Is perceptual acuity asymmetric in isolated word recognition? Evidence from an
ideal-observer reverse-engineering approach
Nathaniel J. Smith and Wen-Hsuan Channjsmith@cogsci.ucsd.edu, whchan@ucsd.edu
University of California at San Diego, Department of Cognitive Science
9500 Gilman Drive #515, La Jolla, CA 92093-0515 USA

Roger Levy
rlevy@ling.ucsd.edu
University of California at San Diego, Department of Linguistics
9500 Gilman Drive #108, La Jolla, CA 92093-0108 USA
Abstract
An asymmetrical optimal viewing position (OVP) effect in isolated word recognition has been well documented, such thatrecognition speed and accuracy are highest when the pointof fixation within the word is slightly to the left of center.
However, there remains disagreement as to the source of theasymmetry in the OVP effect. One leading explanation is thatperceptual acuity in isolated word recognition is asymmetric,
falling off more rapidly to the left than to the right. An alternative explanation is that of lexical constraint: perceptual acuity
may be symmetric, but that the distributional statistics of thelexicon are such that the letters near the beginning of a word
are on average of greater value in discriminating word identity than the letters near the end. On both these accounts, aleft-of-center fixation point optimizes the efficient accrual ofperceptual input from the word string, but for different reasons. These accounts have been difficult to tease apart experimentally due to the ubiquitous potential influence of lexicalconstraint. Here we take a novel approach, constructing an
ideal-observer model of isolated word recognition which takes
into account word frequency information and thus intrinsicallyaccounts for the role of lexical constraint. Within this model,
the shape of the perceptual acuity curve is governed by freeparameters that can be estimated from purely behavioral response data from word recognition experiments. Fitting ourmodel to the experimental data of Stevens & Grainger (2003),
we find that the asymmetric version, in which perceptual acuity
can differ to the left and to the right, fits human behavioral responses significantly better than symmetric versions in which
the perceptual acuity curve is constrained to be the same tothe left and to the right. Furthermore, in both parametric andnonparametric versions of the asymmetric model, perceptual
acuity falls off more rapidly to the left than to the right. These
results support the position that the perceptual acuity curve inisolated word recognition is indeed asymmetric.
Keywords: Psychology, Cognitive Science, Perception, Language Understanding, Decision Making, Bayesian modeling

Introduction
Literate native speakers are exquisitely adapted to the visualand linguistic processing of written text in their language.
The naturalistic task underlying most of this adaptation isreading (Rayner, 1998). In the study of eye movements inreading, one of the most striking examples of this adaptationin the last several decades has been discovery of asymmetryof the perceptual span: in languages which are written fromleft to right, readers are more sensitive to material to the rightof the center of fixation on the page than they are to materialon the left. In languages which are written from right to left,

however, this sensitivity is reversed (Rayner, Well & Pollatsek, 1980). Since visual acuity per se is not itself asymmetric(as evidenced by experimental work on perception of nonlinguistic visual inputs), the most intuitive interpretation ofthis finding is that, in ordinary progressive reading, because
readers of languages written left to right have already seen
what lies to the left of their eyes, they differentially attend towhat lies to the right (and vice versa for languages written
right to left).
However, the discovery by O’Regan, Lévy-Schoen, A.,
Pynte, J. & Brugaillére (1984) of the optimal viewing position (OVP) in isolated word recognition makes the picture
more complex. The OVP in isolated word recognition can be
succinctly described as follows: word recognition is fastest
and most accurate when the initial fixation point of the eyes
is slightly to the left of the center of the word (Figure 1a). The
discovery of the OVP launched considerable discussion as toits nature and implications, since it cannot be obviously accounted for by an asymmetry in perceptual acuity that would
be adaptive for the task.
At present, there are two leading explanations that havebeen proposed for the OVP in isolated word recognition. One
is that the asymmetry of perceptual acuity to the left and to
the right within reading may affect all processing of visual
linguistic input. If acuity drops off more rapidly to the leftof the fixation point than to the right, then the best strategy
to recognize a word would be to fixate at a left-of-center
location, maximizing average acuity across the word as awhole. Evidence supporting this position has been adduced
by Nazir, O’Regan & Jacobs (1991) and Nazir, Heller & Sussman (1992), who demonstrated left-right acuity differences
in tasks involving the detection of a target letter at a variable position within a masking letter string (e.g., kkkkkykk).
They found that the drop-off in performance was a monotonic
function of visual eccentricity, and the left visual field showed
steeper drop-off than the right visual field (Figure 1b).
The other leading explanation was put forth by Clark &O’Regan (1999), who argued that a better way to understandthe contributions of these different mechanisms may lie in the
distributional statistics of the written lexicon itself. They investigated the contributions of orthographic constraints, con-(a) OVP effect (Stevens & Grainger, 2003)(b) Asymmetric perceptual acuity in letter-inmask recognition (Nazir et al., 2003)(c) Asymmetry of lexical constraint (Clark& O’Regan, 1999)

