How Data Drive Early Word Learning: - Ricerca sull'intelligenza artificiale specializzata al MIT

How Data Drive Early Word Learning:
A Cross-Linguistic Waiting Time Analysis

1
Francis Mollica

and Steven T. Piantadosi

1Brain & Cognitive Sciences, University of Rochester

Keywords: word learning, rational construction, waiting time models

ABSTRACT

The extent to which word learning is delayed by maturation as opposed to accumulating data
is a longstanding question in language acquisition. Further, the precise way in which data
inﬂuence learning on a large scale is unknown—experimental results reveal that children
can rapidly learn words from single instances as well as by aggregating ambiguous
information across multiple situations. We analyze Wordbank, a large cross-linguistic dataset
of word acquisition norms, using a statistical waiting time model to quantify the role of data
in early language learning, building off Hidaka (2013). We ﬁnd that the model both ﬁts and
accurately predicts the shape of children’s growth curves. Further analyses of model
parameters suggest a primarily data-driven account of early word learning. The parameters of
the model directly characterize both the amount of data required and the rate at which
informative data occurs. With high statistical certainty, words require on the order of ∼ 10
learning instances, which occur on average once every two months. Our method is
extremely simple, statistically principled, and broadly applicable to modeling data-driven
learning effects in development.

The ﬁrst year of life is an incredibly productive time for language learners. Babies discover
which sounds are in their language (Eimas, Siqueland, Jusczyk, & Vigorito, 1971; Kuhl,
Williams, Lacerda, Stevens, & Lindblom, 1992), how speech is segmented (Saffran, Aslin,
& Newport, 1996), what common words refer to (Bergelson & Swingley, 2012), E, toward
the end of the ﬁrst year, how to produce their ﬁrst word (Brown, 1973; Schneider, Daniel,
& Frank, 2015). This growth is a complex endeavor that requires relying on abilities in many
domains—social and pragmatic understanding, conceptual representation, joint attention, E
acoustic and motor systems. Tuttavia, little is known about how the development of nonlin-
guistic factors inﬂuences language growth. For instance, is the timing of language growth
locked to factors like the maturation of cognitive and motor systems (per esempio., memory and atten-
zione), or to the growth of children’s conceptual repertoire? Or, alternatively, is early language
learning primarily limited by the amount of data that children receive about language itself?

Evidence for a data-driven view of the timing of language learning comes from stud-
ies showing the importance of linguistic input for early learning (Hoff, 2003; Huttenlocher,
Haight, Bryk, Seltzer, & Lyons, 1991; Shneidman, Arroyo, Levine, & Goldin-Meadow, 2013;
Weisleder & Fernald, 2013). Tuttavia, there are complications for the view that data are
all that matters. Maturational constraints are often thought to play an important role in lan-
guage learning (Borer & Wexler, 1987; Newport, 1990). Many words like function words
(per esempio., “the”) and number words (per esempio., “two”) are learned surprisingly late for their frequency,

a n o p e n a c c e s s

j o u r n a l

Citation: Mollica, F., & Piantadosi, S. T.
(2017). How data drive early word
apprendimento: A cross-linguistic waiting
time analysis. Open Mind:
Discoveries in Cognitive Science,
1(2), 67–77. https://doi.org/10.1162/
opmi_a_00006

DOI:
https://doi.org/10.1162/opmi_a_00006

Supplemental Materials:
www.mitpressjournals.org/doi/suppl/
10.1162/opmi_a_00006

Received: 13 May 2016
Accepted: 3 Gennaio 2017

Competing Interests: The authors
declare no competing interests.

Corresponding Author:
Francis Mollica
mollicaf@gmail.com

Copyright: © 2017
Istituto di Tecnologia del Massachussetts
Pubblicato sotto Creative Commons
Attribuzione 4.0 Internazionale
(CC BY 4.0) licenza

The MIT Press

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How Data Drive Early Word Learning Mollica, Piantadosi

suggesting that the number of times a word is heard by a child is not a deﬁnitive predictor of
apprendimento. This fact has motivated hypothetical processes, including maturational constraints
on function words or syntax (Borer & Wexler, 1987; Modyanova & Wexler, 2007) and con-
ceptual or linguistic constraints in the case of number words (Carey, 2009).

