Optimal Processing Times in Reading: a Formal Model and Empirical Investigation
Nathaniel J. Smith (njs@pobox.com)
Department of Cognitive Science, 9500 Gilman Drive #515
La Jolla, CA 92093-0515 USA

Roger Levy (rlevy@ling.ucsd.edu)
Department of Linguistics, 9500 Gilman Drive #108
La Jolla, CA 92093-0108 USA
Abstract

noisy samples from which to (Bayes-optimally) average out
the noise and extract the signal; if we are optimal perceptual
discriminators then the number of samples we must gather(and thus how long we must wait) depends on the form of thesignal, of the noise, and of our prior beliefs. (For one example of this approach in the context of word recognition, see
Norris, 2006.) It seems unlikely, however, that the sensory
system is to blame for all response delays; we are reminded
every time we start up our computers that computation qua
computation takes time.
Furthermore, while such models seem plausible for visual
perception, it is unclear how they might apply to, for instance,
audition. We have reasonable control over how long we lookat a scene, but very little control over how long we listen to anutterance. Yet, tasks using single auditorily presented words
as stimuli find systematic variation in reaction time — and,in fact, these variations are similar to those observed in corresponding visual presentation paradigms (Goldinger, 1996).
This is one reason that we believe language to be a fruitfularea in which to investigate alternative approaches to modeling processing time as an optimal behavior. While language
can be presented in the written modality, which is very convenient for experimentation, the bulk of our exposure is to
spoken language, and the spoken modality has primacy both
evolutionarily and developmentally. To the extent, then, that
sensory sampling approaches couched in the framework of
optimal perceptual discrimination are implausible for spoken
language comprehension, these approaches are unlikely toprovide the full story for general language processing. On theother hand, language processing is highly practiced and very
efficient, which suggests that some other kind of optimality
approach would still be valuable.
In this paper, we present a new model of optimal responsetime couched in a framework of optimal preparation that
we believe may be more appropriate to domains such language processing in which we cannot always control timeof sensory exposure. This model is motivated by the wellestablished fact that processing times in language comprehension are probability-sensitive: in a given context, words
which are more predictable are also read more quickly (e.g.
Ehrlich & Rayner, 1981). This is intuitively sensible — certainly we would prefer it to the reverse! — but it is, as yet,inadequately theorized. Our model explains this result as optimal behavior under a cost function which trades off preparation costs versus processing time; one would like to pro-

It is widely known that humans can respond to events they expect more quickly than to unexpected events, but we still have apoor understanding of why. Models exist that derive a relation
between subjective probability and response time on the basisof optimal perceptual discrimination, but these models rely on
the ability of the responder control over perceptual samplingof the environment, rendering them problematic for some domains, such as auditory language processing, in which there
are nevertheless clear dependencies between probability andresponse time. We present a new model deriving the relationship between probability and reaction time as a consequenceof optimal preparation. This model is valid under very general conditions, requiring only that the results of optimization
are invariant across scale of input stimulus granularity. The
model makes the strong prediction that response times should
scale linearly with the negative conditional log-probability ofthe stimulus. We present evidence for this prediction in ananalysis of an existing database of eye movements in the reading of naturalistic texts.
Keywords: Optimal behavior; Language; Response time modeling; Surprisal; Sentence comprehension; Eye movements;
Reading

Introduction
It takes time to perform computation using a physical device,and the human brain is such a device. While obvious, thispoint is worth revisiting in light of the recent surge of interestin rational models of optimal behavior (Chater et al., 2006;
Todorov, 2004). Such models have provided elegant explanations for many aspects of behavior, but processing time provides a particular challenge for this approach. In general, humans respond to different stimuli within any given class withdifferent speeds, and response times are a large part of thestock and trade of experimental cognitive psychology. Froman optimality perspective, the difficulty is that it is unclear
why the time to spend on performing a computation should
ever be larger than the physical minimum. Yet, from a theoretical point of view, response times seem like the perfect candidate for an optimality approach, because they are so clearly
relevant to evolutionary fitness. We are all real-time organisms who must react quickly and correctly in a wide varietyof circumstances. So why are we still so much slower thanwe could be?
The primary approach to these problems deployed within
the Bayes-optimal framework has been to ascribe reaction
times and other such delays to the sensory system. The argument is that we require accurate information about the world
to act, but our sensory system is noisy. Therefore, to acquire accurate information, we must wait and gather multiple

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cess quickly, but this requires preparation, and preparation
is expensive in its own right. This model makes strong predictions about the relationship between probability, optimal
preparation costs, and optimal reading times. We then test themodel’s prediction about probability and reading time against
a corpus of naturalistic language processing data.