Figure 1: The OVP effect in isolated word recognition and possible explanations for it.
structing a simple measure of residual lexical ambiguity thatwould hold assuming only that two letters near the fixation
point and at the end of the word are known. For a range ofword lengths in both French and English, this ambiguity measure is minimized just to the left of the word’s center, capturing the OVP effect through lexical constraint without resorting to an asymmetric perceptual acuity curve (Figure 1c).
It is difficult to adjudicate between these two possible explanations through purely experimental means, because wellestablished effects on word recognition such as those of word
frequency and neighborhood density make it is fairly clearthat lexical constraint plays a ubiquitous role in the process
but varies slightly for every word in the lexicon, making it difficult to design a word-recognition experiment that controls
for lexical constraint while testing perceptual acuity. Conversely, our understanding of precisely how the perceptual
acuity curve affects word recognition remains limited, making it difficult to hold it constant and test only lexical constraint as a possible source of OVP effects. In this paper, wetake an alternative approach, constructing an ideal-observer
model (Marr, 1982; Anderson, 1990) of isolated word recognition in which lexical constraints are assumed to be available. The perceptual acuity curve in this model is determinedby a set of parameters which are free and can be fit to behavioral data using well-established techniques of statistical inference. Our ideal-observer model thus allows us to “reverseengineer” the perceptual acuity curve active in isolated word
recognition on the basis of a lexicon (with word frequencies)
and a behavioral data set, and assess whether and how thereverse-engineered acuity curve may be asymmetric.

Data
The dataset used in our study is from Experiment Two of
Stevens & Grainger (2003). In this experiment, words were
presented for 50ms each at various positions relative to thecenter of the fixation. After presentation of each word, theparticipant was asked to type the presented word back into
the computer.. The dataset contains 75 five-letter words and105 seven-letter words and the human performance (correctness of the response) for each word from seventy subjects, fora total of 12,600 observations balanced across word identityand fixation position. We obtained a word frequency databasefrom the Agence France Presse (AFP) French Corpus, whichcontains French journal articles from 1993-1996. Since thehuman dataset has both five-letter and seven-letter Frenchwords, we extracted frequency counts for all five-letter andseven-letter words in the corpus. The resulting lexicon contains 14,379 five-letter French words and 21,569 seven-letter
French words.

Model
The Bayesian Reader
We use an ideal-observer model of isolated word recognition,
the Bayesian Reader (Norris, 2006, 2009). Our version of theBayesian Reader introduces several key assumptions regarding the nature of word recognition which determine the formof the probabilistic model:• Word recognition is a Bayesian hypothesis test in whichprior expectations regarding what word is likely to be presented are combined with perceptual evidence d to determine posterior beliefs about what word w is being seen:1

P(w|d) =P(d|w)P(w);P(d)• The prior probability of each word P(w) is proportional toits corpus frequency of occurrence;• Perceptual evidence consists of a sequence of independent
identically distributed (i.i.d.) input samples d (1) , . . . , d (N)drawn from a NOISE DISTRIBUTION P(d|w), where samples accrue at a constant rate over time;• If we denote the letters of a word w of length L as
w1 , . . . , wL , then an input sample d can be decomposed into
L samples d1 , . . . , dL , with di conditionally independent ofd j given wi and w j for all i 6= j, so that
1 Tasks in which the decision to be made is something other than
the identity of the word—e.g., a lexical decision about whether the
input string is a word in the participant’s language—can be formulated as Bayesian hypothesis tests among the possible choices(Norris, 2006; del Prado Martı̀n, 2008).

