Infant Categorization is a Discrete Learning Process
Abe Huijbers
University of Amsterdam
Infants as young as 10 months old show an increase in prediction success when a human moves the object, but not when the same action sequence is self-propelled (Hunnius, Meyer, Mandell, & Bekkering, 2012). Moreover, they can acquire these skills in the course of a short lab visit. What remains largely unclear, is how it is that infants acquire their increasing success in catego-rization tasks. In the present study we therefore examine infants’ learning curves of an earlier experiment. Most importantly, we discovered that infant categorization learning is a discrete rather than an incremental process. Also, a 2- or 3-state model with a guessing state seems to fit the data best. It appears that infants learn to categorize by discovering rules, although they seem to shift between a guessing state and a learned state even after they have discovered the rules.
Introduction
Human infants experience a daunting amount of environ-mental experiences when they grow up. Categorization is the primary means of coding those experiences; it underlies perceptual and reasoning processes, as well as inductive in-ference and language (Rakison & Oakes, 2003). Infants can learn to sort categories based on different types of features, as has been demonstrated in various eye-tracking studies.
For example, at the age of 12 months, infants show not only the ability to discriminate between different objects, but also to predict goal-directed actions accordingly (Henrichs, Elsner, Elsner, & Gredebäck, 2012). When infants were shown a large grip aperture in a human hand, they would predict the hand to grasp a large object rather than a small ob-ject. Around the same age, infants can even anticipate where objects will reappear, depending on their visual properties (Mandell & Raijmakers, 2012). Researchers showed stim-ulus material of an object disappearing behind an occluder stimulus, and infants were able to predict where the object would reappear, based on the color or shape of the object. There is evidence to suggest that multiple distinct learning modes are involved in these processes (Schmittmann, Visser, & Raijmakers, 2006), although these results were obtained from children of ages 4 years and older.
However, relatively little is known about the way in which categorization is learned. Previous studies have investigated categorization learning in the fields of visual anticipation and goal prediction in infants (Mareschal & Quinn, 2001; Plun-kett, Hu, & Cohen, 2008; Bomba & Siqueland, 1983). Such studies have used eye-tracking analysis to investigate infant development of cognition, or more specific, infant catego-rization learning: infants’ increasing understanding of cat-egories (e.g., white vs. black, square vs. triangle). They
found that infants as young as 3-4 months old were capable to learn a categorisation task in the course of a short lab visit (Bomba & Siqueland, 1983). Unfortunately, most of them used one-category habituation tasks. The downside to such tasks is that they cannot be used to study the infants learning process, because one-category tasks are subject to criterion effects; the willingness of the observer to judge two stimuli as different can vary, thereby rendering a decision for cate-gory membership uncertain (McMurray & Aslin, 2004).
Two-category learning tasks, in contrast, force the ob-server to actively choose between the two categories, thereby eliminating criterion effects. They make it possible to study not only learning outcomes, but also the process that under-lies those outcomes. Studying the learning process is essen-tial in understanding the development of infant cognition.
In the present study, Hidden Markov Models (HMMs) were used to analyse the response patterns. This approach identifies the number and nature of underlying learning pro-cesses in a response sequence. HMMs are used when the marginal distribution of the data is thought to be a mixture distribution (i.e., the data are drawn from two or more distri-butions, with different parameter values). Therefore, a Hid-den Markov Model is a model of one or more separate states, with each state producing a typical behaviour (Visser, 2011). For example, if a person has to complete 10 trials and re-sponds incorrectly on the first 7 trials, and correctly on the last 3 trials, it seems likely that this person underwent two separate states, making the switch of one state to the other after the 7thtrial.