At the heart of data-driven accounts is an ambiguity about how much data are required.
Experimental studies of word learning have revealed children’s ability to acquire word mean-
ings from single instances (Carey & Bartlett, 1978; Heibeck & Markman, 1987; Markson
& Bloom, 1997; Spiegel & Halberda, 2011), as well as from the aggregation of word usage
across multiple contexts (Smith & Yu, 2008). It is not known which of these regimes governs
the majority of lexical acquisition: Are most words learned by aggregation of tens, hundreds,
or thousands of examples, or from a single informative instance?

Here, we develop a novel data analysis of word learning across 13 languages in order
to address two questions about early word learning: When does it begin and how much data
does it require? These questions turn out to be interrelated—they are coupled together by
quantitative predictions that they make about the distribution of ages at which children learn a
word. To illustrate this, consider a simpliﬁed picture of learning: Suppose that a word is learned
by age 2. This could occur under many different situations. Three illustrative examples are: (UN)
the child could start accumulating data at birth, require about 24 cross-situational examples of
the word, and receive them about once a month; (B) the child could start accumulating data
at birth, require 4 examples, and receive them on average once every 6 months; (C) the child
could start accumulating data at 12 months, require 12 cross-situational examples, and receive
them once a month.

The central idea of our approach is that although (UN), (B), E (C) predict the same mean
age of learning, they critically predict different distributions of ages at which acquisition suc-
ceeds due to the statistics of waiting for data (Guarda la figura 1). Empirical measurement of the
distribution shape could in principle distinguish these hypotheses, informing us about how
data inﬂuence the process of word learning. For instance, if the distribution supported (B), we
might infer that there are few early constraints on learning since data accumulation begins at
birth, and that learning required few examples. If the data supported (C), we might infer that
cognitive or maturational constraints delayed the accumulation of data substantially, and that
word learning required aggregating information across contexts.

The logic of our approach is to formalize the process of learning by accumulating data.
Following Hidaka (2013), we assume that learners successfully acquire a word after k effective
learning instances (ELIs), or instances of the word that contribute to the learner’s accumulating
an amount of information about the word and we assume that ELIs arrive with an average fre-
quency of λ per month.1 However, unlike previous work, we also infer the age s at which data
accumulation begins and implement our analyses in a Bayesian data analysis that is capable
of inferring the likely ranges of parameter values from children’s data. This Bayesian approach
comes with several distinct advantages (Kruschke, 2010; Wagenmakers, Lee, Lodewyckx,
& Iverson, 2008), including the ability to determine all three variables simultaneously, con
our uncertainty in each correctly inﬂuenced by uncertainty in the others. Così, our inferences

1 Hidaka (2013) compares three different generative models for AoA distributions including one with a chang-
ing rate. In this analysis, we extend on his best-ﬁtting model for the greatest amount of words, which has a ﬁxed
rate. As this might seem counterintuitive, we summarize the models he suggested and justify our choice of
model in Appendix A of the Supplemental Materials.

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How Data Drive Early Word Learning Mollica, Piantadosi

Figura 1. Example acquisition ages under 3 example assumptions: (UN) children receive learning
instances once a month from birth and require 24 total, (B) children require 4 examples and receive
one every 6 months on average, (C) children require 12 instances, coming once every month, Ma
only begin accumulating data at 12 months. Each predicts the same mean of 24 months (dotted
line), but different shapes and variances in the timing of acquisition.

about the amount of data required to learn a word are statistically adjusted for our uncertainty
over when learning that word began, and vice versa. The analysis also has the potential to
reveal that the data are not informative about these variables, in which case we would ﬁnd
high uncertainty in the parameters given children’s data. The advantage of our analysis com-
pared to Hidaka’s (2013) model comparisons is that we can conﬁdently focus on interpreting
the parameter estimates.