L

P(d|w) = ∏ P(di |wi ).
i=1

We will refer to the term P(di |wi ) as the noise distribution
for letter wi .
Thus far, this version of the Bayesian Reader is simpler andmore general than that introduced by Norris (2006), who assumed (a) a specific representation of each sample d as a point
in a high-dimensional space; (b) a multivariate Gaussian form
for the noise distribution P(d|w); and (c) a specific estimate of
the noise variance used by the ideal observer in computing theposterior distribution over words given perceptual evidence.

P(r|w∗ , k) = Ed|k,w∗ [P(r|d, k)]

Adaptation to modeling visual acuity curves
In order to adapt the Bayesian Reader to the task of estimating
visual acuity curves, however, we need to introduce a dependence of the noise distribution for each letter on its physicalpositioning. In particular, we assume that the noise distribution for each letter is dependent on its eccentricity as measured in number of characters from the point of fixation, withnegative values corresponding to left-of-fixation and positive
values to right-of-fixation; and on its proximities vL and vR tothe left and right edges of the word, also measured in characters. If we define these physical positioning characteristics
of a letter wi as ki ≡ hei , vLi , vRi i, then its position-contingent
noise distribution can be denoted as P(di |wi , ki ).
For the empirical modeling studies presented here, we
make the additional assumption that the value of
Edi |w∗i ,ki [P(di |wi , ki )] ,was the participant’s guess as to which word they saw. Our
goal is to use these behavioral responses to learn the dependence of the visual acuity of a letter—as quantified by confusability in (1)—on its physical positioning.
We model the naming task using the assumptions outlinedin the previous two sections. In general, the experimental participant must choose their response r through some possibly
stochastic decision process based on their posterior beliefs
P(w|d, k) about what word they saw. We further assume thatthe participant makes their choice of response through probability matching, so that the probability of any response r given
the word w∗ actually being presented is given by its expected
posterior probability:(1)

where Edi |w∗i ,ki denotes expectation under the conditional distribution P(di |w∗i , ki ) for the true letter w∗i being presented,
depends only on k and on whether wi = w∗i . This assumption can be interpreted as stating that every letter is equallyconfusable with all letters other than itself; the quantity in (1)for wi 6= w∗i can be interpreted as the level of confusabilityof a letter as a function of its physical positioning. This assumption is not necessary within the overall framework, andindeed could be relaxed in order to incorporate letter confusability matrices (Engel, Dougherty & Jones, 1973; Geyer,1977) into the model and even to learn them directly from behavioral word-recognition data. In the present studies, however, this assumption greatly simplifies and facilitates boththe statistical learning problem and its computational implementation.

Learning visual acuity from word identification data
Recall that in their word-identification study, Stevens &Grainger (2003) presented experimental participants with
five- and seven-letter words one at a time for a brief, fixedinterval too short to permit refixation, with fixation position
varying across trials. The behavioral response r in each trial(2)

where Ed|k,w∗ represents the expectation marginalizing overpossible perceptual input samples given the true word and its
physical positioning. For notational simplicity, we omit thesubscript on the expectation whenever it is clear from context.
Equation (2) can be rewritten using Bayes’ rule asP(r|w∗ , k) = P(r)E
P(d|r, k).
P(d|k)(3)

Equation (3) is the expectation of a ratio of random variables
E[Y /X], an expression which cannot in general be manipulated exactly. Using the method of propagation of error, however, a second-order approximation for the expectation of aratio can be found (Rice, 1995):

 


