Optimal policies for free recall.

THEORETICAL NOTE
Optimal Policies for Free Recall
Qiong Zhang1, 2, 3, Thomas L. Grif /uniFB01ths4, 5, and Kenneth A. Norman4, 6
1Department of Psychology, Rutgers University –New Brunswick
2Department of Computer Science, Rutgers University –New Brunswick
3Center for Cognitive Science, Rutgers University –New Brunswick
4Department of Psychology, Princeton University
5Department of Computer Science, Princeton University
6Princeton Neuroscience Institute, Princeton University
There is rich structure in the order in which studied material is recalled in a free recall task ( Howard &
Kahana, 2002a ). Extensive effort has been directed at understanding the processes and representations that
give rise to this structure; however, it remains unclear why certain types of recall organization might be
favored in the /uniFB01rst place. We provide a rational analysis of the free recall task, deriving the optimal policy for
recalling items under the internal representations and processes described by the context maintenance and
retrieval (CMR) model of memory search ( Polyn et al., 2009a ). Our model, which we call rational-CMR,
shows that the optimal policy for free recall is to start from the beginning of the list and then sequentially
recall forwards, providing a rational account of the primacy and forward asymmetry effects typically
observed in free recall. In addition, when recall is not initiated from the beginning of list, it is optimal during
recall transitions to minimize the amount of forward asymmetry. Predictions from the rational model are
con/uniFB01rmed in human behavioral data: Top-performing human participants demonstrate a stronger tendency
to initiate recall from the beginning of the list and carry forward recalls, and the amount of forward
asymmetry in participants depends on whether they start recall from the beginning or end of the list. We
discuss the resemblance of optimal behavior in free recall to participants ’behavior when applying
mnemonic techniques such as the method of loci.
Keywords: free recall, memory search, rational analysis, computational modeling
Psychologists have used the free recall task for decades to gain
insight into the processes and representations underlying memory
search ( Murdock, 1960 ,1962 ;Roberts, 1972 ;Standing, 1973 ). In a
free recall task, participants /uniFB01rst study a list of items and are later
asked to freely recall as many items as they can from the list. One of
the key bene /uniFB01ts of the free recall task is that the data support a wide
range of analyses. In addition to tracking overall performance
(operationalized as the number of recalled items), studies have
also looked at serial position effects (showing primacy —enhanced
recall of items from the start of the list —and also recency —enhanced
recall of items from the end of the list; Murdock, 1962 ) and contiguity
effects (regularities in which items tend to be recalled together). Early
studies of contiguity focused on semantic clustering, where studied
items that are drawn from the same semantic category are recalled
successively ( Bous /uniFB01eld, 1953 ;Bous /uniFB01eld & Sedgewick, 1944 ;Cofer et al., 1966 ;Howard & Kahana, 2002b ). The temporal
contiguity effect is also a ubiquitous property of the recall sequences:
Items studied in nearby serial positions tend to be recalled successively
regardless of their degree of semantic association ( Kahana, 1996 ).
These temporal contiguity effects are bidirectional (participants show
an enhanced probability of recalling items that preceded and
followed the item at study) and typically show a forward asymmetry
(i.e., forward transitions are more likely than backward transitions;
Howard & Kahana, 1999 ;Kahana, 1996 ).
Several computational models have been developed to account for
these regularities of free recall, including the temporal context model
(TCM; Howard & Kahana, 2002a )a n di t ss u c c e s s o r ,t h ec o n t e x t
maintenance and retrieval model (CMR; Lohnas et al., 2015 ;Polyn
et al., 2009a ). We focus on CMR here, but our results should also
generalize to TCM ( Howard & Kahana, 2002a ;Sederberg et al., 2008 ),This document is copyrighted by the American Psychological Association or one of its allied publishers.
This article is intended solely for the personal use of the individual user and is not to be disseminated broadly.
Qiong Zhang
https://orcid.org/0000-0001-9062-9571
Thomas L. Grif /uniFB01ths
https://orcid.org/0000-0002-5138-7255
Kenneth A. Norman
https://orcid.org/0000-0002-5887-9682
Preliminary versions of this work have been presented in a talk at the
Annual Meeting of the Society for Mathematical Psychology in 2020 and a
poster at the Annual Meeting of the Psychonomic Society in 2020. A
preprint of the article is available at https://psyarxiv.com . This study was
not preregistered. All data sets analyzed in this work are from
publicly available data sets, which can be accessed from: http://memory.psych.upenn.edu/Data_Archive . All code and analysis can be
accessed from: https://github.com/qiongzhang/rationalCMR . This work
was supported by the National Science Foundation, DRL-2024587 awarded
to Kenneth A. Norman and Qiong Zhang; the U.S. Air Force, FA9550-18-1-
0077 awarded to Thomas L. Grif /uniFB01ths; and a C.V. Starr Fellowship awarded to
Qiong Zhang.
Correspondence concerning this article should be addressed to Qiong
Zhang, Department of Psychology, Rutgers University –New Brunswick,
152 Frelinghuysen Road, Piscataway, NJ 08854, United States. Email:
qiong.z@rutgers.eduPsychological Review© 2022 American Psychological AssociationISSN: 0033-295Xhttps://doi.org/10.1037/rev0000375
1
given the high degree of similarity between the TCM-A model
(Sederberg et al., 2008 ) and the variant of CMR considered here (see
the Background section, for details). CMR posits that a slowly
changing internal context representation is associated with each of
the studied items and is then used to guide memory search at recall.
In CMR, recency arises because the time-of-test context at the
beginning of the recall overlaps with the contexts associated with
the most recent list items. Temporal contiguity arises because
recalling one item leads to the retrieval of its contextual state at
study, which subsequently cues retrieval of adjacent items (see the
Background section, for detailed discussion of how CMR accounts
for these and other /uniFB01ndings).
The architectural assumptions of CMR are important in account-
ing for these regularities. Nevertheless, while CMR can be parame-
terized to show all of these regularities, it can also be parameterized
so that these regularities do not appear. This leaves open the
question of why people show these regularities in the /uniFB01rst place.
To our knowledge, our work is the /uniFB01rst to address the latter question
by showing how these characteristics of free recall result from the
cognitive system optimizing recall performance. This is achieved by
applying rational analysis, which explains human behavior as an
optimal solution to computational problems posed by the environ-
ment ( Anderson, 1990 ;Anderson & Milson, 1989 ). While CMR
(parameterized to /uniFB01t human data) provides an explanation at Marr ’s
algorithmic level ( Marr, 2010 ), our work provides additional insight
about the free recall task at the computational level. This distinction
is in spirit similar to the contribution of the original rational analysis
of human memory: Though recency and frequency effects had been
known to memory researchers since the work of Ebbinghaus (1885/
1913) ,Anderson (1990 ) was the /uniFB01rst to demonstrate that recency
and frequency effects arise from human memory optimally adapting
to the statistics of the environment. Similarly, models of categori-
zation can capture human behavior in categorization tasks by
making different assumptions about whether categories are repre-
sented by prototypes or exemplars ( Medin & Schaffer, 1978 ;
Nosofsky, 1984 ;Reed, 1972 ), a rational model of categorization
can justify these assumptions by explaining how they are useful in
achieving the computational goal underlying the categorization task
(Anderson, 1990 ;Grif/uniFB01ths et al., 2007 ).
For our rational analysis, we de /uniFB01ne the goal of free recall to be
correctly recalling as many items as possible. Our basic architectural
assumptions about internal representations and processes during
memory search are based upon the CMR model ( Lohnas et al.,
2015 ;Polyn et al., 2009a ). That is, we treat the core architecture
of the CMR model as a hard constraint on how memory works. This
approach inherits from rational analysis the idea of explaining human
behavior in terms of optimal solutions to problems posed by the
environment ( Anderson, 1990 ), but follows recent extensions of this
approach to the case where that environment is partly speci /uniFB01ed by the
cognitive processes and internal representations of the agent (cogni-
tive-bounded rational analysis: Howes et al., 2009 ; see also resource-
rational analysis: Lieder & Grif /uniFB01ths, 2020 ). In contrast to the model-
/uniFB01tting procedure typically used in CMR, which obtains a set of
parameters to best /uniFB01t human behavioral data, our model (which we
call rational-CMR) obtains a set of parameters to maximize the total
number of items recalled. The derived optimal behavior provides a
rational account of why certain patterns in recall organization arise
(i.e., because they lead to better task performance). The rational
analysis also leads to testable predictions about the relationshipbetween recall organization and overall performance (i.e., people
whose behavior most closely approaches the optimal parameteriza-
tion identi /uniFB01ed by rational-CMR are predicted to perform better).
In the remainder of the article, we /uniFB01rst give some background on
the CMR model of human memory search, then we describe the steps
to carry out a rational analysis on CMR. We show that —under the
assumption that people can choose their entry point into the list —the
optimal policy is to always start recall from the beginning of the list
and then sequentially recall forwards, providing a rational account of
primacy and forward asymmetry effects typically observed in free
recall. To explain why people do not always start recall from the
beginning of the list, we highlight the fact that —under CMR —people
do not have the ability to simply choose their entry point into the list.
Rather, the /uniFB01rst item needs to be recalled based on cues available at the
end of the list, which tend to match end-of-list items more than start-
of-list items; this results in participants sometimes initiating recall
with end-of-list items (recency) instead of start-of-list items (primacy).
We show that the optimal policy during recall is different when
participants happen to initiate recall from the end of the list. In this
case, the optimal policy after retrieving an item is to retrieve the state
of temporal context associated with that item at study and then to cue
with this retrieved context; as described in the next section, retrieved
temporal context is a symmetric cue that allows participants to move
backwards as well as forward in time —this backward movement is
especially useful when participants start from the end of the list. We
also discuss how the optimal parameters to use at encoding differ
depending on whether —at recall —participants end up initiating recall
from the start of the list (primacy) or the end of the list (recency); given
that it is not knowable at encoding whether participants will succeed
in getting back to the start of the list at recall, the optimal strategy at
encoding is for participants to interpolate between parameters that
yield optimal performance given primacy versus recency.
After describing the model ’s predictions, we compare the theo-
retical predictions from the optimal policy with human behavioral
data, /uniFB01nding that top-performing human participants are using this
policy more often than the rest of the participants. Finally, we
discuss the implications of the optimal policy beyond free recall task
and its connection to mnemonic techniques; among other things, we
argue that common mnemonic techniques such as the method of loci
help participants to get back to the start of the list, thereby giving
them access to the bene /uniFB01ts (noted above) of initiating recall at the
start of the list.
Background: Context Maintenance and
Retrieval Model
It has been long recognized that associations are made not only
among items but also between items and context ( Anderson & Bower,
1972;Bower, 1967 ;Estes, 1955 ). Building on the classic stimulus-
sampling theory developed by Estes (1955) ,Bower ’s( 1 9 6 7 ) model
posits a context that slowly /uniFB02uctuates in a context space and binds with
any to-be-remembered external or internal experiences (see also ran-
dom context models; Murdock, 1997 ). As a consequence, experiences
that are encoded close together in time also share similar context
vectors. This accounts for the recency effect in free recall, where items
at the end of the list are better recalled since they are close in time and
share similar context with when the recall starts.
These earlier formulations of context models provide a sound
theoretical footing for more recent computational models such asThis document is copyrighted by the American Psychological Association or one of its allied publishers.
This article is intended solely for the personal use of the individual user and is not to be disseminated broadly.2 ZHANG, GRIFFITHS, AND NORMAN
TCM and CMR ( Howard & Kahana, 2002a ;Lohnas et al., 2015 ;
Polyn et al., 2009a ). In addition to having a slowly drifting context,
these more recent models also allow for the retrieval of previously
encountered contextual states, which can in turn serve as cues for
subsequent recalls ( Howard & Kahana, 2002a ). The state of context
at time t, denoted as ct, follows the process:
ct=ρct−1+βcIN, (1)
where, cINrepresents the retrieved context induced by an encoun-
tered experience, βis the parameter that determines the rate of
contextual drifting (0 ≤β≤1), and ρis set to a scalar that ensures
/uni2016ct/uni2016=1. When an item is presented during the encoding phase, the
retrieved context cINcorresponds to the preexperimental context of
the item —a representation of the item ’s features based on all of the
encounters with the item prior to the start of the experiment. This can
be expressed as follows:
cIN=MFCpreft, (2)
where, MFCpreencodes the contexts that were associated with different
items prior to the experiment and ftis the binary vector that is all
zeros except at the position that represents the presented item at time
t; therefore, MFCpreftis the context previously associated with the
presented item. In addition to a /uniFB01xedMFCprethat represents item-
context associations prior to the experiment, there are also MFCexpand
MCFexpthat capture the item-to-context associations and context-to-
item associations acquired during the experiment. These matrices
are initially set to zero and are incremented during the encoding
process based on the Hebbian outer-product learning rule
/uni0394MFCexp=/uni0394MCFexp=ftcT
t−1to instantiate the associative learning
that takes place between a presented item ftand the current context
ct−1. The overall effect of having context drift toward the retrieved
context of presented items, as described in Equations 1 and2,
together with associative learning, is that each item is embedded at a
location in the context space corresponding to the representations of
recently encountered items. With each new presented item, the
context drifts slightly toward the preexperimental context associated
with that item, so that items presented close in time to each other are
linked to similar contextual states. We followed the order-of-opera-
tions consistent with Howard and Kahana (2002a) ,Sederberg et al.