Even though there have been accounts of learning mod-els in the literature for both infants (Mandell & Raijmakers, 2012) and children (Schmittmann et al., 2006), there is little consensus on the development of infant categorization. Both Schmittmann et al. (2006) and Mandell & Raijmakers (2012)
2 ABE HUIJBERS concluded that multiple modes of responding - or states - are
involved in infant categorization learning. We tried to shed light on the matter by conducting a similar experiment. Here, we took the example of Schmittmann et al. (2006) and Man-dell & Raijmakers (2012) by fitting several Hidden Markov models to the data of 129 infants. The data were provided by Hunnius and her colleagues (2012), who had conducted a visual anticipation experiment on a group of 10-month-olds. The experiment that was re-examined consisted of a vi-sual anticipation task in which subjects eye movements were followed using an eye tracker. Specifically, the researchers speculated that seeing another human perform goal-directed actions would enhance the infants’ learning process, as previ-ous neurocognitive research showed motor system activation when observing others’ actions (van Elk, van Schie, Hun-nius, Vesper, & Bekkering, 2008). Hunnius and her col-leagues constructed three different stimulus conditions: one with a human performing the action, one with the human present but a propelled action, and one with only a self-propelled action and no human present. They were able to demonstrate a steeper learning curve for the condition with a human performing the action, but they left the learning pro-cess itself unattended. Our subsequent analyses focused on the learning process for each of these conditions, as well as for all conditions together. It was expected that the condi-tion with the human performing the accondi-tion would facilitate categorization learning best, with the other two conditions showing fewer increase.
The analysis for this study consisted of four parts. First, the best out of four dependent variables for infants’ visual anticipation in the categorization task was sought. Second, we investigated whether infant categorization learning is best described as a discrete process or an incremental process. Third, we set out to unveil the model that describes this learn-ing process best, by applylearn-ing statistical constraints to the models at hand. Fourth, and last, we examined whether the preferred model showed different fit to the different stimulus conditions that were used in the experiment.
Methods Procedure
Participants. We examined data of 130 ten-month-old infants. One infant was dropped from all analyses because it was unable to complete any of the trials. The final sample consisted of the remaining 129 infants1.
Stimuli. Infants were shown short audio-visual clips of a table with two cups on it, one on the left and one on the right. An example of a trial is depicted in Figure 1. A ball would appear in the middle of the table, which marked the start of the trial (Figure 1a, approximately 0-2002 ms). The ball would be lifted straight up (Figure 1b, approx. 2003-3670ms), remain in mid-air for a moment (Figure 1c, approx.
3671-4205ms), and then move into one of the cups (Figure 1d and 1e, approx. 4206-6342ms). After the ball had found its destination, a reward stimulus2 would be shown in front of the cup (Figure 1f, approx. 6343-7608ms).
(a) A ball appears on the table (start of lifting phase).
(b) The ball is lifted up into the air.
(c) The ball hangs still in midair (end of lifting phase).
(d) The ball travels either to the left (red) or right (blue).
(e) The ball ends up in the cup. (f) A reward stimulus is shown. Figure 1. An example of the sequence of events in a trial. This example shows a trial in the Human Active condition, see Figure 2 for the other conditions.
In each trial, the ball would be either red or blue. If the ball was red, it would end up in the left cup; if the ball was blue, it would end up in the right cup. The trials lasted a total of 7341 milliseconds (blue trials) or 7875 milliseconds (red trials). All stimulus videos (conditions) had the exact same durations.
Each trial consisted of 2 separate phases. The first phase was named the Lifting Phase and consisted of the period of time between the start of the trial, and the moment that the ball started moving to either lateral side of the table (Figure 1a through 1c). This phase lasted from 0 to 4137 ms (red tri-als) or from 0 to 3203 ms (blue tritri-als). The second phase was named the Target Phase and consisted of the period of time between the moment that the ball started moving to either side of the table, and the end of the trial (Figure 1d through 1f).
1There was no reason to exclude any other infants because
un-completed trials did not pose a problem to our analyses.
INFANT CATEGORIZATION LEARNING 3 Trials. Each infant participated in 20 trials, unless the
experiment was stopped earlier due to ongoing fussiness. Of 129 infants, 107 completed at least 15 of the 20 trials, and the mean number of completed trials was 16.95. More in-formation on the nature of uncompleted trials is given in the Results section.
Trials 1 through 14 were learning trials in the original experiment. In these trials, the moving object was a ball. Trials 15 through 20 were generalisation trials. In those tri-als, the moving object was a triangle. This difference was not deemed an important factor in the current study and therefore disregarded in analyses. Each trial was randomly conducted as either a red trial (50.4%) or a blue trial (49.6%). All infants were shown an attention getter after trials 2, 4, 7, 9, 14 and 17, to shift their attention to the monitor screen if they had allocated it elsewhere. The experiment took approximately 5 minutes per infant to conduct.
