The variance theory of the mirror effect in recognition memory

Transcription

1 Pychonomic Bulletin & Review 001, 8 (3), The variance theory of the mirror effect in recognition memory SVERKER SIKSTRÖM Stockholm Univerity, Stockholm, Sweden The mirror effect refer to a rather general empirical finding howing that, for two clae of timuli, the cla with the higher hit rate alo ha a lower fale alarm rate. In thi article, a parimoniou theory i propoed to account for the mirror effect regarding, pecifically, high- and low-frequency item and the aociated receiver-operating curve. The theory i implemented in a recurrent network in which one layer repreent item and the other repreent context. It i hown that the frequency mirror effect i found in thi imple network if the deciion i baed on counting the number of active node in uch a way that performance i optimal or near optimal. The optimal performance require that the number of active node i low, only node active in the encoded repreentation are counted, the activation threhold i et between the old and the new ditribution, and normalization i baed on the variance of the input. Owing to the interference caued by encoding the to-be-recognized item in everal preexperimental context, the variance of the input to the context layer i greater for highthan for low-frequency item, which yield lower hit rate and higher fale alarm rate for high- than for low-frequency item. Although initially the theory wa propoed to account for the mirror effect with repect to word frequency, ubequent imulation have hown that the theory alo account for trength-baed mirror effect within a lit and between lit. In thi cae, conitent with experimental data, the variance theory ugget that focuing attention to the more difficult cla within a lit affect the hit rate, but not the fale alarm rate and not the tandard deviation of the underlying denity, leading to no mirror effect. The mirror effect refer to a general cla of empirical phenomena regarding the order found in hit and fale alarm rate for two clae of tudied item, when one of the two clae i eaier to recognize (A) and produce a larger d than doe the other (B). For intance, in the cae of a free-choice ye no recognition tak, the mirror effect for new (N) or old (O) item i een in the following order in term of increaing probabilitie (P) of ye repone: eay new, difficult new, difficult old, and eay old [P(AN) < P(BN) < P(BO) < P(AO)]. The phenomenon i named the mirror effect becaue the order for the old item i the revere of the order of the new item. The mirror effect ha been tudied extenively in different context and ha been hown to be a rather univeral, robut, and baic recognition memory phenomenon. It ha been found for many clae of timuli that differ in everal timulu dimenion (e.g., word frequency, concretene, material, and trength; for more detailed review, ee Glanzer & Adam, 1985, and Glanzer, Adam, Thi reearch wa upported by Grant APA 146 from NSERC to Ben Murdock and by a potdoctoral grant from STINT to the author. I thank Ben Murdock for inightful advice and careful comment during the writing of thi paper. I alo thank Tim Curran, Kiok Kim, Matthew Duncan, Dave Smith, and Carola Åberg for comment on the paper. A pecial thank to Shu-Chen Li for helpful comment and upport during the lat tage of the reviing proce. Correpondence concerning thi article hould be addreed to S. Siktröm, Department of Pychology, Stockholm Univerity, S Stockholm, Sweden ( verker@ pych.utoronto.ca). & Kim, 1993). Furthermore, a few quantitative theorie have alo been propoed to explain the pychological mechanim cauing the mirror effect (e.g., Glanzer et al., 1993; Greene, 1996; McClelland & Chappell, 1998; Shiffrin & Steyver, 1997). However, a will be dicued later, thee current model do not predict ome apect of the mirror effect. Thi paper preent a new theory whoe aim i to extend the previou model in order to overcome thee challenge. Different Facet of the Mirror Effect Firt, I review a few major empirical finding regarding the mirror effect. For intance, Glanzer and Adam (1985) found a material-baed mirror effect with repect to concrete veru abtract word and picture veru word. In term of the concretene-baed mirror effect, the hit rate of the concrete word are larger, but alo the fale alarm rate are lower. With repect to picture veru word, the d and the hit rate of the picture are larger and higher, but alo the fale alarm rate are lower. The frequency-baed mirror effect i, however, mot commonly tudied. In thi cae, low-frequency word are eaier to recognize and produce lager d, and the mirror effect i thu a follow: Hit rate are higher for low-frequency word than for high-frequency word, wherea the oppoite i true for fale alarm rate. The underlying denity ditribution for old and new high- and low-frequency word correponding to thi phenomenon are a hown in Figure 1. Copyright 001 Pychonomic Society, Inc. 408

2 VARIANCE THEORY OF MIRROR EFFECT 409 Figure 1. Hypothetical underlying ditribution along the deciion axi for low-frequency new item (AN), high-frequency new item (BN), high-frequency old item (BO), and lowfrequency old item (AO). For illutrative purpoe, the variance are plotted in thi graph to be equal. The horizontal axi how the recognition trength that the deciion i made on, and the vertical axi the denity of thi recognition trength. The mirror effect ha alo been tudied with twoalternative forced-choice recognition tet. There are everal poible two-alternative comparion between old and new item. For intance, when high- and low-frequency (better recognized) word are ued, there are four tandard pairing of old and new by high- and low-frequency comparion, plu the two null-choice condition (Glanzer & Bowle, 1976) in which either a pair of new item or a pair of old item are compared. According to the mirror effect, the order for the four tandard pair i P(BO, BN) < P(AO, BN), P(BO, AN) < P(AO, AN), where P(O, N) i the probability of chooing old over new. Regarding the null-choice condition, the mirror effect ugget that P(BN, AN) >.50, P(AO, BO) >.50. Another general finding of the mirror effect i that trength manipulation that affect the poition of the new ditribution alo affect the old ditribution when the manipulation are made between condition (Glanzer et al., 1993). Example of frequently ued experimental manipulation of trength are the duration of tudy time, encoding condition, and peed veru accuracy intruction. Thee experimental