In any case, we are done as soon as we work out what formr(t) takes. This function summarizes the costs involved inmany kinds of preparation occurring over many time-scales.
For instance, these might over the short term involve the attentional resources required to speculatively pre-compute responses to stimuli that are especially likely in this context;
over the medium term, the metabolic costs of maintaining
more or less precise cortical circuit tuning; and over the longterm, the allocation of limited cortical area to items which
prove themselves to be reliably common. We therefore do
not assume or attempt to derive any particular functional form
for r(t) from first principles, and limit ourselves to two simple assumptions: (i) that it depends only on the chosen time
ti and not on any other specific properties of the context or
stimulus; and (ii) that it is some smooth and monotonic decreasing function. We turn instead to the particular attributesof our system of interest, language.

Model
How much preparation is too much?
Our main idea is that in general, the nervous system does notoperate at the fastest possible speed, and that the reason forthis is that operating at the limits of efficiency is very expensive. Instead, it adjusts its performance on particular tasks
to optimize a composite cost function that balances the speed
achieved against the costs of achieving that speed.
We further assume that this optimization occurs before
each stimulus is actually encountered, because once the stimulus is encountered it is too late to reallocate resources — one
has whatever resources one has, and all there is to do is to process the stimulus as fast as possible given those constraints.
Thus there are two stages, each with a cost: the pre-stimulusor “preparatory” period, where optimization occurs, and thepost-stimulus or “processing” period, lasting from stimulus
onset until an appropriate response can be made. The duration of the latter is evolutionarily relevant and what we commonly measure experimentally, but the costs incurred in theformer are presumably just as important to the brain.
Formally, assume that for a given context there are n possible stimuli that we may possibly encounter (in the case ofreading, these stimuli could be the words that may occurnext), and we may freely choose how long each will take us
to process, subject to an additive global cost function:

Scale-free assumption
One of language’s most celebrated properties is that it hashierarchical structure, with regularities occurring at all levels of granularity from, e.g., sentences to clauses to words tomorphemes to letters or phonemes. In our experiments wemay choose to measure processing time at word granularity,
but we have no reason to believe that this is a uniquely preferred scale for the brain. Spoken language is essentially acontinuous auditory stream, and clearly there are many options for how to break it into discrete, enumerable ‘stimuli’
as required by our model. Therefore, instead of trying to derive r(t) directly, let us require that our model give the sameanswer regardless of the temporal granularity we use to divide
our stimuli.
Formally, suppose we have some item i (e.g., a word)
which we can partition into smaller items i1 , . . . , im (e.g., thephonetic segments in that word). Let pi j denote the conditional probability that i j appears given that i1 , . . . , i j−1 haveappeared previously — i.e., pi j = P(i j |i1 , . . . , i j−1 , context)— while pi as defined above can be rewritten as
P(i1 , . . . , im |context). Applying the chain rule to these formulas shows that probability of the larger item is simply the product of the probabilities of the smaller items:
pi = ∏mj=1 pi j . On the other hand, the time taken to process
the larger item is the sum of times taken to process its partsti = ∑ j ti j . By (2′ ), ti j = f (pi j ). Substituting into (2′ ), we findnC(t) = ∑ r(ti ) + E(tI |context).