Y
E[Y ]
1
E[Y ]
E
≈
+
Var[X]− Cov[X,Y ] .
X
E[X] E[X]2
E[X]
We now turn our attention to the rightmost part of thisexpression, the covariance between the numerator and thedenominator—in Equation (3), these terms are P(d|r, k) andP(d|k) respectively. Insofar as any individual word plays
only a small part in the calculation of the marginal probability P(d|k), we would expect the covariance of this marginal
probability with P(d|r, k) to be small (with the importantcaveat that because words tend to look more like each otherthan like non-words, there will generally be some positive covariance, and its magnitude may depend on the orthographically typicality of w∗ and r). Dropping the covariance fromthe above approximation allows us to approximate our posterior probability as

P(r|w∗ , k) ≈ P(r)E [P(d|r, k)]
E [P(d|k)] + Var[P(d|k)]3
E[P(d|k)].

Ignoring the denominator (which is constant with respect tor) and decomposing the perceptual input d into its component
independent samples at each time j and letter position i, we
obtainhi( j)P(r|w∗ , k) ∝ P(r) ∏ E ( j) ∗ P(di |ri , ki ) .
i, j

di |ki ,wiWe take advantage of the identical distribution of the N samples to obtain the approximate unnormalized probability
N
P(r|w∗ , k) ∝ P(r) ∏ Edi |ki ,w∗i [P(di |ri , ki )] .
i

We are now ready to take advantage of our assumption
from the previous section that each letter is equally confusable with all letters otherthan itself—thatis, the value of eachNof the above terms Edi |ki ,w∗i [P(di |ri , ki )] depends only on
ki and on whether wi = w∗i . For each ki , let us denote the value
taken when wi = w∗i as pi , the value taken when wi 6= w∗i as
qi , and the ratio qpii as li . Substituting these terms in and dividing the entire expression by q1 . . . qL gives us our final approximate expression for the probability of the participant’s
response:
P(r|w∗ , k) ∝ P(r)

∏ li

(4)

i:wi =w∗iwhere li is dependent on the physical positioning of the letter in question. On the original assumption of the Bayesian
Reader that the number of input samples N accumulates at aconstant rate over time, the value log li can be interpreted as
the average rate at which perceptual information accrues at
position i.

Model parameterization and estimation
Within the context of our model, the goal of inferring a visual acuity curve from behavioral word-recognition data entails estimating these input-accrual rate parameters log li . In
the studies presented here, we assume as stated before thatthe log li for each letter is a function of three properties of its
physical position: its eccentricity from the fixation point, its
proximity from the left edge of the word, and its proximityfrom the right edge of the word. All measurements are madein characters. We consider two functional forms for the eccentricity parameters: a PARAMETRIC form in which the values of log li is assumed to follow a Gaussian curve centered at
the fixation point with maximum value α and standard deviation σ; and a NONPARAMETRIC form in which each eccentricity has its own arbitrary parameter value. For each form,
we consider a SYMMETRIC version in which the eccentricity parameters log li are determined by the absolute eccentricity, and an ASYMMETRIC version in which the parameters for negative and positive eccentricity values of the same
magnitude can be different (in the parametric Gaussian case,the asymmetric model allows different standard deviations σLand σR to the left and the right of the fixation point). Additionally, all models include one left-edge and one right-edge“bonus” parameter, bL and bR , added to the eccentricity parameters to determine the input-accrual rate parameters log li
for the first and last letters of a word respectively. That is, if
the fixation position is on the f -th character of the word andthe function e(·) maps eccentricities to values in log li space,
we havelog l1 = e(1 − f ) + bL

(first letter)log li = e(i − f )log lL = e(L − f ) + bR

(middle letters; i ∈/ {1, L})(last letter)

In all cases, we fit the parameters of our models using maximum likelihood estimation. Fortunately, the gradient of ourmodel is readily calculable and allows estimation using standard gradient-descent techniques.