manipulation affect both the new and the old ditribution by moving the ditribution cloer together, an effect known a concentering, or by moving the ditribution apart, an effect known a diperion. However, if thee trength manipulation are applied differently to item within a condition or in a ingle lit, quite different reult are found. For intance, recently Stretch and Wixted (1998a, 1998b) differentially trengthened high-frequency word by preenting them with a higher preentation frequency than that for lowfrequency word. Thi manipulation affected the hit rate of the high-frequency word, wherea fale alarm rate were largely unaffected. Thu, thi manipulation did not how a tandard mirror effect. In addition to the baic data with repect to hit and fale alarm rate, data on z-tranformed (or normalized) receiveroperating characteritic curve (z-roc) repreent another regularity of the ditribution underlying the mirror effect. The z-roc-curve are obtained by plotting the z-tranformation of hit rate againt the z-tranformation of fale alarm rate while varying the recognition criterion. According to ignal detection theory, the lope of thi curve i equal to the tandard deviation of the newitem ditribution divided by the tandard deviation of the old-item ditribution. Data on the mirror effect ugget that the tandard deviation of the underlying item ditribution are ymmetrical with repect to the clae (Glanzer & Adam, 1990; Glanzer et al., 1993). The tandard deviation () of the recognition trength of the olditem ditribution for the difficult cla i maller than the old-item ditribution for the eaier cla [i.e., (BO) < (AO)]. The tandard deviation of the new-item ditribution for the difficult cla i maller than the tandard deviation for the eaier cla [i.e., (BN) < (AN)]. However, thi ymmetry i tilted, o that the tandard deviation of the new ditribution i maller than the tandard deviation of the old ditribution [i.e., (N) < (O); Ratcliff, Sheu, & Gronlund, 199]. Previou Theorie of the Mirror Effect In addition to it empirical generality, the mirror effect i an important benchmark phenomenon becaue it ha challenged many global matching memory model, uch a SAM (Gillund & Shiffrin, 1984), MINERVA (Hintzman, 1988), CHARM (Metcalfe, 198), TODAM (Murdock, 198), and the matrix model (Humphrey, Bain, & Pike, 1989). The main limitation of the global memory model in capturing the mirror effect arie from their lack of mechanim for predicting differential fale alarm rate for BN and AN item, given that, by definition, thee are the new item and neither cla of item i preented during the tudy phae. Confronted with the limitation of the global memory model, the mirror effect had fotered the development of everal recent theorie to account for the pychological mechanim that give rie to the phenomenon itelf, but a will become clear later a thee theorie are preented, many controverie till remain. Repone trategy account. For intance, Greene (1996) propoed a repone-trategy-baed account for the mirror effect regarding order and aociative information. Specifically, it wa uggeted that if ubject equate the number of repone made to each cla of word (auming that the number of lure and target are approximately equal), thi imple repone trategy would produce repone ditribution that give rie to the mirror effect. However, Glanzer, Kiok, and Adam (1998; ee alo Murdock, 1998) argued againt thi account by howing that the mirror effect i till preent in the abence of a ditinctive et of data and in the abence of repone equalization. Furthermore, Greene account pre-

3 410 SIKSTRÖM dict a mirror effect when ubject are made to focu the attention on the more difficult cla. However, increaing the number of hit for the difficult cla actually diminihe the fale alarm rate. Thu, the repone trategy account i at odd with the experimental data. Attention-likelihood theory. Another attempt to account for the mirror effect i the attention-likelihood theory (Glanzer & Adam, 1990; Glanzer et al., 1993). Thi theory ay that the difference between BO and AO occur becaue ubject pay more attention to item in Cla A (i.e., the better-recognized cla) than to the item in Cla B. However, the four ditribution (i.e., AN, BN, BO, and AO) along the deciion axi reflecting the mirror effect occur mainly becaue ubject tranform the recognition trength of the item to log-likelihood ratio and ue that a the bai for their deciion. More pecifically, the attention-likelihood theory conit of the following aumption. (1) Stimuli conit of N number of feature. The feature are either marked or unmarked. A marked feature indicate that it wa in the tudy lit. () Some proportion p(new) of feature are already marked in a new timulu, which reflect the noie level, with the rationale that feature in a new item (which i not tudied during encoding) hould be marked becaue of the random noie in the deciion proce. (3) Different clae of timuli (i) evoke different amount of attention [n(i)]. (4) During tudy, n(i) feature are ampled and marked. The proportion of feature ampled for a given timulu i n(i) /N. Given that when the noie level i not zero, ome proportion of the new item will alo be marked, the probability that a feature i marked become n(i) p(i, old) 5 p(new) + [1 p(new)]. N (5) During recognition, ubject randomly ample a et of n(i) number of feature. Given that the ampling i independent of whether the feature were marked at tudy, the number of marked feature i thu binomially ditributed, with parameter n(i) and p(i, old) for old timuli and n(i) and p(new) for new timuli. (6) During tet, ubject count the number of marked feature (x). They note the difference between the ample ize and the number of marked feature [i.e., n(i) x]. Ye no deciion are then baed on the log-likelihood ratio given a cla [l(x i)], and it i defined jointly by x, n(i), p(new), and p(i, old): Thi prediction contrain the tandard deviation of the ditribution of the difficult cla to be maller than the tandard deviation of the ditribution of the eay cla [i.e., (BN) < (AN) and (BO) < (AO)]. However, it doe not contrain the tandard deviation of the new ditribution to be maller than the tandard deviation of the old ditribution [i.e., (BO) may be maller than (BN)]. There are 4 poible way (i.e., 4! 