Conditions. The participants were randomly assigned to one of three stimulus conditions. In the Human Active conditionthe ball appeared to be moved by a person sitting behind the table (Figure 2a). In the Human Passive condition the person behind the table remained inactive throughout the trial, with the ball appearing to move by itself (Figure 2b). In the Self Propelled condition, there was no person behind the table and the ball appeared to move by itself (Figure 2c). The number of infants in the conditions was 44, 44 and 41 respectively.
Materials
Eye tracker and monitor. A Tobii Corneal Reflection eye tracker with a frequency of 50 hertz and average accu-racy 0.5◦ visual angle was used (Tobii 1750, Tobii Technol-ogy, Stockholm, Sweden). Stimuli were shown on a 17” TFT flat-screen monitor.
Calibration procedure. A standard 9-point calibration task (3x3-grid) was used. The eye tracker determines which eye movements correspond to which looking locations by putting highly salient stimuli in each of the 9 points in the grid and recording the corresponding gazes.
Experimental setup. Infants were seated in a baby seat. The baby seat would be placed on the parents’ lap, with the parent sitting in a chair. The viewing distance was approxi-mately 60cm for all infants.
Analysis
Areas of Interest. The screen of the monitor consisted of 1024x1280 pixels3. Three non-overlapping Areas of In-terest (AOIs) were defined to enable analyses of looking be-haviour. The first and second area of interest (AOI 1 and AOI 2) were rectangular areas around the left and right cup respectively, both with some margin around them. The third area of interest (AOI 3) was the rectangular area around the ball throughout the lifting phase. Figure 3 shows an example
Categorization
learning
- From early in life, infants learn about objects and
begin to organize them in terms of categories.
- Various studies demonstrate that how categories
are presented has an impact on how successful
infants are in acquiring new categories (Mandell &
Raijmakers, 2010) .
- Neurocognitive research, on the other hand, has
shown that the infants’ motor system gets activated
when observing others’ actions (van Elk, van Schie,
Hunnius & Bekkering, 2008) . This action simulation
mechanism is thought to play an important role for
cognitive processing (Iacoboni et al., 2005) .
-Yet, it is unknown whether action simulation
enhances learning, especially categorization
learning. We hypothesized that
presenting infants with categories in terms of
actions would facilitate their learning of novel
categories.
Action simulation facilitates
categorization learning in infants
Sabine Hunnius
1
, Marlene Meyer
1
, Dorothy J. Mandell
2
, & Harold Bekkering
1
1
Radboud University Nijmegen, The Netherlands
2
University of Amsterdam, The Netherlands
Method
- 3 between-subjects conditions
- The colour of the object (blue; red) was predictive of
which bucket (right; left) served as the target location.
- 14 learning trials (7 for each colour), followed by
6 generalization trials (3 per colour) with a differently
shaped object
Fig. 1: Infants watched the
stimulus movies on a Tobii
eye-tracker.
Fig. 2:
The Human Action, Non-Human Movement, and Self-Propelled condition
- Eye-tracking study with
10-month-old infants (N=113)
- To assess categorization
learning, infants’ anticipatory
eye-movements to the target
locations were measured.
Fig. 3: Percentage of
correct target
anticipations during
the first and second
half of the learning
phase.
Learning phase:
- Only the infants in the Human Action condition tended
to anticipate more often to the correct target in the
second compared to the first half of the training trials.
- A GEE with the factors Condition and First/Second
half of the learning phase revealed a significant
interaction effect (
2
(2)= 7.95, p=.02).
Generalization phase:
-During the generalization trials, only infants in the
human action condition tended to anticipate more
frequently than chance to the correct target
(t(17)=1.59, p=.065).
Fig. 4: Percentage of
correct anticipations
during the generalization
phase.
Human Action Non-Human Movement Self-Propelled
Human Action Non-Human Movement Self-Propelled
%
c
o
rr
ec
t
an
ti
ci
p
ati
o
n
s
%
c
o
rr
ec
t
an
ti
ci
p
ati
o
n
s
Results
References
Iacoboni, M., Molnar-Szakacs, I., Gallese, V., Buccino, G., Mazziotta, J.C., Rizzolatti, G. (2005).
Grasping the intentions of others with one's own mirror neuron system. Plos Biology, 3, 529-535.