(2008) ,Lohnas et al. (2015) , and Rouhani et al. (2020) , which is to
associate an item with the context /uniFB01rst, before the context vector
drifts toward the item ’s preexperimental context.
During recall, when memories of these items are needed again,
the memory search process is driven by the current state of the
context representation ct. The support for recalling each item
depends on how much the current context matches the item ’s study
context. The starting context at recall is close in time to (and thus
similar to) the end-of-list context during encoding; this gives rise to
better recalls for items studied at the end of the list (i.e., the recency
effect). As recall continues, context evolves under the same process
as it did during the encoding phase, ct=ρct−1+βcINas inEquation 1 ,
but with the retrieved context cINintroduced differently. The key
difference is that items are encountered for the /uniFB01rst time during the
experiment in the encoding stage, whereas —in the recall phase —
items are encountered for the second time when they are recalled. The
/uniFB01rst time that an item is encountered during its presentation at t=i,
there is no association between this item and the experimental contextyet; therefore, the retrieved context cINconsists of solely the pre-
experimental context associated with this item, expressed in
cINenc=MFCprefi; however, when the same item is encountered a second
time when it is recalled at t=j, the retrieved context cINreccan come
from both the preexperimental context associated with the item
MFCprefjand the experimental context associated with the item
MFCexpfj, which has been acquired through Hebbian learning during
the encoding phase. The extent of retrieving the preexperimental
context versus retrieving the experimental context is regulated by the
parameter γfc∈[0, 1],
cINrec=ð1−γfcÞMFCprefj+γfcMFCexpfj=ð1−γfcÞcINenc+γfcci−1:(3)
The value of γfchas important implications for the recall transition
patterns. When γfc=0, the retrieved context entirely consists of
preexperimental context associated with the item MFCprefj, which is
identical to the retrieved context cINencwhen the item was /uniFB01rst
encountered during encoding. When γfc=1, the retrieved context
entirely consists of experimental context associated with the item
MFCexpfj, which is essentially the context ci−1that was associated with
the item at t=iduring encoding. Both cINencandci−1are part of the
original study context, as illustrated in Figure 1 , so reinstating them
into the current context has the effect of mentally “jumping back in
time. ”As a consequence of this “jumping back in time, ”items that
were studied close in time to the just-recalled item have a higher
chance of being recalled next, because of the similarity between
their study context and the current context. This contributes to the
temporal contiguity effect commonly observed in free recall.
Though both contribute to temporal contiguity, reinstating cINenc
and reinstating ci−1bias forward recalls and backward recalls
differently. This is illustrated in Figure 1A : During encoding, at
t=i, the drifting part of the study context ρci−1(in orange) is similar
to contexts both before and after t=i, and has the chance to be
associated with items that come before and after t=i. Therefore,
reinstating ci−1gives rise to both forward and backward recalls.
However, since an item ’s preexperimental context is incorporated
into temporal context after the item is presented, the retrieved
context cINenc(in blue) does not share any similarity with contexts
before t=i, and only has an opportunity to be associated with items
that come after. Therefore, reinstating cINencgives rise to only forward
recalls, which accounts for the forward asymmetry commonly seen
in free recall experiments. Figure 1B illustrates the effect of γfcon
the recall transition patterns. When γfc=0, only cINencis reinstated,
which leads to asymmetry in forward and backward recalls; this
asymmetry can be seen in the conditional response probability
curve, which is computed by dividing the number of times a
transition of that lag is actually made by the number of times it
could have been made ( Kahana, 1996 ). When γfc=1 only ci−1is
reinstated, which leads to symmetric forward and backward recalls.
Reinstating cINencandci−1both contribute to temporal contiguity; for
distinguishing between the two in the rest of the article, we refer to
the asymmetric part of the temporal contiguity as “asymmetric
contiguity, ”which is also the forward asymmetry effect, and the
symmetric part as “symmetric contiguity. ”In practice, experimental
participants adopt an intermediate value of γfc, demonstrating a
combination of both effects.
At this point, we have described the key properties of CMR
(Howard & Kahana, 2002a ;Lohnas et al., 2015 ;Polyn et al., 2009a ),This document is copyrighted by the American Psychological Association or one of its allied publishers.
This article is intended solely for the personal use of the individual user and is not to be disseminated broadly.OPTIMAL FREE RECALL 3
which closely resembles the TCM-A model ( Sederberg et al., 2008 )
in terms of the features relevant for this rational analysis. To be able
to fully simulate the behavioral patterns, the model also needs to be
equipped with a retrieval rule and a stopping rule. Here, we use the
softmax function as the retrieval rule pi=eksiP
jeksj, with sias the
support to retrieve item iand parameter kgoverning the amount of
noise during the retrieval ( Howard & Kahana, 2002a ;Sederberg et
al., 2008 ). The stopping probability is a function of summed support
for the already recalled items srand the not-yet-recalled items snr,
expressed as pstop=e−εdsnr=sr(Kragel et al., 2015 ). Importantly, the
key model properties reviewed above are capable of explaining
recency and temporal contiguity, but they do not account for the
primacy effect, where early items in the list are better recalled than
items in the middle of the list. This has been captured by assuming
that there is increased attention during recall to beginning-of-list
items, with the support to each item iscaled by ϕi=ϕse−ϕdði−1Þ+1
(Lohnas et al., 2015 ;Polyn et al., 2009a ). This introduces an
additional two parameters, ϕsandϕd, into the model.
A Rational Analysis of the Free Recall Task
In this section, we derive an optimal solution to the free recall task
via rational analysis. We adopt the steps laid out in Anderson (1990) :
1. Precisely specify the goals of the cognitive system.
2. Develop a formal model of the environment to which the
system is adapted.3. Make minimal assumptions about computational limitations.
4. Derive the optimal behavior function, given items 1
through 3.
5. Examine the empirical evidence to see whether the pre-
dictions of the behavior function are con /uniFB01rmed.
Anderson (1990) argues that human memory is adapted to the
statistical patterns in the physical environment; in our analysis,
we commit to the architectural assumptions in CMR, which has been
shown to capture human behavior in memory search. We specify the
corresponding components for each step with respect to the free
recall task and the CMR model and organize the rest of the article
based on these steps.
First, we specify the goals of the cognitive system. In the task of
free recall, the goal is to recall as many items as possible from the
studied list within the time constraints imposed by the recall task
(e.g., participants might be told that they have to complete recall
within 2 min), while at the same time minimizing recall errors such
as those from extra-list intrusions. There is more than one way to
combine these goals. However, for the current analysis, we set the
goal as maximizing the number of items correctly recalled from the
list. This de /uniFB01nition of the goal is consistent with the instructions
participants commonly receive in free recall experiments, where
they are told to recall as many items as possible from the list without
penalizing them for recall errors or rewarding them for their speed.
Next, we build a formal model of the environment based on CMR,
making minimal additional assumptions about computationalThis document is copyrighted by the American Psychological Association or one of its allied publishers.
This article is intended solely for the personal use of the individual user and is not to be disseminated broadly.Figure 1
Illustration of How the Value of γfcRegulates the Extent of Forward Recalls Versus Backward Recalls
Note. (A) Context can be updated in two ways at encoding: Using the preexperimental context cINencintroduced by the just-
encoded item ft, and using the context drifting from the previous time step ct−1. From the /uniFB01gure, it is clear that only the items that
come after cINenchave a chance to be associated with it. Therefore, when cINencis reinstated at recall, it only gives rise to forward
recalls. In contrast, the drifting part of the context is associated with both items before time tand items after time t, therefore,
giving rise to both forward and backward recalls. (B) The value of γfcregulates the amount of retrieved context from cINencversus
ct−1when item ftis encountered again during recall. When γfc=0, only cINencis reinstated, which leads to forward asymmetry (i.e.,
asymmetric contiguity) as seen from the conditional response probability. When γfc=1, only ct−1is reinstated, which leads to
symmetric forward and backward recalls (i.e., symmetric contiguity). Parameters used in the simulation were obtained from the
CMR /uniFB01t inFigure 2 . CMR =context maintenance and retrieval. See the online article for the color version of this /uniFB01gure.4 ZHANG, GRIFFITHS, AND NORMAN
limitations under this environment. In contrast to previous studies of
rational analysis, the environment here incorporates not only the
local task environment of free recall but also incorporates the
architectural assumptions from CMR. Our approach thus closely
resembles cognitive-bounded rational analysis ( Howes et al., 2009 )
and resource-rational analysis ( Lieder & Grif /uniFB01ths, 2020 ), which
identify optimal cognitive processes given a utility function that is a
result of both constraints imposed by their cognitive architecture and
the local task environment. CMR describes how the cognitive
system navigates a context space to encode and later recollect a
list of items. The space of possible behaviors during memory search
is fully speci /uniFB01ed in the architectural assumptions of CMR and its
parameters. For example, one of the main assumptions is how the
cognitive system updates the current context in a drifting process
based on a just-encoded or recalled item, which has been shown to
capture human behavioral patterns in empirical data ( Howard &
Kahana, 2002a ;Lohnas et al., 2015 ;Polyn et al., 2009a ). Specifying
an environment without these assumptions, such as one that allows
traversing freely to any position in the context space regardless of
previous context, can easily achieve the goals in memory search but
does not provide a realistic model of how human memory functions.
For the current rational analysis, we divide the set of CMR
parameters into two distinct categories. Some parameters, includ-
ingkin the retrieval rule and εdin the stopping rule, correspond to
the core architectural constraints of the memory system. We
assume that these constraints that describe the noise level in the
memory retrieval process ( k) and the extent that one can still
continue retrieval given the memory strengths of different items
(εd) are not themselves subject to optimization. The rest of the
parameters in CMR specify the space of possible encoding and
recall behavior during memory search. The goal of the current
work is to examine how these parameters can be optimized by the
cognitive system while keeping the core architectural constraints of
the memory system /uniFB01xed. Parameters that fall into the second
category include βencand βrec, which describe the amount of
drifting during encoding and recall, γfc, which describes the
proportion of preexperimental and experimental associations
used to update context and cstart(a newly introduced parameter
to replace ϕsandϕd; see text below), which describes the amount
of primacy. Our rational analysis consists of /uniFB01xing the /uniFB01rst set of
parameters to some reasonable values to capture the architectural
constraints during the retrieval process, and then optimizing the
parameter values in the second set that describe the space of
encoding and recall policies to achieve maximum free recall
performance. Essentially, our analysis will provide a rational
account of the following three issues:
1. What amounts of drifting during encoding ( βrec) and
during recall ( βrec) lead to optimal performance?
2. What is the ideal balance between retrieving preexperi-
mental context versus experimental context through γfc?
3. How does regulating the amount of primacy affect overall
free recall performance?
By searching the space of these encoding and recall policies through
these parameters, the cognitive system has the opportunity to /uniFB01nd
the optimal solution in the current task. Since these components are
optimized jointly, not separately, there will be an opportunity toidentify the subspace of encoding and recall policies where different
components interact to maximize overall recall.
As noted above, the standard way of addressing primacy in
CMR is to provide a boost to early-list items at recall via the
parameters ϕsandϕd; while this parameterization provides a way
of descriptively /uniFB01tting the primacy effect, it does not correspond to
a mechanistic claim about how recall works, and thus we do not
consider it one of the “constraints on the recall process ”provided
by CMR. To provide a mechanism for primacy, Kragel et al. (2015)
argue (following Laming, 1999 ) that primacy arises because
participants partially reinstate the start-of-list context when they
initiate recall. We instantiate this in our model by replacing ϕsand
ϕdwith a different parameter cstartthat speci /uniFB01es which study
context (by serial position) to reactivate at the beginning of recall.
When cstart=0, the starting context at recall is set to the beginning-
of-list context that was active before the presentation of the /uniFB01rst
item; when cstart=10 (given that the list length is 10), the starting
context at recall is set to the end-of-list context that was active after
the last item was presented. This parameterization assumes that
participants have the ability to reinstate the context associated with
any serial position at the outset of recall; this is, of course, a major
simpli /uniFB01cation —just because participants want to recall starting at a
particular point does not mean they will succeed. We will address
this issue (i.e., what if participants do not succeed at reinstating the
context associated with their desired starting point) later in the
article.
Before we can proceed to obtain the optimal encoding and recall
policy in the free recall task, we need to /uniFB01xt h ep a r a m e t e r st h a t
characterize the cognitive constraints during the retrieval process.