5 4) to order the four tandard deviation. Equation allow 3 out of the 4 ordering namely, (BO) < (BN) < (AN) < (AO), (BN) < (AN) < (BO) < (AO), and (BN) < (BO) < (AN) < (AO). However, not all 3 ordering are found in the empirical data. A will be preented later, the variance theory propoed in thi paper i more contraining than the attention-likelihood theory, and it permit only thoe of the ordering that are in line with the empirical data. The attention-likelihood theory ha recently inpired at leat two model, which alo are baed on log-likelihood ratio namely, the ubjective-likelihood account of recognition memory (McClelland & Chappell, 1998) and REM (Shiffrin & Steyver, 1997). Below, they are preented in turn. The ubjective-likelihood model. The ubjectivelikelihood model for recognition memory (McClelland & Chappell, 1998) ha been applied to the mirror effect with repect to lit length, lit trength, z-roc curve, and other recognition phenomena. A major difference between McClelland and Chappell approach and the attentionlikelihood theory i that the ubjective-likelihood account ue one detector for each old and each new item. Thee detector contrat the probability that a to-be-recognized item i an old item with the probability that it i a new item to form the odd ratio. The model make an old deciion if at leat one of the logarithm of the odd ratio i larger than a certain fixed criterion; otherwie, a new deciion i made. Differing from the attention-likelihood theory, the odd ratio in the ubjective-likelihood model i calculated on etimate of the probabilitie, rather than on the true value that a recognition ytem (or ubject) might not have acce to, but wa nonethele ued in Glanzer et al. (1993) model. The uage of a limited number of detector, one for each item, in the ubjective-likelihood theory i a central mechanim, ued to account for everal phenomena. For example, the trength-baed mirror effect i accounted for by the aumption that detector for trong item, when trengthened, are le likely to be activated by new item during recognition, which then lower the fale alarm rate. Thi mechanim work when the number of target i reaonably large in relation to the number of potential ditractor, which wa aumed in McClelland and Chappell (1998) imulation. However, in reality, the num- é p( i, old ) ( 1- p( i, old) l( x i) = ln ê ëê p( new) ( 1- p( new) x n( i) - x x n( i) -x (1) Therefore, in contrat to trength theorie, which include mot of the global memory model, the recognition deciion i made along the log-likelihood dimenion, rather than the dimenion of item trength. In addition to the mirror effect with repect to the ordering of repone ditribution of different clae of timuli, the attention-likelihood theory predict the following inequalitie for the lope (defined by the ratio between. the tandard deviation of the repone ditribution for different timulu clae) of the z-roc curve: (AO) /(BN) < (AO) /(AN) < (BO) /(BN) < (BO) /(AN). ()

4 VARIANCE THEORY OF MIRROR EFFECT 411 ber of target in a typical recognition tet (for example 50) i negligible in comparion with the number of poible ditractor word (the number of word in the ubject vocabulary e.g., 0,000). Thu, the ubjective-likelihood account for the lit-trength effect may encounter a problem when the number of detector aociated to each ditractor increae to a more realitic number. A imilar problem may alo be apparent with repect to the litlength effect, which i accounted for by the aumption that the number of detector i proportional to the lit length. Arguably, the ubjective-likelihood theory would not account for the lit-length effect, given a more plauible aumption that the number of detector i related to the ize of the ubject vocabulary. REM. REM (Shiffrin & Steyver, 1997) i another model whoe aim i to account for the mirror effect, ROC curve, lit trength, and other phenomena in recognition and recall. Similar to the ubjective-likelihood theory, REM i baed on likelihood ratio and ue noiy vectorbaed repreentation in memory. Although REM alo ue likelihood ratio, REM differ in the ene that it ue true probabilitie in calculating the likelihood ratio, wherea the ubjective-likelihood theory ue etimate. Furthermore, in REM the value aigned to the model repreentation on a particular feature dimenion i determined when the dimenion i ampled the firt time. In the ubjective-likelihood theory, the repreentation of the item are refined ucceively over preentation. In REM, everal factor are combined together to produce the frequency-baed mirror effect. (1) The likelihood ratio are aumed to be maller for old high-frequency word, becaue high-frequency feature are le diagnotic. () Thi factor i larger than the compenating factor that high-frequency word have lightly more matching feature value, becaue error in torage tend to produce common value, increaing the probability of accidentally matching a high-frequency feature value. (3) New high-frequency word have a light increae in the number of chance matche, which increae the likelihood ratio. Limitation of current theorie. Although thee theorie account for everal apect of the data regarding the mirror effect, they have been ubjected to a few general criticim. Perhap the mot obviou problem with thee model i that they predict that trengthening of a cla yield a mirror effect. Although thi prediction i upported by data in tudie in which the trength manipulation were applied between condition (Glanzer et al., 1993), it i certainly inconitent with the data when the trength manipulation were applied within condition (Stretch & Wixted, 1998a; ee alo Stretch & Wixted, 1998b). In addition, there are a few other more pecific criticim about thee theorie. Here are ome problem regarding the attention-likelihood theory. Firt, calculating the log-likelihood ratio require knowledge of the cla (l dependent on i). Thu, it i neceary to claify the to-be-recognized timuli into ditinct categorie (i) or, at leat, to etimate the timuli along ome dimenion, uch a frequency. Glanzer et al. (1993) noted that the attentionlikelihood theory predict the mirror effect even though the etimate of p(i, old) are not accurate for example, when thi probability i et to the average of the two clae. Thu, the etimate of p(i, old) are not critical to the prediction. However it i neceary to etimate the number of feature ampled at recognition [n(i)] in Equation 1 to make the correct prediction, and thi proce require knowledge of the cla of timuli. Second, knowledge of everal variable i required. Depending on the claification, the attention paid to thi category [n(i)] mut be etimated. The probability that a feature in a new item i marked mut alo be etimated. Third, the attention-likelihood