PROBABILISTIC ASSUMPTIONS

Our model requires three primary assumptions: (io) age of acquisition (AoA) consists of two
periods of time: a start time s before learning a word begins and an accumulation time t,
during which children are waiting for data; (ii) children learn a word after observing a number
k of ELIs of the word; E (iii) these ELIs occur stochastically, but at a ﬁxed rate λ (measured here
in ELIs per month). For instance, s = 0, k = 24 and λ = 1 in example (UN) above. Note that the
model infers these parameters from learning curves, not from counting putative ELIs in child-
directed data. It is likely that a constellation of factors are involved in determining whether
any given instance contributes to learning (counts as an ELI). Allo stesso modo, start time s could reﬂect
several processes, including when children develop the ability to track and remember the data
that they need to learn a word, or when their conceptual repertoire is ready to begin learning
a word.

When data are observed stochastically with a rate λ that is uniform in time, the number
of ELIs actually received in a month will follow a Poisson distribution with rate λ. Under these
assumptions, the distribution of times t children must wait before receiving k ELIs follows a
Gamma distribution Γ(k, λ) with density,

F (T; k, λ) =

tk−1e−t·λ · λk
Γ(k)

(1)

Così, f describes the distribution of time children must wait before observing enough data
to learn a word. The curves in Figure 1 are Gamma distributions with the appropriate values

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How Data Drive Early Word Learning Mollica, Piantadosi

Figura 2. Graphical model notation for our model. Nodes denote variables of interest. Shaded
nodes are observed variables. Plates denote groups of variables over age (UN) and words (W). Nel
testo, we provide equations for a single word and omit the subscript w.

for k and λ. Note that in a Gamma, the mean scales linearly in the variance, meaning that if
acquisition is driven by accumulating data, children’s variance in learning times should scale
with their mean learning time. Gamma-shaped learning time distributions should be taken as
a hallmark of data-driven, constructivist accounts of learning (Xu, 2007; Xu & Kushnir, 2012)
that applies to any theory of development in which accumulating data is the primary force
advancing learners’ knowledge.

THE DATA ANALYSIS MODEL
Our data analysis model uses Bayesian techniques to recover k, λ, and s from empirically
measured learning curves. To do this, we require one data-analysis assumption that the pop-
ulation of children studied is relatively homogeneous, meaning that we may extend a word’s
single s, k, and λ across children.2 In this case, the proportion of children who know a word
at accumulation time T will approximate the cumulative distribution function of (1) at time T,

F(T; k, λ) =

(cid:2) T

F (T; k, λ) dt.

(2)

Figura 2 shows a graphical model of the relationships between these variables and the observed
dati. At each age a, Na children were measured and xa of them reported having learned the
word to either production or comprehension.3 We model the number of children producing/
comprehending the word xa as being drawn from a binomial distribution with Na trials and a
probability of success equal to the proportion of children who know the word given by (2) at
time t = a − s:

xa ∼ Binom(F(a − s, k, λ), Na)

(3)

We assume uniform priors on these variables: k ∼ Uniform(0, 10,000) ELIs, λ ∼ Uni-
form(0, 10,000) ELI(S)/month and s ∼ Uniform(0, 1,000) months. Bayesian inference in this
generative model allows us to take the empirical acquisition curves and determine posterior
distributions for k, λ, and s for each word in each language.

2 Our conclusions hold even if we relax this assumption (see Appendix B of the Supplemental Materials).
3 We ﬁt the comprehension and production data separately.

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How Data Drive Early Word Learning Mollica, Piantadosi

RESULTS

The Cumulative Gamma Matches Observed Word Learning Curves

Figura 3 shows a general visualization of the model ﬁt across a variety of English words. Despite
its simplicity, the model closely accounts for the empirical learning trajectories across word
types for both comprehension and production. Quantitatively, correlations between predicted
values and the behavioral data are near 1.0 for each language (see Supplemental Figure S1 in
our Supplemental Materials [Mollica & Piantadosi, 2017]) meaning that the model is able to
capture the overall shape of acquisition across languages. More importantly, the model is able
to more successfully predict learning than more standard alternatives: a probit (McMurray,
2007) and a logistic model. To test this, we divided the learning curve for each word into
two halves, where we ﬁt k, λ, and s for each word on the ﬁrst half and then computed the
correlation between model and human data across words and ages on the full curves. IL
Gamma distribution ﬁt quantitatively outperforms either the probit or the logit across most
languages (Guarda la figura 4).