To obtain a set of reasonable values for these parameters, we /uniFB01t
CMR to an immediate free recall data set ( Kahana et al., 2002 ). We
use Bayesian optimization to search the space of CMR parameters
in order to minimize the normalized root-mean-square error
between the CMR simulations and the data. Results from the
obtained CMR model can be found in Figure 2 .I tc a p t u r e st h e
probability of recall by serial position as shown in the serial
position curve, the probability of the /uniFB01rst recall by s erial position,
and the conditional response proba bility, computed by dividing
the number of times a transition of that lag is actually made by the
number of times it could have been made. More details on
the CMR /uniFB01tting procedure and the data set used can be found
in the Appendix .
Deriving the Optimal Behavior Function
Next, having /uniFB01xed the parameters that specify hard constraints on
the retrieval process, εdandk, based on the model /uniFB01ts shown in
Figure 2 , we ran a second round of Bayesian optimization on the
remaining parameters, which determine encoding and retrieval
policies. The goal of this second round of Bayesian optimization
was to implement our rational analysis: That is, we set out to identify
the con /uniFB01gurations of these parameters (i.e., the encoding and
retrieval policies) that maximize recall performance under the
constraints imposed by the CMR architecture and the other ( /uniFB01xed)
parameters. We refer to the resulting model as rational-CMR. The
experiment in the simulations was set up based on the stimuli and
trial structures in Kahana et al. (2002) . More details on obtaining the
rational-CMR parameters can be found in the Appendix . This study
was not preregistered. All data sets analyzed in this work are fromThis document is copyrighted by the American Psychological Association or one of its allied publishers.
This article is intended solely for the personal use of the individual user and is not to be disseminated broadly.OPTIMAL FREE RECALL 5
publicly available data sets, which can be accessed from: http://me
mory.psych.upenn.edu/Data_Archive . All code and analysis can be
accessed from: https://github.com/qiongzhang/rationalCMR .
Results from rational-CMR can be found in Figure 3 .Figure 3A
shows that the optimal policy is one that always starts recalling from
the beginning of the list according to the probability of /uniFB01rst recall, and
then sequentially recalls in the forward direction according to the
conditional response probability ( cstart=0,βenc≈1,βrec≈1,γfc≈0;
there are no alternative policies with an equal level of recall perfor-
mance, as the optimal policy identi /uniFB01ed here consistently arises over
multiple runs of Bayesian optimization with differently initialized
parameters). This result provides a rational account of why it is
bene/uniFB01cial to have primacy ( cstart=0) and forward asymmetry
(γfc≈0) during free recall.
The optimal policy corresponds to a strategy that chains items
through context ( Figure 3B ). With βenc=1 (the optimal value
identi /uniFB01ed by rational-CMR), when an item is studied (call it item n),
the model drifts context all the way to the preexperimental context of
that item; once the next item (item n+1) is presented, the just-
retrieved context from item nis associated with item n+1. This
process iterates until the encoding reaches the end of the list. During
the recall phase, the optimal starting context identi /uniFB01ed by rational-
CMR is the context at the beginning of the encoding list ( cstart=0).
Since the start-of-list context was associated with the /uniFB01rst item on
the study list, reinstating the start-of-list context makes it possible to
recall the /uniFB01rst item reliably. From this point on, the optimal recall
policy is to only retrieve the preexperimental context but not the
experimental context of a just-recalled item (i.e., setting γfc=0), and
then have the current context drift all the way to the just-retrieved
context (i.e., setting βrec=1). This way, retrieval follows the same
chain that was traversed during encoding: Retrieving item n’s
preexperimental context triggers recall of item n+1 via the
associative link that was forged at study, leading to retrieval of
itemn+1’s preexperimental context, which cues recall of item n+2,
and so on. We explored how sensitive the optimal policy is to the
speci /uniFB01c variant of CMR that was used. A similar optimal policy was
obtained when we allowed intrusions from other lists in the simula-
tions, in a different variant of CMR (CMR2; Lohnas et al., 2015 ).We also found that the optimal policy was qualitatively identical
when we used an alternative primacy mechanism: the attention-
based account instantiated in TCM and CMR ( Lohnas et al., 2015 ;
Polyn et al., 2009b ). Full results of these analyses can be found in the
Appendix . When obtaining the optimal policy, we assumed that the
/uniFB01xed parameters that characterize the cognitive constraints during
the retrieval process are shared across all participants. We observe
that splitting all participants into two performance groups when
/uniFB01tting CMR, that is, assuming that better performing subjects and
worse performing subjects are different in their cognitive con-
straints, does not change our main conclusions about the optimal
policy (see more details in the Appendix ).
The derived optimal policy is nontrivial for several reasons.
First, it is not obvious that the optimal policy is to serially recall in a
task where one can recall items in any order; second, even if one
/uniFB01gures out intuitively that chaining (in this case, indirectly through
context) is a useful strategy becau se it minimizes the odds that an
item will be inadvertently skipp ed, it is not obvious that one should
carry out forward chaining rather than backward chaining, given
that the active context at the start of recall is the end-of-list context.
We will come back to the comparison between forward and
backward chaining in the General Discussion section. Interest-
ingly, the optimal policy resembles the stereotyped walk that
characterizes the method of loci, a mnemonic technique commonly
adopted for memorizing a long list of information ( Yates, 1966 ). In
the method of loci, one mentally walks through a prede /uniFB01ned route,
associating each item with loci along this route during encoding;
when it comes to recall, one simply mentally walks through the
same route and uses loci along the route as memory cues to
subsequently retrieve each item. We will come back to the com-
parison with mnemonic techniques in the General Discussion
section.
There are two features of the optimal policy that might appear
nonintuitive and require closer examination. First, under the strategy
to drift all the way to each encoded item ( βenc=1), each item is
associated with the item presented right before it but not any other
items. This appears to contradict the compound cuing effect com-
monly observed in free recall experiments ( Lohnas & Kahana, 2014 ),This document is copyrighted by the American Psychological Association or one of its allied publishers.
This article is intended solely for the personal use of the individual user and is not to be disseminated broadly.Figure 2
Behavioral Patterns in the Data Set in Kahana et al. (2002) and the CMR Fit to These Patterns
Note. From left to right are the serial position curve, probability of the /uniFB01rst recall, and conditional response probability. CMR parameters obtained using
Bayesian optimization: βenc=0.79, βrec=0.59, γfc=0.40, ϕs=4.66, ϕd=2.73, k=5.38, and εd=2.72. CMR =context maintenance and retrieval.6 ZHANG, GRIFFITHS, AND NORMAN
where multiple prior items combine to form a compound cue for the
next response. To obtain the compound cuing effect in CMR, βenc
must be set to a value less than one. Here, we show the conditions
when it is optimal to have βenc<1 instead of βenc=1. With increasing
noise during encoding ( Figure 4B ) or during recall ( Figure 4C ), the
optimal value of βenc, expressed as β/C3
enc, decreases. Intuitively, this
makes sense —when there is more noise in the system, “chaining ”
one item to the next through context is suboptimal because of the
high likelihood that the chain will be broken. Though having βenc
values smaller than 1 can cause occasional skipping of an item
during recall, which is not ideal, it is more robust to such an
interruption: When an item is associated with other nearby items
in addition to the immediate adjacent item, occasionally failing to
encode or recall the next item in the chain due to noise will not
prevent other items from being recalled, as shown in Figure 4A .In contrast, when βenc=1, failing to encode or recall the next item in
the chain will prevent all of the successive items from being recalled.
In the simulations, encoding noise is implemented as the percentage
of time when there is a failure to associate an item with the current
context; the recall noise is controlled by the scale parameter kin the
softmax retrieval rule during retrieval.
The second feature of the optimal policy that merits closer
examination is the optimal starting context. It is optimal to initiate
recall with the beginning-of-list context during study. If humans are
indeed rational, one might wonder why participants often start
recalling items from the end of the list. This can be explained based
on the idea that participants start recall with the end-of-list context
active in their mind; their ability to adopt the optimal policy is
constrained by how much they are able to suppress the end-of-list
context and reactivate the beginning-of-list context.This document is copyrighted by the American Psychological Association or one of its allied publishers.
This article is intended solely for the personal use of the individual user and is not to be disseminated broadly.Figure 3
A Rational Account of Primacy and Forward Asymmetry
Note. (A) Proportion of items correctly recalled, optimized over encoding and recall policy. The optimal policy always starts by recalling from the begin ning
of the list with cstart=0 and then sequentially recalls forwards. The optimal parameters are: β/C3enc=0.99, β/C3rec=0.99, and γ/C3
fc=0.001. The /uniFB01xed parameters are:
k=5.38 and εd=2.72. (B) This optimal encoding and recall policy can be understood as a strategy that chains items through the context. With βenc=1,βrec=1,
andγfc=0, when an item is presented (at encoding) or retrieved (at test), context always drifts all the way to that item ’s preexperimental context. For example, at
encoding, when a duck is studied, context drifts fully to the preexperimental context for duck, which is then associated with the next item (apple). At test, when
duck is retrieved, context fully drifts (once again) to the preexperimental context for duck, which triggers recall of apple. This way, the same chain is traversed
during encoding and recall. See the online article for the color version of this /uniFB01gure.OPTIMAL FREE RECALL 7
To account for this cognitive constraint in the model, we derive
the optimal policy under the scenario when one is not able to
reactivate the beginning-of-list context. Figure 5A shows the
proportion of items recalled, as a function of the starting study
context, assuming (for illustrative purposes) that participants can
initiate recall at any serial position. We are most interested in two
scenarios: one is when the participant is able to reactivate the
beginning-of-list context ( cstart=0), and the other is when the
participant is occupied with the end-of-list context ( cstart=10).
When starting recall from the beginning of the list, the optimal
value of βencis large ( Figure 5B ) for reasons described in the
previous section; by contrast, if starting recall from the end of
the list, the optimal value of βencis small ( Figure 5C )—in this case,the model can no longer rely on the strategy of chaining items
through context in a forward manner, and instead the model makes
items accessible at retrieval by placing them close by in the context
space during encoding. The challenge here is that —at encoding —
the participant does not know whether they will succeed or fail at
initiating recall from the start of the list. To address this challenge,
an optimal participant would need to choose the single value of βenc
that yields the highest expected value of recall performance,
factoring in the relative probabilities of the two outcomes (i.e.,
initiating recall from the start vs. the end of the list). Put another
way: While a participant cannot anticipate whether they will
succeed at initiating recall from the start of the list on a particular
trial, it is reasonable to think that the participant would know howThis document is copyrighted by the American Psychological Association or one of its allied publishers.
This article is intended solely for the personal use of the individual user and is not to be disseminated broadly.Figure 4
The Effect of Encoding and Retrieval Noise on the Optimal Policy
Note. (A) When βenc=1 in the optimal policy, each item is only associated with the item before and after
(chained through context); when βenc<1, items are also connected with items not immediately adjacent.
(B) Encoding noise is increased by introducing the possibility that, on a percentage of trials (horizontal axis), an
item fails to be associated with the current context. The optimal drift rate during encoding β/C3encdecreases when
encoding noise increases. The optimal drift rate during recall β/C3recalso decreases accordingly from βrec=1 to
βrec=0.90 at the highest level of encoding noise. Optimal γfcstays at 0. The /uniFB01xed parameters in the simulation
are:k=5.38 and εd=2.72. (C) Retrieval noise is increased through the parameter kin the softmax retrieval rule.
Smaller kvalues correspond to larger retrieval noise. β/C3encdecreases when retrieval noise increases. β/C3recalso
decreases accordingly from βrec=1 toβrec=0.56 at the highest level of retrieval noise. Optimal γfcstays at 0. For
this simulation, εd=2.72. See the online article for the color version of this /uniFB01gure.8 ZHANG, GRIFFITHS, AND NORMAN
successful they are on average , and they can use this information
when deciding how to parameterize encoding.
To capture this, Figure 5D shows how overall recall accuracy
varies as a function of βencand also the average proportion of trials
where the participant succeeds at initiating recall from the start (vs.
end) of the list. To obtain the results shown in Figure 5D , we/uniFB01xed
the values of these parameters ( βencand the probability of initiating
recall from the start of the list), and then selected values of the
parameters that control retrieval policy ( γfcandβrec) in order to
optimize recall performance. Importantly, at test, participants have
the option of parameterizing recall differently depending on whether
they were successful (on that trial) at initiating recall from the
beginning of the list. Accordingly, the model was allowed to select
different values of γfcandβrecon trials where recall was initiated
from the start versus the end of the list. Figure 5D shows that, as the
probability of initiating recall from the start of the list increases, the
optimal value of βencincreases.