theory involve everal tep of complex calculation. Although thi may not be the reaon for dimiing the theory (ee Murdock, 1998, for a dicuion of thi topic), it would be preferable to have a impler account. Given that the ubjective-likelihood account (McClelland & Chappell, 1998) and REM, like the attention-likelihood theory (Shiffrin & Steyver, 1997), are alo baed at their core on variation in log-likelihood ratio, thee criticim would alo apply to them. Reearch Goal and Organization of the Paper Given the limitation of current theorie, the purpoe of thi paper i to propoe a new account of the mirror effect that can avoid mot of thee criticim. The theory i propoed pecifically for the frequency-baed mirror effect, but it alo account for the trength-baed mirror effect within a lit, the trength-baed mirror effect between lit, z-roc curve aociated with the mirror effect, and the lit-length effect. The paper i organized a follow. Firt, a brief overview of the theory i preented, which i then followed by an in-depth preentation. Second, the theory i implemented in a connectionit model of memory previouly developed by the author (i.e., TECO, the target, event, cue, and object theory; Siktröm, 1996a, 1996b). Third, mechanim of the theory reponible for capturing the mirror effect are preented in detail. Fourth, a ection preent the theory application with repect to the variou claical empirical regularitie, uch a between-lit trengthening, the lit-trength effect, z-roc curve, and repone criterion hifting. Fifth, prediction regarding a recently dicovered lack of the mirror effect in the context of within-lit trength manipulation are preented (e.g., Stretch & Wixted, 1998a, 1998b), and an experiment i carried out to evaluate the prediction. For reader who are intereted in the analytic olution of the theory, mathematical derivation of thee olution are preented in the ixth ection, and an analyi of the model optimal performance i conducted. Finally, implication for ditributed model of memory and the relation between the variance theory and previou theorie of the mirror effect are dicued. THE VARIANCE THEORY OF THE MIRROR EFFECT Overview of the Theory In a nuthell, the variance theory propoed here i imilar to previou model of the mirror effect, in the ene

5 41 SIKSTRÖM that it alo ue vector-feature repreentation of the item and etimate (via imulation) the repone probabilitie of old (target) and new (lure) item during a recognition tet. However, the variance theory i driven by different conceptual and technical conideration. At the conceptual level, the variance theory et out to capture the mirror effect mainly in term of the relation between the tudy material and the natural preexperimental context aociation the item may have. Thi i conceptually quite different from all previou theorie eeking to explain mirror effect primarily in term of the individual deciion proce. Rather, the approach taken here conider the context in which the individual recognition deciion procee take place. The natural frequencie of event occurring in the individual day-to-day context may be reflected in recognition deciion procee, and the individual may or may not know (or be conciouly aware of) thee procee. At the technical level, the variance theory alo differ from previou theorie in a ignificant way. Intead of directly computing ratio between probabilitie, a new way of computing recognition trength i propoed by normalizing the difference between the repone probabilitie for the target and the lure item with tandard deviation of the underlying repone ditribution. Specifically, in dealing with the frequency-baed mirror effect, a rather plauible key aumption of the variance theory i that high-frequency word are aumed to have appeared in more naturally occurring preexperimental context than have the low-frequency word. Thi aumption i implemented in connectionit network imulation in a rather traightforward way by aociating the imulated high-frequency item with more context than the low-frequency item during a imulated preexperimental phae. In implementing the theory, item and context are repreented by two eparate array (vector) of binary feature, with each feature being repreented by a node (or element of the vector), a i hown in Figure. A particular item, uch a CAR, activate ome feature at the item layer. A particular context, uch a REPAIR SHOP, activate ome feature at the context layer. Different item and context may or may not activate ome of the ame feature. The item and context feature are fully interconnected with weight. When an item appear in a particular context and certain feature are activated, the weight that reciprocally connect the item feature to the context feature are adaptively changed according to a pecific learning rule, decribed later. Note that in the implementation, two type of context namely, the preexperimental context and the tudy context (repreenting internally generated time or lit information aociated with an item during the actual experimental epiode) are repreented in the network, uing one common context layer. But thee two type of context information are differentiated by two imulation phae namely, the preexperimental phae and the actual encoding and teting phae. A will be mathematically outlined later, the tandard deviation of the input at the context layer increae when an item i aociated with everal context. Therefore, high-frequency item (aociated with more preexperimental context) will have larger tandard deviation than will low-frequency item in their activation pattern, which are ubequently propagated to the item layer. However, the expected value of the input i equal for high- and low-frequency item. During the recognition tet, an item vector i preented to reintate the activation of the item feature. The feature of the tudy context are reintated by preenting the network with the context pattern encoded during the tudy-encoding phae (but not from other preexperimental context that the network wa trained with during the preexperimental phae). The degree of recognition trength i determined by firt calculating the net input to the context and the item node. The net input i the ignal a node receive from other active node connecting to it, and the trength of the ignal determine whether the node will be activated or not at retrieval. The net input of a given item node i imply computed a the um of all weight connected to active node. Similarly, the net input of a given context node