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Figura 3. Points shows the proportion of English-speaking children (y-axis) who know a word
at each age (x-axis) as measured by comprehension (blue) and production (green). Lines show
the posterior mean parameters in the model (2), and X and O show the posterior mean start time
of data accumulation for each word. This generally shows good model ﬁts, early start times for
comprehension, and somewhat later times for production.

OPEN MIND: Discoveries in Cognitive Science

How Data Drive Early Word Learning Mollica, Piantadosi

Figura 4. Model comparison of the Logit, Probit, and Gamma models when trained on the ﬁrst half
of comprehension and production learning curves and tested on the full trajectory. Across words
and languages, the correlations between observed data and model predictions for the full curve are
close to 1 with the Gamma model showing the best ﬁt.

On the Order of 10 ELIs Are Needed to Learn a Word

The order of magnitude of the estimated parameters are informative about the underlying mech-
anisms of learning, as they characterize when learning starts (S), how many ELIs are needed
(k), and how frequently they occur (λ). Figura 5 shows the mean values of k, λ, and s for each
lingua. The box plots for English further broken down based on MacArthur-Bates Com-
municative Development Inventory (MCDI) semantic category are similar (see Supplemental
Figure S2 in our Supplemental Materials [Mollica & Piantadosi, 2017]).

Figure 5a and 5d show that, across languages, the order of magnitude of k is around 10 for
production, with slightly lower values for comprehension. It is important to focus on the order

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Figura 5. Box plots of the distribution of k, λ, and s across words in each language.

OPEN MIND: Discoveries in Cognitive Science

How Data Drive Early Word Learning Mollica, Piantadosi

of magnitude, not the exact numerical values, because the order of magnitude of our parameter
estimates are robust to noise (see Appendix B of the Supplemental Materials). The important
issues in language development can still be distinguished based on order of magnitude. Noi
primarily interpret Figure 5 as showing that languages agree in order of magnitude of their
estimates.4 Thus, children do not require hundreds or thousands of instances of a word to learn,
even for words that may be very frequent, nor do they learn from a single instance. Invece,
learning is likely focused around ten critically informative learning instances. These ﬁndings
demonstrate the importance of cross-situational statistics over single examples and is consistent
with the ﬁnding that children do not retain fast-mapped meanings (Horst & Samuelson, 2008).

ELIs of a Word Occur Roughly Every Two Months
The variable λ characterizes the estimated rate at which ELIs of a word occur. Figures 5b and
5e show that ELIs of a word occur once every two months (λ ≈ 0.5), indicating that ELIs are
relatively infrequent for an individual word. Tuttavia, because children learn many words
simultaneously, ELIs of any word may in fact be quite frequent. For instance, if children track
statistics on 1,000 early words, and observe an ELI for each word on average once every two
months, they will receive around 17 ELIs per day.

Data Accumulation Starts Around Two Months

The start times in Figures 5c and 5f show that learning begins early: approximately by two
months in the case of comprehension measures. The starting age is somewhat later when
curves are ﬁt to production measures, possibly because production may require motor and
speech systems to be working before production can progress. This may indicate that although
maturational factors play little role in learning as measured by comprehension, production
depends on the development of other cognitive or motor systems.