So far, we have provided a rational account of primacy and
forward asymmetry. As discussed earlier, forward asymmetryarises from updating context at retrieval using the preexperi-
mental context associated with the retrieved item ( γfc=0). Next,
we identify the conditions where it is optimal to update context at
retrieval using the context that wa sa s s o c i a t e dw i t ht h er e t r i e v e d
item at study ( γfc=1), which gives rise to symmetric contiguity
effects (i.e., backward recalls in addition to forward recalls).
Figure 5E shows that, on trials when one is not able to jump back
to the beginning of the list (i.e., the starting context is the end-of-
list context), it is optimal to use both backward and forward
recalls by setting γfc=1, providing a rational account of the
symmetric contiguity effect .T h i sp a t t e r nh o l d sw h e n βencis
smaller than 0.8: When βencis too large, the context at encoding
ct=ρct−1+βenccINis dominated by the preexperimental context
induced by newly encoded items βenccIN,l e a v i n gav e r yw e a k
amount of experimental context ct−1to generate backward recalls
even if one can fully reinstate this experimental context with
γfc=1(βenc=0.6 in the speci /uniFB01ce x a m p l eg i v e ni n Figure 5E —
this is the optimal βencvalue for when the proportion of start-of-
list trials is 80%).This document is copyrighted by the American Psychological Association or one of its allied publishers.
This article is intended solely for the personal use of the individual user and is not to be disseminated broadly.Figure 5
Optimal Drift Rate at Encoding Varies According to How Recall Is Initiated
Note. (A) Proportion of items correctly recalled, optimized over encoding and recall policy, assuming (for illustrative purposes) that participants ca n start recall by
reinstating the context associated with an arbitrary serial position in the list. (B) The optimal drift rate at encoding β/C3encis large if starting recall from the beginning of
list. (C) β/C3encis small if starting recall from the end of the list. (D) β/C3encis constrained by how much one is able to initiate recall with the beginning-of-list context. The
plot shows the effect of βencon overall recall accuracy as a function of the proportion of trials in which participants succeed at starting recall from the beginning of
the list (each colored line corresponds to a different proportion), under noisy retrieval conditions ( k=3). A total of /uniFB01ve parameters are optimized in the simulations:
βencfor all trials, and different βrecandγfcfor trials that start from the beginning or start from the end; the /uniFB01xed parameters are k=3,εd=2.72. (E) On trials where
the starting recall context is the end-of-list context, it is optimal to use symmetric contiguity (i.e., large γfc) to recall backwards. The plot here assumes that the
probability of initiating recall with the beginning-of-list context is 80%, in which case, the optimal parameters for trials with the end-of-list co ntext are γ/C3
fc=0.93,
β/C3enc=0.6, and β/C3rec=0.81. The other /uniFB01xed parameters are k=3 and εd=2.72. See the online article for the color version of this /uniFB01gure.OPTIMAL FREE RECALL 9
Testing the Predictions of the Optimal Behavior
Function on Empirical Data
We have learned from rational-CMR that the optimal behavior
function in free recall is to start recalling from the beginning of the
list and then sequentially recall forward, relying on the forward
asymmetry induced by updating context with the recalled item ’s
preexperimental context ( γfc=0). In addition, when recall cannot be
initiated at the beginning-of-list context, it is optimal to minimize the
amount of forward asymmetry and instead rely on symmetric
contiguity, induced by updating context with the context that
was associated with the recalled item at study ( γfc=1). These
observations serve as predictions for human behavioral data. Here,
we test these predictions at the level of individuals, using a large free
recall database: the Penn Electrophysiology of Encoding and
Retrieval Study (PEERS). The present analysis is based on the
behavioral data from the 171 participants (age range: 18 –30) in
Experiment 1 of the PEERS data set, which used an immediate free
recall task. More details of the PEERS data set can be found in
previous studies ( Healey et al., 2018 ;Healey & Kahana, 2016 ;
Lohnas et al., 2015 ). The present analysis also incorporates the
behavioral data from the 61 participants in Howard and Kahana
(1999) with an immediate free recall condition and a delayed free
recall condition. The criteria for including participants and sessions
into the current analysis can be found in the Appendix .
Prediction 1: Better Performing Participants Will
Demonstrate a Stronger Tendency to Initiate Recall
From the Beginning of the List and Recall in a
Forward Direction
According to rational-CMR, participants who perform better in
the free recall task should demonstrate a stronger tendency to initiate
recall from the beginning of the list, and they should make more
forward recalls than backward recalls. Figure 6A –Csummarizes the
serial position curve, the probability of /uniFB01rst recall, and the condi-
tional response probability in the PEERS data set. Participants were
divided into two groups based on their average performance in the
experiment: the better performing participants are from the top 10%
(gray) and the worse performing participants are from the remaining
90% (orange). Dividing the participants at this percentile highlights
the behavior of the participants who completed the free recall task
with high performance (on average, 85% correct recall) versus the
worse performing participants (on average, 59% correct recall).
Figure 6B shows that, consistent with the model ’s predictions, better
performing participants showed a higher tendency to start the recall
process from the /uniFB01rst item compared with the rest of the participants
(Mann –Whitney U=940, n1=17,n2=154, p=.02, two tailed),
and they also showed a lower tendency to start the recall process
from the last item compared with the rest of the participants (Mann –
Whitney U=719.5, n1=17,n2=154, p<.01, two tailed). In
addition, Figure 6C shows that better performing participants were
more likely to carry out forward recalls compared to the rest of the
participants. To further examine this effect, we summarized forward
asymmetry as a single value —the difference between the summed
probability of four forward lags and the summed probability of four
backward lags from the conditional response probability curve.
Figure 6D shows that Spearman ’s correlation between the forward
asymmetry and free recall performance was ρ=0.25 among the 171individuals ( p<.001). Importantly, this correlation between for-
ward asymmetry and performance is not a trivial consequence of
better performing individuals exhibiting higher primacy. When
participants start recall from the beginning of the list, this naturally
leads to a larger number of available forward recalls than backward
recalls; however, it does not necessarily lead to forward asymmetry
since our measure of forward asymmetry (calculated from the
conditional response probability) controls for the availability of
recall transitions (i.e., the conditional probability of a forward
transition at a particular lag is the observed probability of that
transition divided by the number of times when that transition was
possible).
To visualize in more detail how serial positions change by output
position during recall, Figure 6E plots serial position curve for each
output position, averaged across the top 10% and the bottom 90% of
performers. The optimal policy of starting recall from the beginning
of the list and then sequentially recalling forwards corresponds to the
dashed vertical line for each subplot, where the /uniFB01rst recall is at the
/uniFB01rst serial position, and the second recall is at the second serial
position, etc. Consistent with our model predictions, Figure 6E
shows that recall transitions among the top 10% of performers are
more aligned with the optimal policy than recall transitions among
the bottom 90% of performers.
One might worry that higher performing participants will always
show greater consistency with the optimal policy, regardless of
whether our rational-CMR theory is correct; however, this is not the
case. A simple counterexample to this would be if the participants all
follow random policies that demonstrate some variability, but none
of these policies are close to the optimal policy. In this case, even if
we looked at the high-performing participants, those participants
would be following strategies that differ from the optimal policy (see
Appendix , for a simulation that demonstrates this).
Prediction 2: Better Performing Participants Will
Demonstrate a Stronger Effect of Forward Asymmetry
When Recall Is Initiated From the Beginning of the
List and a Stronger Effect of Symmetric Contiguity
When Recall Is Initiated From the End of the List
Participants are not able to activate the beginning-of-list context
on all trials. According to rational-CMR, patterns of recall transi-
tions (operationalized using conditional response probability)
should depend on whether recall can be initiated at the begin-
ning-of-list context. Speci /uniFB01cally, the model predicts that it is optimal
to maximize forward asymmetry in recall transitions if the recall is
initiated from the beginning-of-list context; conversely, it is optimal
to minimize forward asymmetry and maximize the ability to carry
out backward recalls through the symmetric contiguity effect if the
recall is initiated from the end-of-list context. In Figure 7 , trials are
divided into primacy trials (A –C) and recency trials (D –F) based on
patterns of recall initiation. In human data, there is no “ground truth ”
way of measuring the starting context at recall. Consequently, we
sorted trials into primacy trials and recency trials based on behav-
ioral patterns. The most obvious way to de /uniFB01ne primacy trials would
be to select trials where recall started from the /uniFB01rst item in the study
list (Ward et al., 2010 ). However, applying this criterion would
leave us with fewer than 7 out of 96 trials for more than half of the
171 participants and zero primacy trials for 25% of all participantsThis document is copyrighted by the American Psychological Association or one of its allied publishers.
This article is intended solely for the personal use of the individual user and is not to be disseminated broadly.10 ZHANG, GRIFFITHS, AND NORMAN
(who must therefore be removed from the analysis). To improve the
power of our analysis, we relaxed the de /uniFB01nition of the primacy trials
to encompass all trials where the /uniFB01rst item at study was retrieved
during the /uniFB01rst four recalls (we nevertheless examined the results
under both de /uniFB01nitions of primacy trials and observed that the major
conclusions of the analysis did not change). This is based on the
observation that sometimes participants can recall from the begin-
ning of the list —they just cannot do that right from the /uniFB01rst recall. In
the PEERS free recall data set, the /uniFB01rst item during study is retrieved
the most frequently at the fourth position, as the /uniFB01rst three recalls aredominated by the recency effect. Next, we had to operationally
de/uniFB01ne recency trials. Here, we wanted to select trials where perfor-
mance was unlikely to have been in /uniFB02uenced by the beginning-of-list
context at any point during the recall period; to meet this criterion,
we de /uniFB01ned recency trials as those without recalls of the /uniFB01rst item.
Figure 7A plots the conditional response probability for the top
10% and the bottom 90% of performers in the human data, where the
analysis was limited to primacy trials and “top 10% ”and“bottom
90%”were computed based on performance on these primacy trials.
We found that the better performers showed higher levels of forwardThis document is copyrighted by the American Psychological Association or one of its allied publishers.
This article is intended solely for the personal use of the individual user and is not to be disseminated broadly.Figure 6
Better Performing Participants Demonstrate a Stronger Tendency to Initiate Recall From the Beginning of the List and Carry Out
Forward Recalls
Note. (A –C) Behavioral patterns of the top 10% of participants (gray) in terms of performance in PEERS data set compared with the rest of the participants
(orange): serial position curve, probability of /uniFB01rst recall, and conditional response probability. The /uniFB01rst three recalls are removed when computing the
conditional response probability curve as they are heavily in /uniFB02uenced by the recency effect. (D) Correlation between the forward recall asymmetry and
participants ’recall performance. Forward recall asymmetry is calculated as the difference between the summed probability of four forward lags and the summed
probability of four backward lags from the conditional response probability. (E) Serial position curves for each output position for the top 10% and t he bottom
90% of performers. The vertical dashed lines represent the optimal policy. PEERS =Penn Electrophysiology of Encoding and Retrieval Study. See the online
article for the color version of this /uniFB01gure.OPTIMAL FREE RECALL 11
asymmetry. To formally test this, we calculated a behavioral
forward asymmetry score (plotted in Figure 7B separately for the
top 10% and the bottom 90% of performers), operationalized as the
difference between the summed probability of four forward lags and
the summed probability of four backward lags from the conditional
response probability curve in Figure 7A ; this score was higher in the
top 10% of performers (Mann –Whitney U=807,n1=17,n2=154,
p<.01, two tailed) than in rest of the participants.
Figure 7C plots the conditional response probability for the top
10% and the bottom 90% of performers in the human data, where the
analysis was limited to recency trials and “top 10% ”and“bottom
90%”were computed based on performance on these recency trials.
We found that the better performers showed a stronger effect ofsymmetric contiguity. As discussed earlier, CMR posits that forward
transitions (as captured in the conditional response probability curve)
are contributed by both forward asymmetry and symmetric contigu-
ity, whereas backward transitions are uniquely contributed by sym-
metric contiguity; we therefore calculated the behavioral in /uniFB02uence of
symmetric contiguity by computing a backward transition score
(plotted in Figure 7D separately for the top 10% and the bottom
90% of performers), operationalized as the summed probability of
four backward lags from the conditional response probability curve
inFigure 7C . The score was higher in the top 10% of performers
(Mann –Whitney U=641, n1=17,n2=154, p<.001, two tailed).
Though the forward asymmetry in the top 10% of performers is still
sizable and similar to the bottom 90% of performers (Mann –WhitneyThis document is copyrighted by the American Psychological Association or one of its allied publishers.
This article is intended solely for the personal use of the individual user and is not to be disseminated broadly.Figure 7
Better-Performing Participants Demonstrate a Stronger Effect of Forward Asymmetry
Than Other Participants on Primacy Trials (A –B), and a Stronger Effect of Symmetric
Contiguity than Other Participants on Recency Trials, as Characterized by Backward
Transitions (C –D)
Note. The behavioral score of forward asymmetry in B is calculated as the difference between the
summed probability of four forward lags and the summed probability of four backward lags from the
conditional response probability of each participants ( N=171). The behavioral score of backward
transitions in D is calculated as the summed probability of four backward lags from the conditional
response probability curve of each participant ( N=171). In A and B, “Top 10% ”refers to
participants whose performance during primacy trials is in the top 10th percentile of the population.