Results
For each of our nonparametric (−PAR) and parametric (+ PAR)models, we fit a symmetric (+ SYM) and an asymmetric(−SIM) variety to the 12,600-observation dataset of Stevens& Grainger (2003, Experiment 2) estimation. This dataset
contains presentations of both five-letter and seven-letterwords, with every letter of each word serving as the fixation point for an equal number of trials. For each model, we
used a single set of eccentricity and edge-bonus parametersparameters to cover all trials, giving us four parameters in thesymmetric parametric case (one maximum acuity parameter,
one standard deviation, and two edge bonuses), five in theasymmetric parametric case (two standard deviations insteadof one), nine in the symmetric nonparametric case (seven eccentricity parameters and two edge bonuses), and fifteen in
the asymmetric nonparametric case (thirteen eccentricity parameters instead of seven). These models are nested in theclassical statistical sense as follows:[−PAR ,−SYM ]≺{[−PAR ,+ SYM ],[+ PAR ,−SYM ]}≺[+ PAR ,+ SYM ]Since we use maximum likelihood estimation with far more
observations than parameters, we can use likelihood-ratio
tests for pairwise comparisons of all models except between [−PAR ,+ SYM] and [+ PAR ,−SYM]. These tests indicate that asymmetric models explain participant response
behavior far better than symmetric models in both parametric (χ2 (1) = 250, p  0.001) and non-parametric cases(χ2 (6) = 345.7800, p  0.001). Among asymmetric model
variants, the nonparametric model explains participant response behavior significantly better than the Gaussian model(χ2 (10) = 564.7, p  0.001).
For the asymmetric nonparametric model, we estimatedstandard deviations for our parameter estimates using 100
bootstrap replicates. Figure 2 graphs the value of log l as afunction of eccentricity, together with edge-bonus parameter

●

●

3

3
●

●

1

●

2

●

log l

●

●

●
●

●

2

log l

●

●

●

●

●
●

●
●

●

●

1

●
●

●

●
●

●

0

●

0
●

−1

bL

bR

●

−5

0

5

Eccentricity from fixation point (in characters)

Figure 2: Symmetric (dashed blue lines) and asymmetric(solid black lines) non-parametric model parameter estimates.
Error bars on the asymmetric model parameter estimates are
standard deviations estimated from bootstrap replicates. bL
and bR are “edge bonus” parameters added to the appropriate eccentricity parameters to obtain log l for the leftmost andrightmost letters in a word.
estimates (and error bars to indicate bootstrapped standard
deviations for the asymmetric variant), for the two varietiesof the nonparametric model. Figure 3 graphs these quantities for the two varieties of the parametric model. In both
figures, eccentricity falls off to the left more rapidly in theasymmetric model than in the symmetric model, and to theright more rapidly in the symmetric model than in the asymmetric model.2 This consistent pattern suggests that the dataof Stevens & Grainger (2003) provide evidence for an asymmetry in the perceptual acuity curve in visual recognition ofisolated words even when lexical constraint is taken explicitly
into account.

Conclusion
The results of our modeling studies provide additional evidence for the idea that an asymmetric visual acuity curve
contributes to the OVP effect documented in many studies
on isolated word recognition (O’Regan et al., 1984; Vitu,
O’Regan & Mittau, 1990; O’Regan & Jacobs, 1992; Stevens& Grainger, 2003). Even while explicitly accounting for therole of lexical constraint, we consistently found that the models which best accounted for the distribution of response accuracies found in Experiment 2 of Stevens & Grainger (2003)
had an asymmetric perceptual acuity curve in which acuity
dropped off more slowly as a function of visual eccentricity
to the right than to the left. Although Stevens & Grainger(2003) also presented results of another experiment using theletter-within-mask identification task which called into ques2 In the nonparametric model, the most extreme eccentricities
have oddly-behaving parameter estimates that indicate possible
problems with model specification, perhaps because the edge bonus
parameters are so often implicated in model predictions for these extreme eccentricities. We expect that fitting the nonparametric model
to behavioral data involving presentation of words of a larger varietyof lengths would be likely to reduce or eliminate this problem.