Early Word Learning Is Primarily Data-Driven
The model assumes that AoA is the sum of two time periods: start time s and accumulation
time t. There are two measures we derive from these parameters to quantify the extent to which
early word learning is data-driven: the percent of total AoA time spent accumulating data, E
the percent of variance in AoA explained by variance in accumulation times. If early word
learning is primarily constrained by maturation, the majority of acquisition time should not
be spent accumulating data and the majority of the variance in acquisition times should be
explained by the variance in start times s. D'altra parte, a data-driven account of early
word learning would expect the majority of acquisition time to be spent accumulating data and
the majority of the variance in acquisition times to be explained by variance in accumulation
times t. Figura 6 shows the proportion of total acquisition time and the variance in acquisition
times that is due to t (accumulating data) rather than s (start times). We ﬁnd that generally
the majority of acquisition time is spent accumulating data and the variance in accumulation
times explains the majority of the variance in acquisition times. Taken together, this indicates
that data-driven factors are the primary drivers of early word learning.

4 We suspect that the greater uncertainty around estimates for Hebrew and Swedish is due to data sparsity

(see Supplemental Figure S4 in our Supplemental Materials [Mollica & Piantadosi, 2017]).

OPEN MIND: Discoveries in Cognitive Science

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How Data Drive Early Word Learning Mollica, Piantadosi