In C and D, “Top 10% ”refers to participants whose performance during recency trials is in the top
10th percentile of the population. Error bars indicate the standard error of the mean. CRP =
conditional response probability. See the online article for the color version of this /uniFB01gure.12 ZHANG, GRIFFITHS, AND NORMAN
U=1,235, n1=17,n2=17,p=.35, two tailed), there is decreased
forward asymmetry in the top 10% of performers during recency
trials compared with that in primacy trials (Mann –Whitney U=
1,235, n1=154,n2=154,p=.03, two tailed) but this is not observed
in the bottom 90% of performers (Mann –Whitney U=1,235, n1=
17,n2=154, p=.30, two tailed). We also carried out the same
analysis using an alternative way of scoring forward asymmetry,
calculated as the ratio, instead of the difference, between the summed
probability of four forward lags and the summed probability of four
backward lags from the conditional response probability of each
participant. This does not change the main conclusions.
Analysis of the PEERS data set provided evidence for the model
prediction that it is optimal to maximize forward asymmetry when
the recall is initiated from the beginning-of-list context and maximize
backward recall when the recall is initiated from the end-of-list
context. However, demonstrating that recall transitions are correlated
with how recall is initiated is not the same as demonstrating that recall
transitions depend on how recall is initiated, where the accessibility
of the beginning-of-list context (relative to the end-of-list context) is
experimentally manipulated. The latter is a stronger test of the model
predictions. To investigate this latter scenario, we analyzed publicly
available data sets from Howard and Kahana (1999) , where the recall
phase of the free recall experiments was conducted either immedi-
ately after studying the list, or with a delayed period of distractor
activities. Participants could not predict during encoding whether it
would be an immediate or delayed condition; they were warned that
sometimes there would be a math test between the end of the list and
the signal to recall and sometimes there would not. Adding a period
of delay after studying the list helps with disengaging from the
end-of-list context ( Murdock, 1962 ), and leads to increased accessi-
bility of the beginning-of-list context during recall, as can be seen
from Figure 8B , where there is a higher tendency to start the recall
process from the /uniFB01rst item in the delayed free recall condition than the
immediate free recall condition (Mann –Whitney U=185.500,
n1=65,n2=62,p=.04, two tailed).
Having con /uniFB01rmed that recall initiation is affected by the manipu-
lation of immediate versus delayed recall in this data set, the next step
is to examine whether optimal recall transitions vary in the immediate
versus delayed recall conditions. Consistent with the model predic-
tions, there is stronger forward asymmetry in the delayed free recall
condition than in the immediate free recall condition ( Figure 8C ;
Mann –Whitney U=1188.00, n1=65,n2=62,p=.020, two tailed),
where there is higher accessibility of the beginning-of-list context.
Moreover, we can divide participants into two groups based on their
average performance in the experiment: the better performing parti-
cipants are from the top 20% (gray) and the worse performing
participants are from the remaining 80% (orange). We used a
20/80 split here (instead of dividing the participants into the top
10% and remaining 90% as in the PEERS data set) to make sure that
there were at least 10 participants in each group. Figure 8F shows
that, in the delayed free recall condition (when it is easier to access the
beginning-of-list context), better performing participants demon-
strate numerically (but not signi /uniFB01cantly) stronger forward asymmetry
compared to the worse performing participants (Mann –Whitney
U=198.50, n1=12,n2=41,p=.159, two-tailed; however, at
the+1 lag of the conditional response probability curve, the differ-
ence between better performing and worse performing participants is
signi/uniFB01cant: Mann –Whitney U=187.50, n1=12,n2=41,p=.007,
two-tailed); conversely, in the immediate free recall condition(when the end-of-list context is relatively more accessible), better
performing participants demonstrate stronger backward transitions
compared to the worse performing participants (Mann –Whitney U=
144.00, n1=10,n2=48,p=.025, two tailed). Taken together, these
results provide converging support for the model ’s predictions about
how recall initiation will affect recall policy.
General Discussion
Much of the work in the free recall literature has sought to
understand the processes and representations that give rise to
different patterns of recall organization; however, it remains unclear
why certain types of recall organization arise in the /uniFB01rst place. Our
work provides a rational analysis of the free recall task, deriving the
optimal policy under the internal representations and processes of
memory search described by CMR. Our model, which we call
rational-CMR, demonstrates that the optimal policy in free recall
is to start recall from the beginning of the list and then sequentially
recall forwards, providing a rational account of primacy and forward
asymmetry effects typically observed in free recall. In addition, the
rational model also makes a novel prediction that the optimal policy
in recall transitions should depend on recall initiation. These
predictions from the rational model were con /uniFB01rmed in human
behavioral data, where top human participants demonstrated stron-
ger primacy and forward asymmetry than the rest of the participants,
and the amount of forward asymmetry in participants depended on
the starting context at recall. We now turn to the broader implica-
tions of these results.
Why Is the Optimal Policy Optimal?
We show that the optimal policy in free recall is to start recall from
the beginning of the list and then sequentially recall forwards, which
corresponds to a strategy that chains items through context ( Figure 3B ).
Intuitively, chaining (in this case, indirectly through context) is a
useful strategy because it minimizes the odds that an item will be
inadvertently skipped. However, if the chained structure in the
optimal policy is the key factor for effective recalls, one might
ask why it is necessary to start recall from the beginning and then
recall forwards, rather than starting recall from the end and then
recalling backwards. The latter would be more convenient, because
one already has access to the end-of-list context at the start of recall,
whereas reactivating the beginning-of-list context requires an extra
retrieval step. To understand this better, we need to review the
encoding mechanism in CMR in Figure 1 . In CMR, there is a
reliable way to proceed forward (by reinstating cINenc), but there is not
a reliable way to proceed backward —the only way to promote
backward transitions is to reinstate the drifting contextual state
associated with the item at study, which is equally likely to propel
recall forward and backward. Since there is a reliable way to propel
recall forward but no reliable way to propel recall backward,
proceeding forward yields better results (assuming that the partici-
pant is able to succeed in initiating recall from the start of the list).
A Rational Account of the Primacy Effect
In the free recall task, words presented at the beginning and at the
end of the list are better remembered than those presented in the
middle of the list. This U shape in the serial position curve of freeThis document is copyrighted by the American Psychological Association or one of its allied publishers.
This article is intended solely for the personal use of the individual user and is not to be disseminated broadly.OPTIMAL FREE RECALL 13
recall is one of the most robust /uniFB01ndings in cognitive psychology
(Murdock, 1962 ). There exist a number of theories that account for
primacy. Rehearsal-based explanations of the primacy effect posit
that increased rehearsal opportunities for early items in the list leadto increased encoding in the short-term memory buffer ( Atkinson &
Shiffrin, 1968 ;Brodie & Murdock, 1977 ;Murdock & Metcalfe,
1978 ;Rundus, 1971 ;Rundus & Atkinson, 1970 ). Even if no short-
term buffer is presumed, the repetition, the distribution, and theThis document is copyrighted by the American Psychological Association or one of its allied publishers.
This article is intended solely for the personal use of the individual user and is not to be disseminated broadly.Figure 8
Optimal Recall Transitions Vary According to How Recall is Initiated
Note. Behavioral patterns in the data set of Howard and Kahana (1999) for the delayed free recall condition and the immediate free recall condition (A –C), the
delayed condition split by performance level of participants (D –F), and the immediate condition split by performance level of participants (G –I). From left to
right, the three columns show the serial position curve, probability of the /uniFB01rst recall, and conditional response probability. See the online article for the color
version of this /uniFB01gure.14 ZHANG, GRIFFITHS, AND NORMAN
recency of rehearsals can make primacy items more accessible at test
(Tan & Ward, 2000 ). Despite the success of the rehearsal account in
explaining primacy, there are empirical studies showing that
primacy is still found when rehearsal is minimized, such as when
a/uniFB01ve-item list is presented in less than 1 s ( Neath & Crowder, 1996 ),
when there is semantic judgment required encoding each word
(Howard & Kahana, 1999 ), or when there is an interval of distractor
activity between every two adjacent words that are presented ( Polyn
et al., 2009b ). Other accounts of primacy do not depend on the
rehearsal process. For example, some accounts posit that the /uniFB01rst
items are more distinctive than the other items ( Murdock, 1960 ;
Neath, 1993 ), although this has been critiqued on the grounds that
distinctiveness is an effect rather than an explanation ( Hunt & Lamb,
2001 ). Other accounts posit that primacy is caused by contextual
change at the beginning of the list, induced by either a prediction
error ( Rouhani et al., 2020 ) or novelty ( Davelaar, 2013 ). CMR
captures the primacy effect by assuming a gradient by serial position
without explicitly committing to a particular mechanism of primacy.
In contrast to the debate in the literature on the mechanisms that
give rise to primacy, our work provides a rational account of why
primacy is important to have in the /uniFB01rst place. In a rational analysis
of the free recall task based on CMR simulations, we showed that
initiating recall from the beginning of the list gives better perfor-
mance than initiating recall from other positions in the list. This
model prediction was tested using a large free recall data set.
Consistent with the model ’s predictions, the better performing
participants in the data set showed a higher probability of initiating
recall from the beginning of the list and a lower probability of
initiating recall from the end of the list, compared with those of the
worse performing participants. Notably, the /uniFB01rst recall in the better
performing participants did not always consist of the /uniFB01rst item in the
study list, but rather was dominated by the last few items. This can
be explained in terms of the idea that, at the start of recall, the end-of-
list context is active; to activate the beginning-of-list context,
participants need to suppress the end-of-list context and retrieve
the beginning-of-list context —both of these processes are prone to
failure, and the dif /uniFB01culty of activating the /uniFB01rst part of the list can be
modulated by situational factors (e.g., the presence of a distinctive
event at the start of the list). Thus, the challenging nature of
replacing the end-of-list context with the start-of-list context serves
as a constraint on optimal behavior ( Sahakyan & Kelley, 2002 ). The
second part of the model simulations takes into account this
constraint by deriving what the optimal behavior is when one is
only able to retrieve the beginning-of-list of context on a proportion
of trials. Implications of these results are discussed in the next
section.
There has been other work that has considered how participants
vary how they initiate recall as a function of task instructions. Tan
et al. (2016) found that the tendency to start recall from the /uniFB01rst item
occurred only when participants were required to recall as many
items as possible. When participants were asked to recall only one or
two items, they tended to initiate recall with end-of-list items ( Tan
et al., 2016 ). These /uniFB01ndings show that different retrieval strategies
exist for searching for different numbers of items and that partici-
pants have some control over these retrieval strategies ( Ward & Tan,
2019 ). Our analysis provides a rational explanation for why parti-
cipants would adopt different retrieval strategies under different task
demands. When the goal of the memory task is to retrieve as many
items as possible, recalling from the beginning meets the goal betterthan recalling from the end, as shown in the optimal policy. When
the goal of the memory task is to retrieve only one or two items,
recalling from anywhere in the list can meet the goal equally well;
under this scenario, participants do not have to expend the effort to
reinstate the beginning-of-list context —it is optimal to start recalling
from the end-of-list context, which is already active at the start of
recall.
A Rational Account of Forward Asymmetry
During memory search, forward transitions are more likely than
backward transitions for small absolute values of lag ( Kahana,
1996 ). This effect has been shown in immediate free recall
(Kahana, 1996 ) as well as delayed free recall ( Howard &
Kahana, 1999 ). Unlike primacy, for which CMR does not commit
to a particular mechanism, CMR can account for the mechanism of
forward asymmetry. However, CMR does not address why there is
forward asymmetry. It is possible under the space of CMR param-
eterization to turn off the asymmetry entirely ( γfc=1). Our work
provides a rational account of why forward asymmetry is an
important feature in free recall. By making a connection between
forward asymmetry and overall task performance, we were able to
account for the individual differences in free recall performance
observed in a large free recall database. Speci /uniFB01cally, there was a
signi/uniFB01cant correlation between the amount of forward asymmetry
and overall performance at the level of individuals.
In addition to explaining the forward asymmetry, rational-CMR
can also explain the presence of backward transitions. Our results
show that optimal recall transitions depend on recall initiation;
forward asymmetry is only important when recalls are initiated
from the beginning of the list. When participants are not able to
initiate recall from the beginning of the list, it is optimal to rely on
symmetric contiguity to recall backwards instead of forward asym-
metry. This model prediction was further tested over a large free
recall database and a data set containing both immediate and delayed
free recall conditions. Consistent with the model ’s predictions,
better performing participants demonstrated a stronger effect of
forward asymmetry when recall was initiated from the beginning of
the list and a stronger effect of symmetric contiguity when recall was
initiated from the end of the list.