(1)

i=1

Here ti is the time we will require to process stimulus i if itappears, t = ht1 ,t2 , . . .i is the vector of all such times. Our
goal is to select t in such a way that we minimize our overall cost C(t). The cost is composed of two parts. The firstterm, ∑i r(ti ), corresponds to the preparation cost we incur
before encountering the stimulus; r(ti ) denotes the cost of investing resources to prepare for stimulus i, and we prepare
for all possible stimuli (though, at our option, in differing
amounts). The second term, E(tI |context), corresponds to thetime it will take to process the stimulus that we do, in fact, encounter; since the stimulus has not yet occurred, its identity isa random variable, I, and we can only optimize the expected
time for processing it. Simple calculus then shows that (1) is
minimized whenti = (r′ )−1 (−pi )(2)∑ f (pi j ) = f (∏ pi j ).
jjThat is, the function f turns products into sums. The only
non-trivial functions with this property are logarithms. Working backwards and minding the appropriate monotonicity
conditions, we conclude
ti = − logk pi

where pi = P(I = i|context), and r′ denotes the derivative.
This unwieldy formula becomes clearer if define f (x) =(r′ )−1 (−x):
ti = f (pi ).
(2′ )r(t) =

k−tloge kwhere k > 1 is a free parameter.

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(3)
(4)

Methods

0.5

1.0

1.5

Data The main technical challenge in measuring the shapeof a human response curve is obtaining enough data points
to estimate it reliably. Therefore, rather than attempt to construct a small set of balanced stimuli, we chose to analyze theDundee eye-movement corpus (Kennedy et al., 2003), which
consists of all eye-movements made by 10 subjects while
reading a collection of newspaper articles totaling approximately 50,000 words. In this paper we report results for first
fixation times, a standard reading time measure corresponding to the durations of the first fixation to land on each wordin the text.1 This is not a perfect measure of processing time,and it is not a perfect match to our model (which does notassume that during a fixation centered on some word, subjects will process only that word); these facts will tend to increase the noise in our data. Noise, however, can be overcome
through statistical means, and in return we are able to makeuse of existing methods of word probability estimation, andachieve greater comparability with the existing reading literature.

0.0

Preparation cost (real ms)

Preparation cost curve

0

5

10

15

20

Chosen time (potential ms)

Figure 1: Shape of preparation cost curve r(t) derived in (4),for k = 1.89 (chosen to match reading time data). X axis gives
the different achievable processing times that we select frombefore encountering the stimulus; Y axis gives the cost for
selecting any given time, which in equation (1) is then added
to the average actual processing time to produce the total cost.

Probability estimation Probabilities were estimated by atrigram language model trained on the 100 million word
British National Corpus (BNC) . We estimated the model using SRI Language Modeling Toolkit (Stolcke, 2002), withmodified Kneser-Ney smoothing (Kneser & Ney, 1995).2 A
trigram model approximates the probability of a word in context P(wordi |context) as the probability of the word given two
previous words, P(wordi |wordi−1 wordi−2 ); modified KneserNey is a standard method of smoothing these trigram probabilities, and a standard technology for broad-coverage language modeling (Chen & Goodman, 1998). However, it
should be noted that Kneser-Ney trigram probability (henceforth, KN3-probability) is still a very noisy estimate of true
conditional probability.

Predictions
Our model therefore makes two strong predictions about theprocessing of stimuli which are continuous and have scalefree hierarchical structure in time: that preparation costs drop
off exponentially as the chosen processing time increases (seeFigure 1), and that ultimately processing time of a linguistic
unit should be proportional to the negative log of that unit’s
probability (its surprisal or self-information in informationtheoretic parlance). The latter prediction can be tested using
existing data sources.

Data selection The Dundee corpus contains 307,656 first
fixations; of these, we eliminated all fixations on words that
occurred at the beginning or end of a line, which preceded orfollowed punctuation, that did not occur in the BNC (i.e., unknown words), or that occurred in the BNC but in segmented
form (e.g., the BNC codes don’t as two words, do followed byn’t). This left N = 197, 503 fixations for our analysis, spreadroughly evenly across 10 subjects (range: 16666–22390 fixations per subject).

Empirical validation
While it is generally agreed that more predictable words areread more quickly, previous work on reading time and probability has suggested many functional forms for this relationship: logarithmic (Hale, 2001; Levy, 2008), linear (Reichle etal., 1998; Engbert et al., 2005), or even reciprocal (Narayanan& Jurafsky, 2004). None have been empirically verified;empirical work has been restricted to factorial comparisons(Rayner & Well, 1996), which provide limited insight into
curve shape. In this section, we investigate the relationshipdirectly, using multiple regression techniques on an existing
database of eye movements performed during the reading of
naturalistic text.