−1

bL

bR
−5

0

5

Eccentricity from fixation point (in characters)

Figure 3: Symmetric (dashed blue lines) and asymmetric(solid black lines) parametric model results, assuming that
eccentricity is piecewise Gaussian centered around the fixation point. bL and bR are “edge bonus” parameters added tothe appropriate eccentricity parameters to obtain log l for theleftmost and rightmost letters in a word.

tion the generalizability of the findings of Nazir et al. (1991,
1992) regarding asymmetry of the perceptual acuity curve,
the fact that an asymmetric perceptual acuity curve is required
to provide the best account for their word-recognition databy an ideal observer with the knowledge of lexical statisticsof the language calls into question the strong position stakedout by Clark & O’Regan (1999) that the left-of-center position of the OVP may derive purely from lexical constraint.
In our view, the most plausible theoretical position reconciling our modeling findings with empirical findings from thereading literature on the asymmetric perceptual span (Rayner
et al., 1980) and with those of Stevens & Grainger (2003)
on character-within-mask recognition is that isolated word
recognition does indeed involve an asymmetry of perceptual
acuity, but that it is a parasitic byproduct of cognitive adaptation to the naturalistic task of reading. The character-withinmask recognition task may be sufficiently unlike naturalistic reading that it does not trigger this asymmetric perceptual
span. It is worth noting that the experiments of Nazir et al.
(1991) and Nazir et al. (1992) always involved fixation on
the leftmost or rightmost character of the word, and used a letter (instead of the hash mark # used by Stevens & Grainger)
as the mask character. It is possible that the lower overallvisual acuity due to fixation at the word’s right or left edge
together with the letter mask may have induced the language
and perceptual systems to categorize input in this experiment
more like that obtained in natural reading, inducing asymmetric perceptual acuity.
Although we believe our modeling approach represents animportant step forward in resolving these issues, it is important to emphasize that several simplifying assumptions we introduced which led from the general ideal-observer formulation to the specific, highly tractable model of Equation (4)have the potential to significantly affect our modeling results
and deserve more careful consideration in the future. The firstof these simplifying assumptions was equal confusability ofletter pairs. Of course, it is well known that some letter pairs
are more confusable than others (e.g., o is much more confusable with e than with l for native English speakers; Geyer,1977). It is possible that the assumption of equal confusability could interact with the lexical statistics of French and theitems chosen by Stevens & Grainger (2003) to create a confound in the explanation of our modeling results—for example, such a confound might arise if the letters near the end of
their items were more highly confusable than the letters near
the beginning of their items. This possibility can be exploredin future work by relaxing the assumption of equal confusability and using established letter confusion matrices to scale
our model parameters, or even to allow our model to learn
confusability parameters directly from word-recognition behavioral data.
The second simplifying assumption deserving discussionis that participants’s behavioral responses arose from probability matching. In other cognitive domains, this assumption seems to have reasonable theoretical and empirical support (Vulkan, 2000; Mozer, Pashler & Homaei, 2008; Vul &Pashler, 2008; Vul, Goodman, Griffiths & Tenenbaum, 2009).
That being said, it is entirely possible that participants’ responses may reflect maximization or some other similar decision process, and that most inter-trial response variability
derives from the variation inherent in noisy perception. The
consequences of this possibility may be explored in future
work within our framework by explicit simulation of intertrial noise instead of marginalizing over perceptual input as
we have done here.
The final simplifying assumption is that of minimal covariance between the probability of noisy perceptual samples
given the word under consideration and the marginal probability of those perceptual samples. As discussed earlier, thisassumption is clearly wrong insofar as words in any given
language tend to look more like each other than like nonwords; but the sheer number of words in the lexicon, combined with the considerable variability that does exist among
wordforms, implies that this covariance should in general
be rather small. It is also not obvious to us how this simplifying assumption might introduce a confound to our specific result of an asymmetric perceptual acuity curve in isolated word recognition. Nevertheless, there are two ways that
this simplifying assumption could be relaxed in future work.
First, explicit simulation of inter-trial noise would permit us
to quantify the discrepancy between the simplified results
we report here and the results which would obtain under themodel more generally. Alternatively, we might try to quantify
the covariance between P(d|w, k) and P(d|k) through explicit
simulation, and then use these covariance estimates to adjust
our expectation-based model directly. This latter alternative
might have the added benefit of giving us more direct insight into the full range of top-down effects that are present inword recognition. Intuitively, this covariance should be larger
for more “prototypically word-like” words, which should de-crease the expected posterior belief for such words relative toless word-like words, a sort of second-order neighborhooddensity effect. Further elucidation of all these issues awaits
future research.