Both forward asymmetry and symmetric contiguity contribute to
the temporal contiguity effects commonly observed in free recall
(Kahana, 1996 ), and it has been shown that participants ’temporal
contiguity effects correlate highly with their recall performance
(Sederberg et al., 2010 ). Results from rational-CMR are consistent
with this empirical /uniFB01nding while also providing additional insight
into which type of temporal contiguity is most useful for perfor-
mance, conditioned on how recall is initiated.
Our results also connect to several empirical results in the
literature that examined the relationship between primacy/recency
and memory performance ( Dalezman, 1976 ).Dalezman (1976)
instructed participants to recall either from the beginning, the
end, or the middle of a list before they recalled the rest of the
items. Though the amount of primacy and recency can be reliably
altered by the recall instruction, the overall performance largely
stays unchanged. Why is it the case that altering recall initiation does
not affect the overall performance? Our results emphasize that one
needs to coordinate both recall initiation and recall transitions to
achieve optimal recall. Though the optimal policy is to demonstrateThis document is copyrighted by the American Psychological Association or one of its allied publishers.
This article is intended solely for the personal use of the individual user and is not to be disseminated broadly.OPTIMAL FREE RECALL 15
primacy and forward asymmetry, having primacy alone, as induced
by one of the recall conditions in Dalezman (1976) , does not
guarantee improvement in memory performance.
Other work by Unsworth et al. (2011 ) has looked at individual
differences in primacy/recency and memory performance. Partici-
pants were clustered into three groups according to their serial
position curves: a high-primacy low-recency group, a low-primacy
high-recency group, and a high-primacy high-recency group;
Unsworth et al. (2011) found that the last group demonstrated
the highest level of memory performance. In the Appendix , we
include a “proof-of-concept ”simulation showing one way that our
model can explain this pattern of results. We assume that all
participants have a similar mix of trials where they start recall
from the beginning versus the end of the list, but they vary in how
well they can optimize their recall strategy as a function of how they
initiate recall. In our simulation, we have one group that pursues the
optimal policy (adjusting γfcbased on how recall is initiated),
another group that uses the “optimal-for-beginning-of-list ”γfc
regardless of how recall is initiated, and a third group that uses
the“optimal-of-end-of-list ”γfcparameters regardless of how recall
is initiated. With these assumptions, our model qualitatively repli-
cates the key results from Unsworth et al. (2011) —the“optimal
policy ”group has the best overall performance and demonstrates a
high level of performance for both primacy and recency items
compared with other groups, whereas the other two groups demon-
strate high levels of performance for primacy or recency items (but
not both) and lower overall levels of performance. Details of the
simulation can be found in the Appendix .
Connection to Mnemonic Techniques
Rational-CMR may also help to explain the ef /uniFB01cacy of mnemonic
techniques such as the method of loci. The method of loci was /uniFB01rst
documented in Roman and Greek rhetorical treatises, where orators
adopted this technique to memorize key points in a long speech
(Yates, 1966 ). Practicing this method /uniFB01rst involves familiarizing
oneself with an environment, such as a street or a building, and
mentally constructing an ordered route between well-de /uniFB01ned loca-
tions (loci). When given a list of items to remember, the method of
loci is characterized by a stereotyped walk during both encoding and
recall: During encoding, one mentally walks through the prede /uniFB01ned
route, associating each item with loci along this route; during recall,
one simply imagines oneself walking through the same familiar
route, using loci along the route as memory cues to subsequently
retrieve each item ( Yates, 1966 ).
The stereotyped walk used in the method of loci resembles the
optimal policy derived in rational-CMR, which is to start recall from
the beginning of the list and sequentially recall forwards. We
propose that one of the reasons why the method of loci works so
well is that it enforces the optimal policy of memory search through
explicit instructions and pretraining. During free recall, without
using mnemonic techniques, there are two potential reasons why it is
hard for individual participants to adopt the exact optimal policy
derived in the rational-CMR model. First, participants start recall
with the end-of-list context in their mind. Their ability to adopt the
optimal policy is constrained by how much they are able to drop the
end-of-list context and reactivate the beginning-of-list context.
Second, participants have limited encoding time to fully process
and store an item. Their ability to start recall from the beginning ofthe list and recall sequentially forwards is constrained by the /uniFB01delity
of their encoding. Using the method of loci, one can enforce the
usage of optimal policy during free recall by directing the recall
process to always start from the beginning; also, the method of loci
leverages a prelearned spatial representation so that information can
be more easily encoded in a sequence on the /uniFB02y during the
encoding stage.
Our results also shed light on the recent empirical /uniFB01nding that two
other mnemonics —the peg method and the temporal encoding
method —achieve comparable improvements in recall performance
compared to the method of loci ( Bouffard et al., 2018 ;Caplan et al.,
2019 ). Both methods emphasize a temporal feature as the scaffold.
The peg method works by prememorizing a list of “peg words ”;
during encoding, items to study are associated with this list of words
in the order that they are presented. The temporal encoding method
works similarly to the peg method. Instead of prememorizing a list
of words, participants prede /uniFB01ne a chronological timeline using a list
of their most vivid memories and later associate items to study with
these memories. Despite their differences with the method of loci (in
the exact instructions that are used and their use of a nonspatial
“scaffold ”instead of spatial knowledge), these techniques all share
an encoding and recall policy that resembles the optimal policy in
free recall, which may help to explain why they achieve comparable
improvements in recall performance.
Limitations and Future Work
Committing to the architectural assumptions of CMR, the current
work derived the optimal policy and demonstrated that the derived
optimal policy can explain performance observed in human data. It
is possible that there is additional variance in memory performance
that can be explained by altering the architectural assumptions of
CMR. In future work, we will explore the consequences on the
optimal policy if speci /uniFB01c assumptions of CMR are altered (e.g., if we
allowed parameters to vary dynamically as the recall process
unfolds, instead of being held constant). For simplicity, the current
work assumes that the semantic representations of words are
independent of each other. We acknowledge that semantic associa-
tions between words drive a signi /uniFB01cant amount of the recall
dynamics. Typically, in categorized-list free recall experiments,
with semantically related items located at different locations of
the list, the semantic structure can overshadow the effect of temporal
context, with semantic context driving recalls in different directions
than the temporal context. Therefore, we might not be able to arrive
at a stable solution for optimal policy under the in /uniFB02uence of
semantic associations. The /uniFB01rst step of the rational analysis is to
isolate the two and focus on discovering optimal policy without
effect from semantic associations. Future work will examine the two
components jointly.
Conclusion
The discovery of the primacy effect and the recency effect in free
recall dates back to the times of Ebbinghaus (1885/1913) . Since
then, psychologists have used the free recall task to gain insight into
the processes and representations underlying memory search
(Murdock, 1960 ,1962 ;Roberts, 1972 ;Standing, 1973 ). The /uniFB01eld
has primarily focused on identifying an array of behavioral patterns
that are consistently shown across free recall experiments andThis document is copyrighted by the American Psychological Association or one of its allied publishers.
This article is intended solely for the personal use of the individual user and is not to be disseminated broadly.16 ZHANG, GRIFFITHS, AND NORMAN
building computational models to capture these behavioral patterns.
To our knowledge, our work is the /uniFB01rst to examine why humans
demonstrate these recall patterns in the /uniFB01rst place. Under the
architectural assumptions about internal representations and pro-
cesses during memory search instantiated by the CMR model, we
showed that recall patterns including primacy, forward asymmetry,
compound cuing, and symmetric contiguity can arise naturally from
optimizing the overall recall performance. In particular, we showed
that, although participants are free to recall items in any order (hence
“free”recall), the optimal policy is one where participants recall
items in the same order that items are studied. We connected these
results with the literature on mnemonic techniques and proposed
that mnemonic techniques support superior memory by enforcing
the optimal policy of memory search through explicit instructions
and pretraining. These /uniFB01ndings also offer tantalizing insights into
how to further improve human memory through the intelligent
choice of memory cues.
References
Anderson, J. R. (1990). The adaptive character of thought . Psychology
Press.
Anderson, J. R., & Bower, G. H. (1972). Recognition and retrieval processes
in free recall. Psychological review ,79(2), 97 –123. https://doi.org/10
.1037/h0033773
Anderson, J. R., & Milson, R. (1989). Human memory: An adaptive
perspective. Psychological Review ,96(4), 703 –719. https://doi.org/10
.1037/0033-295X.96.4.703
Atkinson, R., & Shiffrin, R. (1968). Human memory: A proposed system
and its control processes. Psychology of Learning and Motivation ,2(4),
89–195. https://doi.org/10.1016/s0079-7421(08)60422-3
Bouffard, N., Stokes, J., Kramer, H. J., & Ekstrom, A. D. (2018). Temporal
encoding strategies result in boosts to /uniFB01nal free recall performance compa-
rable to spatial ones. Memory & Cognition ,46(1), 17 –31.https://doi.org/10
.3758/s13421-017-0742-z
Bous /uniFB01eld, W. A. (1953). The occurrence of clustering in the recall of
randomly arranged associates. The Journal of General Psychology ,
49(2), 229 –240. https://doi.org/10.1080/00221309.1953.9710088
Bous /uniFB01eld, W. A., & Sedgewick, C. H. W. (1944). An analysis of sequences
of restricted associative responses. The Journal of General Psychology ,
30(2), 149 –165. https://doi.org/10.1080/00221309.1944.10544467
Bower, G. (1967). A multicomponent theory of the memory trace. Psychol-
ogy of Learning and Motivation ,1, 229 –325. https://doi.org/10.1016/
S0079-7421(08)60515-0
Brodie, D. A., & Murdock, B. B. (1977). Effect of presentation time on
nominal and functional serial-position curves of free recall. Journal of
Verbal Learning and Verbal Behavior ,16(2), 185 –200. https://doi.org/10
.1016/s0022-5371(77)80046-7
Caplan, J. B., Legge, E. L., Cheng, B., & Madan, C. R. (2019). Effectiveness
of the method of loci is only minimally related to factors that should
in/uniFB02uence imagined navigation. Quarterly Journal of Experimental Psy-
chology ,72(10), 2541 –2553. https://doi.org/10.1177/1747021819858041
Cofer, C. N., Bruce, D. R., & Reicher, G. M. (1966). Clustering in free recall
as a function of certain methodological variations. Journal of Experimen-
tal Psychology ,71(6), 858 –866. https://doi.org/10.1037/h0023217
Dalezman, J. J. (1976). Effects of output order on immediate, delayed, and
/uniFB01nal recall performance. Journal of Experimental Psychology: Human
Learning and Memory ,2(5), 597 –608. https://doi.org/10.1037/0278-7393
.2.5.597
Davelaar, E. J. (2013). A novelty-induced change in episodic (NICE) context
account of primacy effects in free recall. Psychology ,4(9), 695 –703.
https://doi.org/10.4236/psych.2013.49099Ebbinghaus, H. (1913). Memory: A contribution to experimental psychology
(H. A. Ruger & C. E. Bussenius, Trans.). Teachers College, Columbia
University. (Original work published 1885).
Estes, W. K. (1955). Statistical theory of spontaneous recovery and regression.
Psychological Review ,62(3), 145 –154. https://doi.org/10.1037/h0048509
Friendly, M., Franklin, P. E., Hoffman, D., & Rubin, D. C. (1982). The
Toronto word pool: Norms for imagery, concreteness, orthographic vari-
ables, and grammatical usage for 1,080 words. Behavior Research Methods
& Instrumentation ,14(4), 375 –399. https://doi.org/10.3758/BF03203275
Grif/uniFB01ths, T., Canini, K., Sanborn, A., & Navarro, D. (2007). Unifying
rational models of categorization via the hierarchical dirichlet process .
Cognitive Science Society.
Healey, M. K., & Kahana, M. J. (2016). A four-component model of age-
related memory change. Psychological Review ,123(1), 23 –69.https://
doi.org/10.1037/rev0000015
Healey, M. K., Long, N. M., & Kahana, M. J. (2018). Contiguity in episodic
memory. Psychonomic Bulletin & Review ,26(3), 699 –720.https://doi.org/
10.3758/s13423-018-1537-3
Howard, M. W., & Kahana, M. J. (1999). Contextual variability and serial
position effects in free recall. Journal of Experimental Psychology:
Learning, Memory, and Cognition ,25(4), 923 –941. https://doi.org/10
.1037/0278-7393.25.4.923
Howard, M. W., & Kahana, M. J. (2002a). A distributed representation of
temporal context. Journal of Mathematical Psychology ,46(3), 269 –299.
https://doi.org/10.1006/jmps.2001.1388
Howard, M. W., & Kahana, M. J. (2002b). When does semantic similarity
help episodic retrieval? Journal of Memory and Language ,46(1), 85 –98.
https://doi.org/10.1006/jmla.2001.2798
Howes, A., Lewis, R. L., & Vera, A. (2009). Rational adaptation under task and
processing constraints: Implications for testing theories of cognition and action.