Confounds A number of other linguistic measures are correlated with probability, and also known to be correlated withreading time; in particular, these include word length andword frequency. Since we are using naturalistic data, we must
control for such confounds retrospectively. Word length is
1 Analyses of first-pass reading times led to substantially similar
results.
2 Traditionally, such probabilities are estimated via a cloze norming task, but such behavioral measures are impractical for large numbers of data points or low probability events, both of which are major
considerations for our data-set.

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Potential relationships between
probability and reading time

−6

−4

Reading time−2

Linear−8

KN3−probability (log 10)

0

Spread of frequency vs. probability

Logarithmic

Reciprocal
−8

−6

−4

−2

Frequency (log 10)

Probability (log scale)Figure 2: Scatter-plot of log frequency versus estimated log
probability for the words in our data.

Figure 3: Three possible relationships between probability
and reading time as proposed by different authors, illustrated
in log-probability space. Our model predicts that the middle
curve is correct.

simply the length of each word in letters. Word frequency
is the unconditional probability of a word, P(word), measured simply as the number of times a word appears in a large
body of text, divided by the total number of words in that
text. For each fixated word in our corpus, we calculated word
frequency from the BNC. Unsurprisingly given their closely
related theoretical definitions, frequency is highly correlatedwith the conditional probability of a word (ρ = 0.80), and thelog of word frequency is reported to be correlated to processing time on a wide variety of tasks. Our model, of course,
suggests that such effects are not driven by frequency per se,
but rather by probability;3 however, testing this prediction requires that we analyze our data with respect to both frequency
and probability together. Distinguishing such correlated variables relies on what spread does exist; fortunately, this is nonnegligible (see Figure 2).

vided by generalized additive models (GAMs), a spline-based
extension of the standard linear regression framework (Hastie& Tibshirani, 1990). In our case, we fit a model of the form:4
Timei = α + β · WordLengthi +f (log KN3-probabilityi ) + g(log Frequencyi )where α and β are arbitrary constants, and f and g are arbitrary smooth functions; all are chosen by the fitting process.
Such an approach, of course, is prone to overfitting; to combat this (and make the problem well-posed), the fit penalizes
functions based on how ‘wiggly’ they are, so that there is atrade-off between following the data and avoiding extraneous
bends. The relative weight placed on these goals is determined by cross-validation.
The end result of this process are two functions, f and g
above. Plotting f will show us how fixation duration varies in
response to changes in KN3-probability, after accounting forconfounds. Above, three possibilities from the literature werementioned; the corresponding plots in log-space are shown inschematic form in Figure 3.
The function g is less immediately relevant, but interesting nonetheless; there is a long tradition of frequency effects
in psycholinguistics, but these results are usually confoundedwith any potential effect of probability (though see Rayner etal., 2004). Examining g will show us the residual effect of
frequency after accounting for the effects of probability.

Analysis Analysis was carried out in R (R Development Core Team, 2007), using the package mgcv for nonparametric multiple regression (Wood, 2006) and lme4 formixed-effect linear regression (Bates & Sarkar, 2007).

Results
Curve shape Extracting the shape of an unknown functional relationship from noisy data requires some form of
non-parametric regression; doing so while simultaneously
controlling for confounds requires multiple non-parametric
regression. An elegant framework for such analysis is pro3 Note in particular that in single-word paradigms such as the lex-

4 In this model, logs are taken of KN3-probability and frequency
purely for convenience; invertible transformations of predictor variables have a minimal effect on non-parametric regression.

ical decision task, there is effectively no context, which means thatword probability, P(word|context), and word frequency, P(word),
give identical predictions.

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One fit was performed to the data from each subject individually, with results illustrated in Figure 4. As predicted byour model, the curves in the left column (KN3-probability)
are very close to linear (in log space), and this pattern holds
across at least five orders of magnitude in probability. Nineout of ten subjects show this pattern; the exception is subject
G, whose effect appears to be minimal, if any. The curves on
the right (frequency) are also roughly linear, and of a similarorder of magnitude, though they appear to be less reliable —only seven out of ten subjects show a clear effect.
Now that we have established the shape of these effects,
we can better quantify their strength and significance using
traditional parametric techniques.