Mandell, D., & Raijmakers, M.E.J. (2010). Is more always better? Infants’ performance on a
simuktaneous two-category learning task. Poster presented at XVIIth Biennial International Conference
on Infant Studies, Baltimore, Maryland, USA.
van Elk, M., van Schie, H. T., Hunnius, S., Vesper, C. & Bekkering, H. (2008). You'll never crawl alone:
Neurophysiological evidence for experience-dependent motor resonance in infancy. Neuroimage, 43,
808-814.
s.hunnius@donders.ru.nl
Conclusions & discussion
- In a human action context, 10-month-old infants show
an increase in prediction success over successive trials
which is not observed in the other contexts.
- Learning about categories in terms of human actions
enhances the generalization performance on a novel
category.
(a) Human Active condition
Categorization
learning
- From early in life, infants learn about objects and
begin to organize them in terms of categories.
- Various studies demonstrate that how categories
are presented has an impact on how successful
infants are in acquiring new categories (Mandell &
Raijmakers, 2010) .
- Neurocognitive research, on the other hand, has
shown that the infants’ motor system gets activated
when observing others’ actions (van Elk, van Schie,
Hunnius & Bekkering, 2008) . This action simulation
mechanism is thought to play an important role for
cognitive processing (Iacoboni et al., 2005) .
-Yet, it is unknown whether action simulation
enhances learning, especially categorization
learning. We hypothesized that
presenting infants with categories in terms of
actions would facilitate their learning of novel
categories.
Action simulation facilitates
categorization learning in infants
Sabine Hunnius
1
, Marlene Meyer
1
, Dorothy J. Mandell
2
, & Harold Bekkering
1
1
Radboud University Nijmegen, The Netherlands
2
University of Amsterdam, The Netherlands
Method
- 3 between-subjects conditions
- The colour of the object (blue; red) was predictive of
which bucket (right; left) served as the target location.
- 14 learning trials (7 for each colour), followed by
6 generalization trials (3 per colour) with a differently
shaped object
Fig. 1: Infants watched the
stimulus movies on a Tobii
eye-tracker.
Fig. 2:
The Human Action, Non-Human Movement, and Self-Propelled condition
- Eye-tracking study with
10-month-old infants (N=113)
- To assess categorization
learning, infants’ anticipatory
eye-movements to the target
locations were measured.
Fig. 3: Percentage of
correct target
anticipations during
the first and second
half of the learning
phase.
Learning phase:
- Only the infants in the Human Action condition tended
to anticipate more often to the correct target in the
second compared to the first half of the training trials.
- A GEE with the factors Condition and First/Second
half of the learning phase revealed a significant
interaction effect (
2
(2)= 7.95, p=.02).
Generalization phase:
-During the generalization trials, only infants in the
human action condition tended to anticipate more
frequently than chance to the correct target
(t(17)=1.59, p=.065).
Fig. 4: Percentage of
correct anticipations
during the generalization
phase.
Human Action Non-Human Movement Self-Propelled
Human Action Non-Human Movement Self-Propelled
%
c
o
rr
ec
t
an
ti
ci
p
ati
o
n
s
%
c
o
rr
ec
t
an
ti
ci
p
ati
o
n
s
Results
References
Iacoboni, M., Molnar-Szakacs, I., Gallese, V., Buccino, G., Mazziotta, J.C., Rizzolatti, G. (2005).
Grasping the intentions of others with one's own mirror neuron system. Plos Biology, 3, 529-535.
Mandell, D., & Raijmakers, M.E.J. (2010). Is more always better? Infants’ performance on a
simuktaneous two-category learning task. Poster presented at XVIIth Biennial International Conference
on Infant Studies, Baltimore, Maryland, USA.
van Elk, M., van Schie, H. T., Hunnius, S., Vesper, C. & Bekkering, H. (2008). You'll never crawl alone:
Neurophysiological evidence for experience-dependent motor resonance in infancy. Neuroimage, 43,
808-814.
s.hunnius@donders.ru.nl
Conclusions & discussion
- In a human action context, 10-month-old infants show
an increase in prediction success over successive trials
which is not observed in the other contexts.
- Learning about categories in terms of human actions
enhances the generalization performance on a novel
category.
(b) Human Passive condition
Categorization
learning
- From early in life, infants learn about objects and
begin to organize them in terms of categories.
