumel har t ( Eds. ) Back - pr opagat i on : Theor y, ar chi t ect ur e and

applications . Hillsdale, NJ : Erlbaum.

Shanks, D. R. ( 1985) . For war d and backwar d bl ocki ng i n human

cont i ngency j udgment . Quar t er l y Jour nal of Exper i ment al

Psychology, 37b , 1 - 21.

Shanks, D. R. ( 1991) . Cat egor i zat i on by a connect i oni st net wor k.

Jour nal of Exper i ment al Psychol ogy : Lear ni ng, Memor y and

Cognition, 17 , 433 - 443.

Shanks, D. R. ( 1993) . Human i nst r ument al l ear ni ng : A cr i t i cal

r evi ew of dat a and t heor y. Br i t i sh Jour nal of Psychol ogy,

84, 319 - 354.

Shanks, D. R. ( 1995) . I s human l ear ni ng r at i onal ? Quarterly

Journal of Experimental Psychology, 48a , 257 - 279.

Vallée - Tour angeau, F. , Baker , A. G. , & Mer ci er , P. ( 1994) .

Di scount i ng i n causal i t y and covar i at i on j udgment s. The

Quarterly Journal of Experimental Psychology, 47B , 151 - 171.

Van Over wal l e, F. ( 1996a) . A t est of t he j oi nt model of causal

attribution . Eur opean Jour nal of Soci al Psychol ogy, in



press .

Van Over wal l e, F. ( 1996b) . The r el at i onshi p bet ween t he Rescor l a-

Wagner associ at i ve model and t he pr obabi l i s t i c j oi nt model of

causality. Psychologica Belgica , in press .

Van Over wal l e, F. , & Heyl i ghen, F. ( 1995) . Rel at i ng covar i at i on

i nf or mat i on t o causal di mensi ons t hr ough pr i nci pl es of

cont r ast and i nvar i ance. Eur opean Jour nal of Soci al

Psychology, 25 , 435 - 455.

Van Over wal l e, F. , & Van Rooy, D. ( 1996) . Gener al i zat i on beyond

covar i at i on : A compar i son bet ween pr obabi l i s t i c and

connect i oni st model s of causal at t r i but i on. Manuscr i pt

submitted for publication

Van Rooy, D. & Van Over wal l e, F. ( 1996) . A connect i oni st appr oach

t o di scount i ng and augment at i on i n causal at t r i but i on.

Manuscript submitted for publication

Wal dmann, M. R & Hol yoak, K. J. ( 1992) . Pr edi ct i ve and di agnost i c

l ear ni ng wi t hi n causal model s : Asymmet r i es i n cue

compet i t i on. Jour nal of Exper i ment al Psychol ogy : Gener al ,

121 , 222 - 236.

Wei ner , B. ( 1986) . An at t r i but i onal t heor y of achi evement

motivation and emotion . New York, NJ : Springer - Verlag.



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 <.


Causal Attribution 45 Table 2. Fits of the Models to the Discounting and Augmentation Data Model R Model Parameter


Probabilistic Joint .358

Rescorla - Wagner .854 βw = .80

Summarized Presentation

Probabilistic Joint .270


Note . R = correlation, βw = learning rate parameter.



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.



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)



Table 5. Fits of the Models to the Attribution Data Model R Model Parameters



Configural .742 βw = 1.00; αo = .13

Back - Propagation .152 βw = .81; αm = .52

Summarized Presentation


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).



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.


Figure 1

C X

X

Effect

No Effect

a b

c d


Figure 2

Out put Layer

I nput Layer C X

Information coding : C =X =

Question coding : C =X =

10

10

11

01

Associative links


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


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


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


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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Causal Attribution 1 A Connectionist Approach to Causal ...ritter.ist.psu.edu/misc/dirk-files/Papers/VanOverwalle/pubread.pdf · Causal Attribution 2 Introduction Attributing a cause