Figura 6. The bar plot shows percent of the variance in age of acquisition (AoA) times explained
by accumulation time (suggesting data-driven learning). The triangular points shows the percent
of AoA time spent accumulating data. Error bars and point ranges represent bootstrapped 95%
conﬁdence intervals. Outliers (< 2.5% of the data) were removed for this analysis (see Methods section). Learning Instances Are Weakly Correlated With Log Frequency Under a simple view that most usages of a word are informative about its meaning, our esti- mates of k and λ should be surprising; word frequencies vary over several orders of magnitude (Zipf, 1949), yet the inferred k and λ values do not. This means that ELIs cannot be very strongly correlated with frequency. Most of the time a frequent word is used, it is not an ELI. One possibility is that a single ELI for a word like tiger might be an entire visit to the zoo. To investigate the relationship further, we computed the correlation between the esti- mated k, λ, and s values for each word in English and the log frequency as measured in CHILDES (MacWhinney, 2000). For comprehension, there is only a small correlation be- tween the estimated k parameter and frequency (k : r = −.14, p = .01). For production, there is a modest correlation (k : r = .19, p < .001; λ : r = .32, p < .001; s : r = −.22, p < .001) as observed by Hidaka (2013). But what is notable is the weakness of the correlation (see λ λ Figure 7. Correlations between CHILDES frequency for words in English and estimated parameter values. Top row: For comprehension, there is a small correlation between frequency and k and no correlation between frequency and λ and frequency and s. Bottom row: For production, the correlations between frequency and k, frequency and λ, and frequency and s are very weak and only signiﬁcant when frequency is log transformed. OPEN MIND: Discoveries in Cognitive Science 74 l D o w n o a d e d f r o m h t t p : / / d i r e c t . m i t . / e d u o p m i / l a r t i c e - p d f / / / / / 1 2 6 7 1 8 6 8 2 5 2 o p m _ a _ 0 0 0 0 6 p d . i f b y g u e s t t o n 0 7 S e p e m b e r 2 0 2 3 How Data Drive Early Word Learning Mollica, Piantadosi Figure 7)—it is not as though doubling the quantity of input will double the number of ELIs. This ﬁnding is compatible with ﬁndings of frequency effects in word learning (Ambridge, Kidd, Rowland, & Theakston, 2015; Hoff, 2003; Huttenlocher et al., 1991; Shneidman et al., 2013; Weisleder & Fernald, 2013), but suggests that frequency will be less important than the frequency of ELIs (see also Hoff, 2003). DISCUSSION We view the Gamma model not as a mechanistic learning account, but instead as a scientiﬁc tool for understanding the basic forces in early language acquisition. Unlike characterizations in terms of mean acquisition ages, the parameters s, k, and λ are psychologically meaningful in terms of a causal process that likely supports part of word learning, data accumulation (Hidaka, 2013). Our analysis of empirical learning curves strongly suggests that data accumulation begins very early, that production may be delayed due to maturational factors, and that typical words take on the order of ∼ 10 ELIs to learn, not hundreds of occurrences and not a single occurrence or two. The model also suggests that the informative data points for word learning occur relatively infrequently, about once every two months, and that these occurrences are not strongly related to a word’s overall frequency. Moreover, the mechanisms of data accumulation not only provide the best quantitative ﬁt to learning curves, they explain nearly all of the variance in when children learn a word. This analysis has capitalized on the existence of large corpora of acquisition trajectories across children. In particular, the key variables of interest, data amounts, data rates, and the time at which data are ﬁrst considered, are discovered entirely from children’s acquisition trajectory—not from recordings of children’s input. While it may seem tempting to address these questions of acquisition with an intensive home recording study (Roy et al., 2006) or an evaluation of child-parent interactions (MacWhinney, 2000), these approaches come with the challenge of delineating which instances of a word concretely contributed to learning. For example, a word use might only aid acquisition if the child is attentive and receptive, and the referent is clear, which might not be observable in those datasets. Given that we have found that overall frequency is a weak predictor of the rate of ELIs, the detailed measurement of just parental productions will not fully clarify the relevant data sources for learning. Instead, our work takes a different tack, looking to ﬁnd evidence of data-driven effects writ large in the distribution of learning times for words. This work leaves open a central question: what makes a usage of a word an ELI? The weak correlation between the parameters and word frequency suggests that ELIs are rare—and perhaps even intentional. It is likely that children actively decide what stimuli they engage and deeply process (Kidd, Piantadosi, & Aslin, 2012, 2014), which could place an internal yoke on the rate of ELIs. Extrinsic factors probably also play a role though, as seen by the correlations with frequency. Analogously, these analyses raise the question of what determines differences in k and λ across words and languages. Future research should attempt to characterize the impact of external factors, such as semantic content (Jones, Johns, & Recchia, 2012) and phonotactic probability (Storkel, 2001), on k and λ. Our framework provides the initial step at connecting such factors to the data accumulation process that implicitly supports all existing models of word learning. It is also important to note the limitations of the MCDI data and our model. First, we restrict all of our conclusions to the early learned words covered by the MCDI. It will be important to extend this model beyond the age range of the existing MCDI. Children are ﬂexible learners and it is probable that an older child adopts a variety of strategies, which may inﬂuence OPEN MIND: Discoveries in Cognitive Science 75 l D o w n o a d e d f r o m h t t p : / / d i r e c t . m i t . / e d u o p m i / l a r t i c e - p d f / / / / / 1 2 6 7 1 8 6 8 2 5 2 o p m _ a _ 0 0 0 0 6 p d . i f b y g u e s t t o n 0 7 S e p e m b e r 2 0 2 3 How Data Drive Early Word Learning Mollica, Piantadosi the data-driven process. For example, older children might be able to bootstrap from their existing vocabulary/syntactic constructions or their intuitive theories of the world. Additionally, the lack of variability in the MCDI words constrains the empirical testing of many hypothesized constraints on vocabulary acquisition (e.g., Markman, 1990). Applied to the appropriate data, our approach is a suitable tool to evaluate these constraints at the computational level. Further, we chose to encode maturation as a constant offset from birth to address our main questions. This is an appropriate operationalization but a coarse distinction, and future research should address this. METHODS We ﬁt k, λ, and s within individual words and languages on data retrieved on June 16, 2015, from Wordbank (Frank, Braginsky, Yurovsky, & Marchman, 2016), a repository for MCDI in- struments (Fenson et al., 2007). This yielded cross-sectional data from 13 languages (see Supplemental Figure S3 in our Supplemental Materials [Mollica & Piantadosi, 2017] for fur- ther description). For each word in each language, k, λ, and s were ﬁt using JAGS (Plummer, 2003) and corresponding R packages, rjags and runjags. For every word, four chains were run for a total of 1.25 million steps with a thin of 1, 000 steps between each saved step. The chains converged ( (cid:3)R < 1.2) for all 2, 397 words in the comprehension and 9, 420 words in the production measure. 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