- Various studies demonstrate that how categories
are presented has an impact on how successful
infants are in acquiring new categories (Mandell &
Raijmakers, 2010) .
- Neurocognitive research, on the other hand, has
shown that the infants’ motor system gets activated
when observing others’ actions (van Elk, van Schie,
Hunnius & Bekkering, 2008) . This action simulation
mechanism is thought to play an important role for
cognitive processing (Iacoboni et al., 2005) .
-Yet, it is unknown whether action simulation
enhances learning, especially categorization
learning. We hypothesized that
presenting infants with categories in terms of
actions would facilitate their learning of novel
categories.
Action simulation facilitates
categorization learning in infants
Sabine Hunnius
1
, Marlene Meyer
1
, Dorothy J. Mandell
2
, & Harold Bekkering
1
1
Radboud University Nijmegen, The Netherlands
2
University of Amsterdam, The Netherlands
Method
- 3 between-subjects conditions
- The colour of the object (blue; red) was predictive of
which bucket (right; left) served as the target location.
- 14 learning trials (7 for each colour), followed by
6 generalization trials (3 per colour) with a differently
shaped object
Fig. 1: Infants watched the
stimulus movies on a Tobii
eye-tracker.
Fig. 2:
The Human Action, Non-Human Movement, and Self-Propelled condition
- Eye-tracking study with
10-month-old infants (N=113)
- To assess categorization
learning, infants’ anticipatory
eye-movements to the target
locations were measured.
Fig. 3: Percentage of
correct target
anticipations during
the first and second
half of the learning
phase.
Learning phase:
- Only the infants in the Human Action condition tended
to anticipate more often to the correct target in the
second compared to the first half of the training trials.
- A GEE with the factors Condition and First/Second
half of the learning phase revealed a significant
interaction effect (
2
(2)= 7.95, p=.02).
Generalization phase:
-During the generalization trials, only infants in the
human action condition tended to anticipate more
frequently than chance to the correct target
(t(17)=1.59, p=.065).
Fig. 4: Percentage of
correct anticipations
during the generalization
phase.
Human Action Non-Human Movement Self-Propelled
Human Action Non-Human Movement Self-Propelled
%
c
o
rr
ec
t
an
ti
ci
p
ati
o
n
s
%
c
o
rr
ec
t
an
ti
ci
p
ati
o
n
s
Results
References
Iacoboni, M., Molnar-Szakacs, I., Gallese, V., Buccino, G., Mazziotta, J.C., Rizzolatti, G. (2005).
Grasping the intentions of others with one's own mirror neuron system. Plos Biology, 3, 529-535.
Mandell, D., & Raijmakers, M.E.J. (2010). Is more always better? Infants’ performance on a
simuktaneous two-category learning task. Poster presented at XVIIth Biennial International Conference
on Infant Studies, Baltimore, Maryland, USA.
van Elk, M., van Schie, H. T., Hunnius, S., Vesper, C. & Bekkering, H. (2008). You'll never crawl alone:
Neurophysiological evidence for experience-dependent motor resonance in infancy. Neuroimage, 43,
808-814.
s.hunnius@donders.ru.nl
Conclusions & discussion
- In a human action context, 10-month-old infants show
an increase in prediction success over successive trials
which is not observed in the other contexts.
- Learning about categories in terms of human actions
enhances the generalization performance on a novel
category.
(c) Self Propelled condition
Figure 2. The three conditions of the experiment
screen shot of a trial, and the three Areas of Interest that were used. The following coordinates were used:
AOI1 : x: 280 − 505, y: 498 − 718. AOI2 : x: 775 − 1000, y: 498 − 718. AOI3 : x: 540 − 740, y: 250 − 718.
Figure 3. Areas of interest.
3The x-axis of the screen reached from 0 (left) to 1280 (right),
4 ABE HUIJBERS Preprocessing. Missing values in gaze measurements
were filled using linear interpolation, but only for short se-quences of missing values (<167ms). Infants needed to fix-ate an Area of Interest for at least 150ms to be deemed a valid fixation. If less than two fixations within the screen were recorded during a trial, the trial was deemed invalid and discarded.