i imply the um of all weight connected to active node and that particular context node. The net input, then, denote the retrieved tate of activation pattern at the item and context layer. The ubet of item and context node that have activation level exceeding a particular activation threhold at retrieval and that were alo active during encoding are then ued to calculate the recognition trength. Thoe node whoe activation doe not exceed the threhold or that were inactive during encoding have no influence on recognition trength. For example, aume that the activation threhold i et to 0.5, o that any node (item or context) that wa active during encoding and whoe retrieved activation, during teting, exceeded the value of 0.5 would contribute to recognition trength. Imagine that four node out of a total of eight exceed the threhold and are equal to 0.75, 1.00, 1.5, and The recognition trength of the item i the percentage of above-threhold node (50%) Figure. The variance theory. The upper four circle repreent node in the item layer. The lower four circle repreent node in the context layer. The arrow between the item and the context layer repreent connection.

6 VARIANCE THEORY OF MIRROR EFFECT 413 minu the expected percentage of above-threhold node (e.g., 5%) divided by the tandard deviation of actually oberved above-threhold node (i.e., by the tandard deviation of 0.75, 1.00, 1.5, and 1.50). Why i recognition trength determined by thi rule, a oppoed to, ay, jut the percentage of above-threhold node? A will be hown later, thi way of meauring recognition trength (ubtracting the expected value and dividing with the tandard deviation of the net input) allow the model to perform optimally in term of dicriminability between new and old item when the repone bia to the ye repone i manipulated. And in thi cae, the model account for why the tandard deviation of the repone ditribution of the eay cla i larger than the repone ditribution of the difficult cla. It i plauible to aume that human have evolved to generally repond more or le optimally and that thi i reflected in their performance, a well a in the implementation of the variance theory. Similarly, the activation threhold i et in the model to the value that lead to the highet d (i.e., to optimal performance), which occur when the activation threhold i between the new and the old ditribution. Thi optimal tuning of the model allow it to account for ome rather dramatic reult, uch a concentering, howing that target and lure ditribution from different item clae converge on a trength-of-evidence axi a memory condition woren. Here i a brief example of how the model perform. Conider hypothetical level of activation generated by high-frequency and low-frequency old (target) and new (lure) item. Becaue high-frequency word have appeared in a larger number of context, they have a larger variance of net input. A uch, target and lure will be relatively more confuable and will generate percentage of activated node that are difficult to dicriminate. Aume that the tandard deviation of the net input i.10 and the relevant proportion are.5 for high-frequency target and.15 for high-frequency lure. In contrat, low-frequency word will have occurred in fewer context and will be le confuable. Aume that the tandard deviation of net input i le for low-frequency word for example,.05 and that the percentage of active node i.30 for low-frequency target and.10 for low-frequency lure. Given thee value, what are the recognition trength for high-frequency and low-frequency target and lure? If the expected proportion of above-threhold node i.0, they are Low-frequency lure: (.10.0)/ High-frequency lure: (.15.0)/ High-frequency target: (.5.0)/ Low-frequency target: (.30.0)/ The model account for the variou apect of memory phenomena by potulating a connectionit neural network model with an implementation and parameter etting that allow it to perform at optimal or near-optimal level. When the model i optimized, it behave imilarly to how ubject behave, and when it i not optimized, it doe not fit the experimental data. Thi i true not only for the tandard mirror effect, but alo for exception, uch a the abence of a mirror effect for within-lit trength manipulation (omething all other competing formal model fail to do). Furthermore, it predict key feature of the ROC for new and old item, a well a for highand low-frequency item (omething any viable formal model mut do). Preentation of the Variance Theory In thi ection, the detail of the variance theory are preented. A will become clearer a the theory i unfolded, the theory i analytical, and the analytical olution are elf-contained olvable (reader who are intereted can find the analytical olution in the ixth ection). Although the theory itelf doe not require a particular computational framework, it can be more eaily explained and directly implemented by uing a connectionit network. Therefore, the preentation of the theory below i couched within the framework of a Hopfield neural network (Hopfield, 198, 1984), in order to explicate the theory underlying mechanim that generate the mirror effect. Network architecture. The variance theory may be implemented a a two-layer recurrent ditributed neural network (ee Figure ). There are two layer in the repreentation: one i the item layer, and the other the context layer. Both layer conit of N number of node (i.e., N feature), although it could alo be implemented with an unequal number of context and item node. Thu, the total number of node in the network i N. The item and the context layer are fully connected, o that all the node in one layer are connected through weight to all the node in the other layer. Node within a layer are not connected (i.e., no lateral connection). Stimulu and context repreentation. Context and item are repreented a binary activation pattern acro the node in the context and item layer, repectively. A node i active (or activated) if it correponding feature i preent (i.e., of value 1), and a node i inactive (or not activated) if it correponding feature i abent (i.e., of value 0). There are everal reaon for chooing binary repreentation. For intance, binary repreentation erve to deblur noiy information at retrieval (Hopfield, 1984). Binary repreentation alo allow for a low proportion of active node (pare repreentation), which i known to improve performance (Okada, 1996). It alo introduce nonlinearity, which i neceary for olving ome problem in multilayer network (Rumelhart, Hinton, & William, 1986), and it i perhap the implet repreentation. Furthermore, in the preent model, it i hown that it i neceary for capturing characteritic of the z-roc curve that are aociated with the mirror effect. More pecifically, the tate of activation for the ith node in the item layer at encoding i denoted x i d, where the upercript d denote the item layer. The tate of activation for the jth node in the context layer at encoding i denoted x j c, where the upercript c denote the context layer. Context pattern and item pattern are generated by randomly etting node to an active tate (i.e.,