Psychological review ,116(4), 717 –751.https://doi.org/10.1037/a0017187
Hunt, R. R., & Lamb, C. A. (2001). What causes the isolation effect? Journal
of Experimental Psychology: Learning, Memory, and Cognition ,27(6),
1359 –1366. https://doi.org/10.1037/0278-7393.27.6.1359
Kahana, M. J. (1996). Associative retrieval processes in free recall. Memory
& Cognition ,24(1), 103 –109. https://doi.org/10.3758/bf03197276
Kahana, M. J., Howard, M. H., Zaromb, F., & Wing /uniFB01eld, A. (2002). Age
dissociates recency and lag recency effects in free recall. Journal of
Experimental Psychology: Learning, Memory, and Cognition ,28(3),
530–540. https://doi.org/10.1037//0278-7393.28.3.530
Kragel, J. E., Morton, N. W., & Polyn, S. M. (2015). Neural activity in the medial
temporal lobe reveals the /uniFB01delity of mental time travel. Journal of Neurosci-
ence,35(7), 2914 –2926. https://doi.org/10.1523/JNEUROSCI.3378-14.2015
Laming, D. (1999). Testing the idea of distinct storage mechanisms in
memory. International Journal of Psychology ,34(5–6), 419 –426.
https://doi.org/10.1080/002075999399774
Lieder, F., & Grif /uniFB01ths, T. L. (2020). Resource-rational analysis: Understand-
ing human cognition as the optimal use of limited computational re-
sources. Behavioral and Brain Sciences ,43, Article e1. https://doi.org/10
.1017/S0140525X1900061X
Lohnas, L. J., & Kahana, M. J. (2014). Compound cuing in free recall.
Journal of Experimental Psychology: Learning, Memory, and Cognition ,
40(1), 12 –24.https://doi.org/10.1037/a0033698
Lohnas, L. J., Polyn, S. M., & Kahana, M. J. (2015). Expanding the scope of
memory search: Modeling intralist and interlist effects in free recall.
Psychological Review ,122(2), 337 –363. https://doi.org/10.1037/a0039036
Marr, D. (2010). Vision: A computational investigation into the human
representation and processing of visual information . MIT Press.
Medin, D. L., & Schaffer, M. M. (1978). Context theory of classi /uniFB01cation
learning. Psychological Review ,85(3), 207 –238. https://doi.org/10.1037/
0033-295X.85.3.207
Miller, J. F., Weidemann, C. T., & Kahana, M. J. (2012). Recall termination
in free recall. Memory & Cognition ,40(4), 540 –550. https://doi.org/10
.3758/s13421-011-0178-9This document is copyrighted by the American Psychological Association or one of its allied publishers.
This article is intended solely for the personal use of the individual user and is not to be disseminated broadly.OPTIMAL FREE RECALL 17
Mockus, J., Tiesis, V., & Zilinskas, A. (1978). The application of Bayesian
methods for seeking the extremum. In L. C. W. Dixon & G. P. Szegö (Eds.),
Towards global optimization (Vol. 2, pp. 117 –129). North-Holland.
Murdock, B. B. (1960). The distinctiveness of stimuli. Psychological
Review ,67(1), 16 –31.https://doi.org/10.1037/h0042382
Murdock, B. B. (1962). The serial position effect of free recall. Journal of
Experimental Psychology ,64(5), 482 –488. https://doi.org/10.1037/h00
45106
Murdock, B. B. (1997). Context and mediators in a theory of distributed
associative memory (TODAM2). Psychological Review ,104(4), 839 –862.
https://doi.org/10.1037/0033-295X.104.4.839
Murdock, B. B., & Metcalfe, J. (1978). Controlled rehearsal in single-
trial free recall. Journal of Verbal Learning and Verbal Behavior ,17(3),
309–324. https://doi.org/10.1016/S0022-5371(78)90201-3
Neath, I. (1993). Distinctiveness and serial position effects in recognition.
Memory & Cognition ,21(5), 689 –698.https://doi.org/10.3758/BF03197199
Neath, I., & Crowder, R. G. (1996). Distinctiveness and very short-term
serial position effects. Memory ,4(3), 225 –242. https://doi.org/10.1080/
09658211.1996.9753032
Nosofsky, R. M. (1984). Choice, similarity, and the context theory of classi /uniFB01-
cation. Journal of Experimental Psychology: Learning, Memory, and Cogni-
tion,10(1), 104 –114. https://doi.org/10.1037//0278-7393.10.1.104
Polyn, S. M., Norman, K. A., & Kahana, M. J. (2009a). A context
maintenance and retrieval model of organizational processes in free
recall. Psychological Review ,116(1), 129 –156. https://doi.org/10
.1037/a0014420
Polyn, S. M., Norman, K. A., & Kahana, M. J. (2009b). Task context and
organization in free recall. Neuropsychologia ,47(11), 2158 –2163. https://
doi.org/10.1016/j.neuropsychologia.2009.02.013
Reed, S. K. (1972). Pattern recognition and categorization. Cognitive Psy-
chology ,3(3), 382 –407. https://doi.org/10.1016/0010-0285(72)90014-X
Roberts, W. A. (1972). Free recall of word lists varying in length and rate of
presentation: A test of total-time hypotheses. Journal of Experimental
Psychology ,92(3), 365 –372. https://doi.org/10.1037/h0032278
Rouhani, N., Norman, K. A., Niv, Y., & Bornstein, A. M. (2020). Reward
prediction errors create event boundaries in memory. Cognition ,203,
Article 104269. https://doi.org/10.1016/j.cognition.2020.104269
Rundus, D. (1971). Analysis of rehearsal processes in free recall. Journal
of Experimental Psychology ,89(1), 63 –77.https://doi.org/10.1037/
h0031185
Rundus, D., & Atkinson, R. C. (1970). Rehearsal processes in free recall: A
procedure for direct observation. Journal of Verbal Learning and VerbalBehavior ,9(1), 99 –105. https://doi.org/10.1016/s0022-5371(70)
80015-9
Sahakyan, L., & Kelley, C. M. (2002). A contextual change account of the
directed forgetting effect. Journal of Experimental Psychology: Learning,
Memory, and Cognition ,28(6), 1064 –1072. https://doi.org/10.1037//
0278-7393.28.6.1064
Sederberg, P. B., Howard, M. W., & Kahana, M. J. (2008). A context-based
theory of recency and contiguity in free recall. Psychological Review ,
115(4), 893 –912. https://doi.org/10.1037/a0013396
Sederberg, P. B., Miller, J. F., Howard, M. W., & Kahana, M. J. (2010). The
temporal contiguity effect predicts episodic memory performance. Mem-
ory & Cognition ,38(6), 689 –699. https://doi.org/10.3758/mc.38.6.689
Snoek, J., Larochelle, H., & Adams, R. P. (2012). Practical Bayesian
optimization of machine learning algorithms. In F. Pereira, C. J. C. Burges,
L. Bottou, & K. Q. Weinberger (Eds.), Advances in neural information
processing systems (Vol. 25, (pp. 2951 –2959). Curran Associates. http://
papers.nips.cc/paper/4522-practical-bayesian-optimization-of-machine-
learning-algorithms.pdf
Standing, L. (1973). Learning 10000 pictures. Quarterly Journal of Experi-
mental Psychology ,25(2), 207 –222. https://doi.org/10.1080/1464074730
8400340
Tan, L., & Ward, G. (2000). A recency-based account of the primacy effect in
free recall. Journal of Experimental Psychology: Learning, Memory, and
Cognition ,26(6), 1589 –1625. https://doi.org/10.1037/0278-7393.26.6.1589
Tan, L., Ward, G., Paulauskaite, L., & Markou, M. (2016). Beginning at the
beginning: Recall order and the number of words to be recalled. Journal of
Experimental Psychology: Learning, Memory, and Cognition ,42(8),
1282 –1292. https://doi.org/10.1037/xlm0000234
Unsworth, N., Brewer, G. A., & Spillers, G. J. (2011). Inter-And intra-
individual variation in immediate free recall: An examination of serial
position functions and recall initiation strategies. Memory ,19(1), 67 –82.
Ward, G., & Tan, L. (2019). Control processes in short-term storage:
Retrieval strategies in immediate recall depend upon the number of words
to be recalled. Memory & Cognition ,47(4), 658 –682. https://doi.org/10
.3758/s13421-018-0891-8
Ward, G., Tan, L., & Grenfell-Essam, R. (2010). Examining the relation-
ship between free recall and immediate serial recall: The effects of list
length and output order. Journal of Experimental Psychology: Learning,
Memory, and Cognition ,36(5), 1207 –1241. https://doi.org/10.1037/
a0020122
Yates, F. A. (1966). Art of memory . University of Chicago Press.
Appendix
Additional CMR Simulations
The Effect of CMR Variants on the Optimal Policy
We explored how sensitive the optimal policy is to the speci /uniFB01c
variant of CMR that was used. For simplicity, the variant of CMR
we used only recalls from the current list and does not incorporate
intrusions from other lists in the experiment. It is possible that
intrusions could reduce the ef /uniFB01cacy of the chaining policy in the
optimal strategy (by “breaking the chain ”). To capture intrusion
errors, we repeated the analysis on CMR2, where memories accu-
mulate both within and across lists ( Lohnas et al., 2015 ), enabling
the model to account for recall of both current list items ( “correct
recalls ”) and prior list items ( “intrusions ”). In addition to intrusions,
we also explored the effect of an alternative primacy mechanism:
The attention-based account instantiated in TCM and CMR ( Lohnas
et al., 2015 ;Polyn et al., 2009b ), which posits that there is increasedattention during recall to the early list items, thereby increasing the
encoding strengths of these items. This will provide a constant pull
to the beginning of the list even after recall initiation, which can
potentially affect the success of the chaining strategy.
To incorporate these effects, we /uniFB01tC M R 2t ot h es a m ed a t as e t
(Kahana et al., 2002 ); CMR2 successfully captures the amount of
intrusions observed in empirical r ecalls (6.1% of total recalls in the
data and 6.3% in the model). As with CMR, CMR2 parameters can be
divided into two distinct categories :p a r a m e t e r st h a tc o r r e s p o n dt oc o r e
architectural constraints of the memory system and parameters that
specify the space of possible policies that can be pursued (given these
constraints) during encoding and r ecall. CMR2 adds some additional
parameters to the former ( “core architectural constraints ”)g r o u p :p a r a m -
eters that characterize the retrieval process ( κ=0.31, λ=0.13) andThis document is copyrighted by the American Psychological Association or one of its allied publishers.
This article is intended solely for the personal use of the individual user and is not to be disseminated broadly.
(Appendix continues )18 ZHANG, GRIFFITHS, AND NORMAN
parameters that prevent intrusions and recall repetitions ( cthresh=0.073,
ω=11.89, α=0.68), correspond to the core architectural constraints of
the memory system. Differing fro mo u rp r e v i o u sa n a l y s i s ,w ea l s o
considered the attention-based primacy mechanism to be part of the
constraints of the memory system ( ϕd=0.99, ϕs=1.81). The latter
(“policy specifying ”)g r o u po fp a r a m e t e r si st h es a m ef o rC M R 2a n d
CMR; the /uniFB01tv a l u e sf o rt h e s ep a r a m e t e r sw e r e βrec=0.63, βenc=0.52,
andγfc=0.43. Repeating the rational analysis as before, we /uniFB01xt h e /uniFB01rst
set of parameters to capture the arc hitectural constraints during the
retrieval process, and then optimize the parameter values in the second
set that describe the space of encoding and recall policies to achieve
maximum free recall performance.
We observed a similar optimal policy with large βenc=0.94, large
βrec=0.88, and small γfc=0.17 for CMR2 (which incorporates
intrusions and attention-based primacy), compared with βenc=0.99,
βrec=0.99, and γfc=0.001 for CMR. There are also some differences,
with βrec<1a n d γfc>0i nC M R 2 .A l l o w i n gi n t r u s i o n sd o e sn o t
qualitatively change the optimal policy, for two reasons: First, and most
importantly, intrusions are rare; second, using a βrecvalue that is <1
(but still large) implements a kind of “soft chaining ”that allows for the
chain to sometimes be continued even after an intrusion. Speci /uniFB01cally,
when βrec<1, the current context is not entirely dominated by the
retrieved context from the intrusion —there is still some context from
the item recalled before the intrusion, which can sometimes cue recall
of the correct next item (although, more often, the intrusion results in
stoppage of recall, mirroring experiment results from Miller et al.,
2012). We also found that there is no in /uniFB02uence on the optimal policy
from adding the attention-based primacy effect. Even though there is
stronger encoding of the early list items, we do not observe a constant
pull to the beginning of the list in recall patterns. The strengthening of
early list items is negligible compared with the memory support to
recall the next item in the chaining strategy.