Acknowledgments
We are grateful to Michaël Stevens and Jonathan Grainger for
graciously sharing their experimental data with us; and to Rebecca Colavin for her expertise in French. This research waspartially supported by NIH Training Grant T32-DC000041 to
the Center for Research in Language at UCSD to NJS, by a
Taiwan Graduate Fellowship of government sponsorship foroverseas study to WHC, and by NSF grant 0953870 to RL.

References
Anderson, J. R. (1990). The Adaptive Character of Human Thought.
Lawrence Erlbaum.
Clark, J. J. & O’Regan, J. K. (1999). Word ambiguity and the optimal viewing position in reading. Vision Research, 39, 843–857.
del Prado Martı̀n, F. M. (2008). A fully analytical model of the visual
lexical decision task. In Proceedings of the Cognitive Science
Society.
Engel, G. R., Dougherty, W. C., & Jones, G. B. (1973). Correlationand letter recognition. Canadian Journal of Psychology, 27(3),317–326.
Geyer, L. H. (1977). Recognition and confusion of the lowercase
alphabet. Perception & Psychophysics, 22, 487–490.
Marr, D. (1982). Vision. San Francisco: Freeman.
Mozer, M. C., Pashler, H., & Homaei, H. (2008). Optimal predictions in everyday cognition: The wisdom of individuals or
crowds? Cognitive Science, 32(7), 1133–1147.
Nazir, T. A., Heller, D., & Sussman, C. (1992). Letter visibility andword recognition: The optimal viewing position in printed words.
Perception & Psychophysics, 52(3), 315–328.
Nazir, T. A., O’Regan, J. K., & Jacobs, A. M. (1991). On words andtheir letters. Bulletin of the Psychonomics Society, 29(2), 171–
174.
Norris, D. (2006). The Bayesian Reader: Explaining word recognition as an optimal Bayesian decision process. Psychological
Review, 113(2), 327–357.
Norris, D. (2009). Putting it all together: A unified account of word
recognition and reaction-time distributions. Psychological Review, 116(1), 207–219.
O’Regan, J. K. & Jacobs, A. M. (1992). Optimal viewing position
effect in word recognition: A challenge to current theory. Journalof Experimental Psychology: Human Perception & Performance,
18(1), 185–197.
O’Regan, J. K., Lévy-Schoen, A., Pynte, J., & Brugaillére, B.
(1984). Convenient fixation location within isolated words of different length and structure. Journal of Experimental Psychology:
Human Perception & Performance, 10, 250–257.
Rayner, K. (1998). Eye movements in reading and information processing: 20 years of research. Psychological Bulletin, 124(3),372–422.
Rayner, K., Well, A. D., & Pollatsek, A. (1980). Asymmetry of
the effective visual field in reading. Perception & Psychophysics,
27(537–544).
Rice, J. A. (1995). Mathematical Statistics and Data Analysis (2ed.). Duxbury Press.
Stevens, M. & Grainger, J. (2003). Letter visibility and the viewing position effect in visual word recognition. Perception & Psychophysics, 65(1), 133–151.
Vitu, F., O’Regan, J. K., & Mittau, M. (1990). Optimal landing
position in reading isolated words and continuous text. pp, 47(6),583–600.
Vul, E., Goodman, N. D., Griffiths, T. L., & Tenenbaum, J. B.
(2009). One and done? Optimal decisions from very few samples.
In Proceedings of the Cognitive Science Society Conference.
Vul, E. & Pashler, H. (2008). Measuring the crowd within: Probabilistic representations within individuals. Psychological Science,
19(7), 645–647.
Vulkan, N. (2000). An economist’s perspective on probability
matching. Journal of Economic Surveys, 14(1), 101–118.