7 414 SIKSTRÖM with value of 1) and otherwie to an inactive tate (i.e., with value of 0). Let a be a parameter that determine the expected probability that a node i active at encoding. Thi parameter doe not change during the imulation and i aumed to be relatively mall (for purpoe of pare repreentation). Note that a i the expected probability of active node, wherea the real percentage of active node for pecific item or context varie around a. The encoding-tudy phae. Encoding occur by changing the connection weight between the item and the context layer. The weight (or the trength of the connection) contain information of what ha been tored in the network. The weight between item node i and context node j i denoted a w ij, and it i initialized to zero. The weight change (Dw ij ) i calculated by the learning rule uggeted by Hopfield (198; ee alo Hertz, Krogh, & Palmer, 1991, for additional detail). Thi i eentially a Hebbian learning rule that increae connection weight between imultaneouly activated node. Thi rule i choen here becaue it i more biologically plauible than other rule, uch a the delta or the gradientdecent learning rule (e.g., Rumelhart et al., 1986) ued in back-propagation network. However, the variance theory can alo be implemented with other learning rule. According to the Hopfield (198) learning rule, the weight change i computed a the outer-product between the item and the context vector of activation pattern, with the parameter a firt ubtracted from both vector. Thi ubtraction i mathematically neceary to keep the expected value of the weight at zero. Uing the notion for item and context activation defined above, the weight change between thee two element of the item and context vector can be written a Dw ij 5 (x i d a)(x j c a). (3) Word frequency a the number of aociated context at the preexperimental phae. An item may be encoded more or le frequently and, hence, be aociated with more or le different preexperimental context, depending on how often the item occur in the ubject environment. In the model, at the preexperimental tage of the imulation, an item frequency i imulated by the number of time the item i encoded in different context. A low-frequency item i encoded le often and i aociated with a maller number of context, wherea a highfrequency item i encoded more often and i aociated with a larger number of different context. For intance, one might form a preexperimental aociation between SAAB and repair hop after experiencing the rare event of a new expenive SAAB port car breaking down halfway through a honeymoon trip to the Grand Canyon, with the SAAB having to be towed to a repair hop omewhere out in the deert! In implementing the variance theory, the relationhip between word frequency and preexperimental item context aociation can be imulated traightforwardly. At the preexperimental tage of the imulation, a low-frequency item, SAAB, may be imulated by aociating one context item, REPAIR SHOP, with it during encoding. A high-frequency word, CAR, may be imulated by aociating three different context, REPAIR SHOP, TAXI RIDE, and DRIVING TO WORK, with it during encoding. The recognition tet phae. At recognition, an item i preented to the network, the repreentation of thi item i reintated a a cue to the item layer, and the repreentation of the tudy context (imulating an internally generated context regarding lit or time information during the recognition experiment) i reintated a a cue to the context layer. For example, the repreentation of the word CAR i reintated at the item layer. Furthermore, the repreentation of the tudy context STUDY LIST i reintated at the context layer. The ubject mut have thi information (or cue) in order to recognize an item from the particular tudy context (and not recognize the item from all the other preexperimental context). In the actual experimental etting, thi information i uually conveyed to the ubject by the explicit intruction to recognize from the tudy context (e.g., Do you recognize the word CAR from the lit you jut tudied? ). At recognition, each node receive a ignal (called the net input), which i computed on the bai of other active node connecting to it. Item node receive net input from active context node, and context node receive net input from active item node. The net input to a given node i imply the um of the weight of all other active node connected to that node. For example, if item node 1 i connected to four context node (1,, 3, and 4), where context node 1 and 3 are active, the net input to item node 1 i w 11 + w 13. Thu, active node end information to node that they are connected to, wherea inactive node do not end any information. Put another way, node receive information or input from active node that they are connected to, but not from the inactive node. Specifically, the net input to item node i i calculated by firt multiplying the activity of the context node (labeled j) connected to thi node by the weight of the connection between the node i and j and then umming over all connected node. In vector formalization, the weight matrix operate on the activation vector, and the output i the net input vector. The net input to node i(h i d ) at the item layer depend on the tate of activation of node in the context layer and the weight connecting thee two layer: h N d c i = å wijx j j= 1 (4) Following the ame principle, a imilar function i ued to calculate the net input to the context node. Specifically, the net input to context node (labeled j) depend on all the tate of activation of node in the item layer and the weight connecting the two layer: h N c d j = å wji x i i= 1 By inerting Equation 3 into Equation 4 and umming over the p number of encoded pattern during lit learning, it i eay to ee that the net input to a node i imply the um of Np number of term; for example, the net input to an item node i..