The Effect of Alternative Assumptions for the
Cognitive Constraints on the Optimal Policy
To identify the optimal encoding and recall policy in the free
recall task, we need to /uniFB01x the parameters that characterize the
cognitive constraints during the retrieval process. To obtain a set
of reasonable values for these parameters, we /uniFB01t CMR to all
participants in an immediate free recall data set ( Kahana et al.,
2002 ). To explore the consequence that participants at high and low
levels of performance demonstrate different cognitive constraints to
start with, we /uniFB01t CMR separately to the top half of participants and
the bottom half of participants in Kahana et al. (2002) . The obtained
/uniFB01xed parameters are εd=3.01, k=5.47 and εd=2.11, k=4.41,
respectively. In other words, the top half of participants are less
likely to terminate recall and have a lower noise level during
retrieval compared with the bottom half of participants.
We repeated the same analysis to optimize recall performance on
the two groups of participants, given the differences in their cogni-
tive constraints. We observe that splitting all participants into two
performance groups when /uniFB01tting CMR, that is, assuming that better
performing participants and worse performing participants are
different in their cognitive constraints, does not change our main
conclusions about the optimal policy.
For the top half of participants, when we allow the model to decide
where to start recall, it is optimal to recall from the beginning-of-listcontext, with γfc=0.007, βenc=0.99, and βrec=0.98. When recalling
from the end-of-list context, under large βenc=0.6, the optimal
parameters for γfc=0.98 and βrec=0.98. For the bottom half of
participants, when we allow the model to decide where to start recall, it
is optimal to recall from the beginning-of-list context, with γfc=0.004,
βenc=1.00, and βrec=0.98. When recalling from the end-of-list
context, under large βenc=0.6, the optimal parameters are γfc=0.99
andβrec=1.00.
Is There Always Greater Consistency With the
Optimal Policy Among Higher Performing Participants?
Our /uniFB01nding that higher performing participants show greater
consistency with the optimal policy is not trivially true. A simple
counterexample to this would be if the participants all follow
random policies that demonstrate some variability, but none of
these policies are close to the optimal policy. To demonstrate
this, we simulated a population of participants who follow random
policies in the part of the parameter space far from the optimal policy
(for each simulated participant, their γfc,βenc,βrec, and the propor-
tion of primacy trials are uniformly sampled from 0.3 to 0.6) and
subjected them to the same analysis as Figure 6 . If we looked at the
high-performing participants in Figure A1 , they follow strategies
(lower primacy and lower forward symmetry) that differ from the
optimal policy (higher primacy and higher forward asymmetry).
These simulation results do not look similar to the patterns we
observe with human participants in Figure 6 .
Simulations of Qualitative Patterns in
Unsworth et al. (2011)
We ran a “proof-of-concept ”simulation to show that our model can
account for the qualitative patterns shown in Unsworth et al. (2011) .
InUnsworth et al. (2011) , participants were clustered into three
groups according to their serial position curves: a high-primacy
low-recency group, a low-primacy high-recency group, and a
high-primacy high-recency group, with the last group demonstrating
the highest memory performance. We simulate three groups of 50
participants. The three groups have a similar proportion of trials in
which they start recalling from the beginning versus from the end of a
list (p=.5), but they differ in how optimal their recall transitions are.
The best performing group is able to optimally adjust the amount of
forward asymmetry given the recall initiation on a given trial ( γfc=0
when recall is initiated from the start of the list and γfc=1 when recall
is initiated from the end of the list). In contrast, the high-primacy low-
recency group fails to increase forward asymmetry during the primacy
trials ( γfc=1 in all cases), and the low-primacy high-recency group
fails to increase backward transitions during the recency trials ( γfc=0
in all cases). Figure A2A plots the proportion correct for all items,
primacy items (the /uniFB01rst three items only) and recency items (the last
three items only) for each group; these results closely resemble the
qualitative patterns from Unsworth et al. (2011) , as shown in Figure
A2B: The “optimal policy ”group has the best overall performance
and demonstrates a high level of performance for both primacy and
recency items compared with other groups, whereas the other two
groups demonstrate high levels of performance for primacy or
recency items only (but not both) and show lower levels of perfor-
mance overall.This document is copyrighted by the American Psychological Association or one of its allied publishers.
This article is intended solely for the personal use of the individual user and is not to be disseminated broadly.
(Appendix continues )OPTIMAL FREE RECALL 19
Parameter Estimation in CMR and Rational-CMR
CMR
There are three sets of behavioral patterns and a total of N=
2l+8d a t ap o i n t sw ea r ei n t e r e s t ed in modeling: the serial
position curve ( S={s1,s1,:::,sl}), the probability of /uniFB01rst
recall curve ( R={r1,r1,:::,rl}), and the cond itional responseprobability curve with −4/+4l a g( C={c1,c1,:::,c8}).lis the
list length of the free recall data. The goal in /uniFB01tting CMR to a
human behavioral data set is to obtain a set of parameter values
θthat minimize the difference between the simulated values s,
r,a n d cas a function of θ,a n de m p i r i c a lv a l u e s s/uni02B9,f/uni02B9,a n d c/uni02B9of
these data points, expressed in the normalized root-mean-
square error (NRMSE):This document is copyrighted by the American Psychological Association or one of its allied publishers.
This article is intended solely for the personal use of the individual user and is not to be disseminated broadly.Figure A2
Proportion Correct for All Items, Primacy Items (The First Three Items Only) and Recency
Items (the Last Three Items Only) for Each Group (n =50), Comparing Current Simulations
(A) and Unsworth et al. (2011) (B)
Note. For our simulations, “Hi Primacy/Hi Recency ”represents the high-primacy high-recency group
that changes γfcbased on where recall starts; “Hi Primacy/Lo Recency ”represents the high-primacy low-
recency group that fails to have optimal transitions when recall starts from the beginning ( /uniFB01xedγfc=1); and
“Hi Recency/Lo Primacy ”represents the low-primacy high-recency group that fails to have optimal
transitions when recall starts from the end ( /uniFB01xedγfc=0). The /uniFB01xed parameters in the simulations are βenc=
0.6,βrec=1.0,εd=4.60, and k=5.09. For the data shown in B, group labels (e.g., “Hi Primacy/Hi
Recency ”) represent the groups identi /uniFB01ed in the clustering analysis conducted by Unsworth et al. (2011) .
Error bars represent one standard error of the mean. Values in B were estimated from Figure 2 and Table 1
inUnsworth et al. (2011) . See the online article for the color version of this /uniFB01gure.Figure A1
Behavioral Patterns for the Better Performing and Worse Performing Simulated Participants
Note. One hundred seventy-one participants are simulated following random policies that are inconsistent with the optimal policy, with their γfc,βenc,βrec,
and the proportion of primacy trials uniformly sampled from 0.3 to 0.6. “Top 10% ”refers to simulated participants whose performance is in the top 10th
percentile of the population. From left to right of the three columns are the serial position curve, probability of the /uniFB01rst recall, and conditional response
probability. The /uniFB01xed parameters in the simulations are εd=4.60 and k=5.09. See the online article for the color version of this /uniFB01gure.
(Appendix continues )20 ZHANG, GRIFFITHS, AND NORMAN
fðxÞ=ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃPl
i=1ðsi−s′iÞ2
lq
s′max−s′min+ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃPl
i=1ðri−r′iÞ2
lq
r′max−r′min+ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃP8
i=1ðci−c′iÞ2
lq
c′max−c′min, (A1)
where, the normalization in each component rescales each set of
behavioral patterns so that they are comparable across each other.
Given some domain of the parameter space Θ, the goal is to /uniFB01nd the
set of parameters θthat minimizes the error function f(θ):
θ/C3=arg min θ∈ΘfðθÞ: (A2)
The exact functional form for fis not available, but rather farises
as a complex function of the free recall behavior simulated based on
the mechanisms speci /uniFB01ed in CMR. Applying a grid search to the
parameter space would not be feasible since it requires a total of m10
evaluations, assuming there are mpossible values for each parame-
ter. Each evaluation is expensive, as it requires simulating a new
data set and calculating behavioral patterns s,r, and cfrom it.
We apply Bayesian optimization to estimate the parameters in
CMR ( Mockus et al., 1978 ), as it has recently become popular for
training expensive machine-learning models whose behavior de-
pends in a complicated way on their parameters, such as convolu-
tional neural networks ( Snoek et al., 2012 ). Bayesian optimization
models fas a Gaussian process, with a prior p(f)=GP(f;/uni03BC,K).
Given existing evaluations of finD=(Θ,f), the posterior of fcan be
written as p(f|D)=GP(f;/uni03BCf|D,Kf|D). Then, a new set of parameters
θnewto evaluate can be proposed by estimating how desirable
evaluating fatθnewis expected to be for the minimization problem,
measured as the probability of improvement function E(u(θ)|θ,D),
where u(θ)=1 iff(θ)≤minfor 0 if f(θ)>minf.
Parameters values reported in the current work are based on 200
iterations of Bayesian optimization, after 500 random initializations,
which sums up to a total of 700 evaluations of f.
Rational-CMR
Instead of /uniFB01tting the model to minimize the discrepancy with the
human behavioral data, the goal in rational-CMR is to /uniFB01nd the set of
parameter values θthat maximize task performance f:
θ/C3=arg max θ∈ΘfðθÞ: (A3)
where, task performance fis a function of θand can be obtained from
the simulated data set as the average proportion of items correctly
recalled. θis then obtained using the same Bayesian optimization
procedure that was described in the previous section.
Kahana et al. (2002) Data Set
Fitting of CMR and rational-CMR is based on free recall data
from 31 university undergraduates (age range: 17 –21) in Experi-
ment 1 by Kahana et al. (2002) . Participants studied 33 lists of words
in an immediate free recall experiment (the /uniFB01rst three lists were
practiced and were not included in the analyses). Lists were
composed of 10 common, two-syllable nouns chosen at random
and without replacement from the Toronto Noun Pool ( Friendly et
al., 1982 ). At the start of each trial, a /uniFB01xation cross appeared for
1,400 ms, followed by a 100 ms blank interstimulus interval. Then,
the computer screen displayed each list item for 1,400 ms, followed
by a 100 ms blank interstimulus interval. During list presentation,participants were instructed to say each word aloud. Immediately
following the list, participants were given 30 s to recall list items. It
was made clear that they need not attempt to recall the items in the
order of presentation. Each session lasted around 1.5 hr.
Penn Electrophysiology of Encoding and
Retrieval Study
We tested the predictions from rational-CMR using the PEERS, a
large database characterizing the electrophysiological correlates of
memory encoding and retrieval; see full details of the experiment in
Lohnas et al. (2015) ,Healey and Kahana (2016) andHealey et al.
(2018) . The present analysis is based on the behavioral data from the
171 young participants (age range: 18 –30) who completed Experi-
ment 1 of the PEERS data set. Participants performed an immediate
free recall experiment. Each participant completed one practice
session and six subsequent experimental sessions, each with 16
lists. Only the experimental sessions are included in the analysis.
Each study list was composed of 16 words and was followed by an
immediate free recall test. Words were either presented concurrently
with a task cue (size judgment or animacy judgment), or with no
encoding task. There were three conditions of study lists: no-task
lists, single-task lists, and task-shift lists. List and task order were
counterbalanced across sessions and participants. Each item was
presented on the screen for 3,000 ms, followed by a jittered
interstimulus interval uniformly drawn from 800 to 1,200 ms. If
the word was associated with an encoding task, participants indi-
cated their response via a keypress. Once the presentation of the last
item was completed, after a jittered delay of 1,200 –1,400 ms, the
participant was given 75 s to recall any items from the just-presented
list. Some sessions were randomly selected for /uniFB01nal free recall task
and a recognition task after the immediate free recall task. Only the
immediate free recall task was considered in the analysis.
Howard and Kahana (1999) Data Set
We tested the predictions from rational-CMR using the data set
from Howard and Kahana (1999) . Sixty-one participants studied
lists of words for a subsequent free recall test. Lists were composed
of 12 common, two-syllable nouns chosen at random and without
replacement from the Toronto Noun Pool ( Friendly et al., 1982 ), and
they were presented visually at a rate of 1 word per second. During
list presentation, participants were required to perform a semantic
orienting task on the presented words. The participants were to press
the left control key if they judged the word to be concrete and the
right control key if they judged it to be abstract. In an immediate
condition, the free recall test was given immediately after the list
presentation. In a delayed condition, participants performed an
arithmetic distractor task for 10 s before recall. Participants were
given 45 s to recall all items in the list in any order. Participants
could not predict during encoding whether it would be an immediate
or delayed condition; they were warned that sometimes there would
be a math test between the end of the list and the signal to recall and
sometimes there would not.
Received April 15, 2021
Revision received February 24, 2022
Accepted May 7, 2022 ▪This document is copyrighted by the American Psychological Association or one of its allied publishers.
This article is intended solely for the personal use of the individual user and is not to be disseminated broadly.OPTIMAL FREE RECALL 21