Human response curves
Probability

Frequency

0

15

A

0

15

B

15

C

0

Significance To analyze significance, we used linear regression of fixation duration on word length, log frequency,
and log KN3-probability, with subject as a random effect.
After controlling for word length and frequency, KN3probability remains highly significant (χ2 (2) = 246.12, p ≪0.001) as a predictor of first-fixation times. After controllingfor word length and KN3-probability, frequency also remains
significant (χ2 (2) = 182.81, p ≪ 0.001).
We can also investigate the relative magnitude and reliability of these effects by fitting a model including both frequency and KN3-probability simultaneously, after standardizing them both to ensure comparability. In such a model, theresponse coefficient for KN3-probability is both larger thanthat for frequency (−4.1 vs. −3.7 in arbitrary units), and lessvariable across subjects (standard deviation 1.5 vs. 2.4).

0

15

D

0
15

F

0

Time (ms)

15

E

G

15

Discussion0

We have presented a model which predicts that in languagelike tasks — those where processing is skilled, speed is important, and stimuli are arranged through time in a continuous manner with no preferred scale — the processing time for
an individual unit should be proportional to the negative logprobability of that unit occurring. Further, we have for the
first time performed a broad-coverage analysis of the functional relationship between probability and reading times, andhave found that over a wide range of probabilities, this effectis both significant and takes the predicted form. Although wehave validated this prediction of the model on language processing, this model could in principle apply to any cognitiveor perceptual domain in which our assumption that there is nopreferred granularity scale of processing is reasonable.
These results suggest that a substantial portion of what
have previously been understood as frequency may, in fact, becontext-sensitive probability effects — which may be problematic for theories that explain log-frequency effects as aresult of the (static, context-insensitive) structure of the lexicon, such as the classic logogen theory of (Morton, 1969)
and its descendants in this respect, such as READER (Just& Carpenter, 1992). It still remains to compare to the othercovariates that have been proposed besides word frequency(e.g., Gernsbacher, 1984; Morrison & Ellis, 1995; McDonald& Shillcock, 2001; Murray & Forster, 2004).

0

15

H

0

15

I

0

15

J

−4 −2

−4

−2

Figure 4: Estimated response curves for probability (left column) and frequency (right column), both in log space, plotted
individually by subject. The X axis in each case ranges from
−5 to the maximum value that occurred in the data set. This
range was chosen to include ≈ 90% of all data points (see Figure 2); outside of this range the fit becomes extremely unreliable. X = −5 corresponds to a 1-in-100,000 event. Dashedlines are bootstrapped 95% confidence intervals.

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On the other hand, frequency does remain significant, and
it is not entirely clear how to interpret this. (No extant theories have proposed explanations for why we would expect
probability and frequency to have separate, independent effects.) One possibility is that this result arises from noisein the probability estimation process: trigram models havefar more parameters than unigram (word frequency) models,
which increases estimation error; furthermore, the theoretical quantity of interest is not trigram probability at all, but‘full’ conditional probability, and this approximation introduces additional error. Taken together, this means that our
probability estimates should be understood as having a muchhigher degree of noise than our frequency estimates. Despite
this noise, however, KN3-probability still marginally outperforms frequency in explaining fixation times. This suggests
the possibility that frequency’s role could diminish or conceivably disappear with improvements in our technical ability
to estimate subjective probability.
However, it is also possible that frequency’s role will not
disappear, and is in fact real. Our model makes the strong
claim that subjective probability is the only determinant ofreading time, so if frequency’s role is validated, there are two
possibilities: either our model is essentially correct but humans are sub-optimal with regards to estimating conditional
probabilities of words in text — perhaps their estimates are
partially biased and smoothed by word frequency, as a quick,dirty, and low-variance approximation — or our model is incomplete.
Finally, we should note that the juxtaposition of our
optimal-preparation model against optimal perceptual discrimination models such as Norris’s (2006) Bayesian Reader
opens up a typology of optimal response time theories.5
Optimal-discrimination and optimal-preparation accounts
make different predictions — notably, the perceptual confusability of the stimulus should have a huge effect on response
times in optimal-discrimination accounts, whereas it does notplay a role in our account. It is also logically possible thatthe truth lies in a combination of both accounts, or in a third
account that lies elsewhere in this typology, in a theory of
optimal response times yet to be constructed.

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Acknowledgments
This work was supported by an NSF graduate fellowship to
NS.

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submitted).

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