8 VARIANCE THEORY OF MIRROR EFFECT 415 p N d d c c hi = å å( x i - a)( x j - a) x j. j= 1 For a 5.5, the net input are binomially ditributed with a certain expected value. Given a certain criterion (i.e., Np(1 a) a > 10), a binomial ditribution can be approximated with a normal ditribution (Feller, 1968). For a ¹.5, there are actually four outcome; however, the ame normal approximation can be ued. Thu, for reaonably large parameter value of Npa, the ditribution of net input to the node can be approximated by a normal ditribution. If the to-be-recognized item ha not been encoded with the context (i.e., a new item), the net input i imply the um of random weight. Becaue the expected value of all weight are zero, the expected value of the net input for new item will alo be zero. If the item ha been encoded with the context (i.e., an old item), the net input i the um of thoe weight connected to that node whoe repective context node were active at encoding. Owing to the adaptive weight change during encoding, thee weight will have an expected value that i larger than zero if both node were in the active tate during encoding [i.e., each weight change at encoding i computed a (1 a) ] and le than zero if one node wa inactive and the other node wa active at encoding [i.e., each weight change at encoding wa a(1 a)]. Of pecific importance for the theory i that the variance of the net input to the context node (from the item node) increae with the number of context that are aociated with the item. Therefore, the variance of the net input i larger for highthan for low-frequency item. Similarly, the variance of the net input to the item node (from the context node) increae with the number of item aociated with one context (i.e., lit length). Therefore, given that the context i contant during a lit preentation, the variance of the net input i larger for a long than for a hort lit. Brief ummary of optimal performance. Given the trong election preure, arguably, human and animal have evolved to achieve good memory performance. Therefore, it i reaonable to aume that mechanim for recognition deciion have evolved to an optimal or nearoptimal performance. Following thi aumption, the parameter value in the model and the implementation of the model are guided by the idea that the model hould perform optimally. A detailed dicuion about the iue of optimal performance with exact derivation of what i optimal performance in the context of the preent model i preented later in the ixth ection. Here, I give a brief ummary explaining the reult from the analyi of optimal performance, without going into the mathematical detail (ee Figure 9A, 9B, 9C, and 9D). The model performance i optimal if the percentage of node active at encoding (a) i low (ee Figure 9A). For a low a, it i optimal to bae the recognition deciion on node that were active at encoding and to ignore node that were inactive during encoding (ee Figure 9A). Alo, for a low a, it i optimal to place the activation threhold of the node between the expected value of the new and the old net input (ee Figure 9B). Finally, it i optimal to normalize the recognition trength with the tandard deviation of the net input (ee Figure 9C and 9D). For a low percentage of active node, it i optimal to bae the recognition deciion on node that were active at encoding (or node active in the cue pattern) and to ignore node that were inactive at encoding. At recognition, the tate of activation of a node may be either active or inactive. Therefore, the node that are active in the cue pattern and have a net input above a certain activation threhold are activated at recognition; otherwie, the node are inactivated. Let z i d denote the tate of activation at recognition for item node i. An item node i activated at recognition (z i d 5 1) if it wa active in the cue pattern (x i d 5 1) and the net input i above the activation threhold (h i d > T ); otherwie, it i inactivated (z i d 5 0): z i d 5 1, if x i d 5 1 and h i d > T; otherwie, z i d 5 0. Similarly, let z j c denote the tate of activation at recognition for context node j: z j c 5 1, if x j c 5 1 and h j c > T; otherwie, z j c 5 0. Thi way of activating node at retrieval differ from how node are activated in a tandard Hopfield (198) network, where the activation threhold i zero and a node i activated if the net input i above zero (independently of the tate of activation in the cue pattern). The way of activating pattern in a Hopfield network i more likely to produce a retrieved pattern that matche the encoded pattern of activation (e.g., the expected value of active node at retrieval will be the ame a the expected value of active node at encoding). However, a will be dicued later, the way uggeted to activate the node here yield better performance in term of dicrimination between a target item and a ditractor item. A i hown in Figure 9B, performance i optimal when the activation threhold i et approximately between the new and the old net input. The activation threhold (T ) i et to the expected value of the net input of node active during encoding (x i d 5 1, x j c 5 1) for old and new item. The averaging i computed over all node (N ) and over all new and old pattern ( p) in the recognition condition. If half of the item are new and half of item are old, the activation threhold i P N N T = 1 é d d c å êåxi hi + åx jh c j. apn ëê i= 1 j= 1 A wa dicued above, the expected net input of new (lure) item i zero. Therefore, the activation threhold i imply half the expected net input for node encoded in the active tate [T 5 m h (O)/, where m h (O) i the expected value of the net input to node encoded in the active tate]. It i eay to ee that the expected percentage of old and new active node at recognition i one half of the percentage of active node at encoding (a / ). That i, the activation threhold divide the old and new ditribution in

9 416 SIKSTRÖM the middle. Old item will have a higher expected percentage active, and new item will have a lower expected percentage active. The activation threhold i contant during recognition in one condition. However, it mut vary between condition, depending on the net input, to yield optimal performance. The percentage of node active at recognition i counted: P N = 1 æ çåz + N è N d c i åz c j. i= 1 j= 1 ø A i hown in Figure 9C, the performance i near-optimal if the recognition deciion i baed on the number of node active at recognition, normalized by the tandard deviation of the net input acro the active feature of thi item ( h ). Thu, thi tandard deviation i calculated over all the node active at encoding (i.e., x d i = 1 and x c i 5 1; node inactive at encoding are not ued when calculating the tandard deviation becaue, for low level of a, thee item carry little to no information of the item). The recognition trength (S ) for an item i calculated by ubtracting the expected percentage of node active at recognition (a/) 1 from the real percentage of node active at recognition (P c ) and dividing by the tandard deviation of the net input of the item ( h ): P a c - S = (5) h The ubtraction of the expected percentage of node active at recognition make the expected value of the recognition trength (S ) zero. Thi ubtraction i neceary for the normalization to work properly. The ubtraction move the recognition trength ditribution ymmetrically, o that the old and the new ditribution move at the ame rate for a given tandard deviation of the net input (without the ubtraction, the old recognition trength ditribution would be more affected than the new ditribution). Thu, the recognition trength i determined by the difference between two probabilitie (the percentage of active node that varie and the expected percentage of active node that i contant) divided by the tandard deviation of the net input. A ye repone (Y ) i given if the recognition trength (S ) i above the recognition criterion (C ). An unbiaed deciion ha a recognition criterion of zero: Y 5 S > C. ö An iue that may be raied i whether it i enible to bae recognition trength on two quite different ource namely, the percentage of active node and the variability of the net input. The immediate anwer i that if it i reaonable to optimize performance, it i alo enible to meaure recognition trength thi way. Another perpective i to note that unbiaed repone can be made only on the percentage of active node that i, a ye repone occur if the percentage of active node i larger than the expected percentage of active node (P c > a /) and the variability of the net input can be ignored. Thu, normally, ubject bae their unbiaed deciion on the percentage of active node, and the variability of active node only become relevant when ubject are biaed. From thi perpective, the percentage of active node i ued for unbiaed repone, and the variability of the net input become relevant for confidence judgment. Therefore, by combining both the percentage of active node and the variability of the net input, the meaure of recognition trength propoed here will alo reflect the confidence judgment. An Example With Step-by-Step Computation To clarify the computational detail involved in the variance theory, a tep-by-tep example i given here. For tractability, a mall network i ued coniting of four item node and four context node (ee Figure ). The actual imulation to be reported later involved a larger network architecture with 30 node at each layer. The percentage of node active at encoding (denoted by parameter a) i et to.50. Let item BN be repreented a {1,1,0,0} and written a the tate of activation of the four node {x 1 d, x d, x 3 d, x 4 d }. Similarly, let {0,0,1,1} repreent item BO, {1,0,1,0} repreent item AO, and {0,1,0,1} repreent item AN. Let context C BN be repreented a {1,1,0,0}, or the tate of activation of the four node {x 1 c, x c, x 3 c, x 4 c }. Similarly, {0,0,1,1} repreent context C BO and {0,1,0,1} repreent experimental tudy context C Exp. Item BN i a high-frequency new word. For implicity, it i here encoded only once with context C BN in the preexperimental phae (in the imulation below, highfrequency word are preexperimentally aociated with three context). The 16 weight between the four item node and the four context node are changed according to the learning rule, where the probability that a node i active at encoding i determined by the parameter a 5.5. For example, the weight change between item node 1 and context node 1 i w 11 5 [x 1 d (BN) a][x 1 d (C BN ) a] 5 (1.5)(1.5) 5 1 4, where BN i item BN and the C BN repreent context C BN. Similarly, item BO i another high-frequency word that, before the experimental phae, i encoded once with context C BO. Item AO and AN are low-frequency old and new word, and they are not encoded at the preexperimental phae. In the experimental phae, item AO i encoded with the experimental context C Exp. Finally, item BO i encoded with the ame experimental context C Exp. For example, the weight w 11 i now equal to [x 1 d (BN) a) (x 1 c (C BN ) a] + [x 1 d (BO) a] [x 1 c (C BO ) a] + [x 1 d (BO) a][x 1 c (C E ) a] + [x 1 d (AO) a][x 1 c (C E ) a] 5 (1.5) (1.5) + (0.5)(0.5) + (0.5)(0.5) + (1.5) (0.5) After encoding, the full weight matrix i {{.5, 1, 1,.5}, {.5, 0, 0,.5}, {.5, 0, 0,.5}, {.5, 1, 1,.5}}, correponding to the weight {{w 11, w 1,... w 44 }}, repectively.