Anticipation. Any trial could result in one of three pos-sible outcomes. For an infant to show a correct anticipation, it had to anticipate the direction of the ball as follows: (a) look at the ball to discover its color, and (b) look at the cor-rect cup thereafter. Therefore, for an infant to have a corcor-rect anticipation, it had to respond as follows: (a) a fixation in AOI 3 at least once during the Lifting Phase, followed by (b) a fixation at the correct cup -AOI 1 for red trials, AOI 2 for blue trials- at least once during the Lifting Phase, and after it had fixated AOI 3. The same logic was applied to define an incorrect anticipation: a fixation in AOI 3, followed by an incorrect fixation at either AOI 1 or AOI 2 during the Lifting Phase, was defined as an incorrect anticipation. Finally, any trial in which infants did not fixate AOI 3 and/or did not fix-ate AOI 1 or AOI 2 during the Lifting Phase, was defined as a no anticipation trial.
Hidden Markov Models. We used Hidden Markov Models (HMMs) to investigate infant categorization learn-ing. A HMM always consists of one or more states, with each of these states producing a typical type of behaviour. There are three types of parameters in the model: initial prob-abilities, transition probabilities and response probabilities. Initial probabilitiesare the probabilities of being in certain states at the first trial. Transition probabilities are the proba-bilities of switching from one state to the other (or staying in the current state). Response probabilities are the probabili-ties of a particular response (e.g., a correct response) in each of the states.
By constraining specific model parameters, it is possible to test hypotheses about the population. We used this method to find the best model for our data, hoping to develop a model that could explain infant categorization learning adequately.
Results
General Trial Statistics. There were 2580 trials in to-tal. Of those, 410 (15.9%) were excluded because the infants had not fixated the screen during the trial. The remaining 2170 trials were successfully completed and therefore used in analysis.
The trials that were successfully completed consisted of 389 (17.9%) correct anticipations, 329 (15.2%) incorrect an-ticipations, and in the remaining 1452 (66.9%) trials the in-fants did not anticipate. Figure 4 shows the habituation over trials. Almost all infants completed the first trial (N=122), but fewer also completed the last trial (N=90). This suggests
some habituation over time, although most infants still par-ticipated in the later trials.
Figure 4. The number of infants that completed each of the trials. The dotted red lines represent the attention getters be-tween trials.
Defining Correct Anticipations
Because the Lifting phase was quite long, an infant could fixate both cups after seeing the color of the ball. When an infant looked at both cups after seeing the color of the ball, it was uncertain whether it did, or did not, categorize correctly. We derived four different definitions of a correct anticipation: (i) the first anticipatory look was to the correct side, (ii) the lastanticipatory look was to the correct side, (iii) the pro-portionof correct anticipatory looking time as a continuous variable ranging from zero to one, and (iv) the proportion of correct anticipatory looking time as a binary variable: either 0 or 1.
Figure 5 shows these different variables and their results as the proportion correct over all infants. For practical reasons, the percentages shown in Figure 5 are the percentages cor-rect responses over the set of trials in which the infants antic-ipated at all. Only the variable last anticipatory look showed an increase in proportion correct over trials, and had a pro-portion correct that was significantly higher than 50% in the Generalisation Phase ( ˆp= 0.626, z = 4.406, p < 0.001). Our findings corroborated the findings in Mandell & Raijmak-ers (2012), of the last anticipation being the best measure of infant anticipation. Subsequently, infants’ last anticipations were used as the dependent variable in all analyses.
Confirmatory Analysis: Incremental Learning vs. Dis-crete Learning
Two modes of learning can be expected of infants cate-gorization learning. The first is that the infant gradually be-comes more aware of the underlying mechanism. If this were the case, we would expect infants having a larger proportion of correct answers in later trials than in earlier trials, with this proportion gradually building per trial. This learning process was named Incremental Learning.
The second is that the infant abruptly becomes aware of the underlying mechanism, via some sort of "Eureka"-moment. If this were the case, we would expect infants to
Figure 5. The proportions correct with 95% Confidence Intervals, over the different phases of the experiment, for all four of the possible dependent variables. The proportions are the proportions of correct responses over trials in which the infants demonstrated anticipatory looks. Only "Last Anticipation Correct" shows the expected growth in proportion correct over time. have some (relatively low) proportion of correct answers in
earlier trials, and another (relatively high) proportion of cor-rect answers in later trials. This learning process was named Discrete Learning.
To find whether infant categorization learning is an in-cremental or discrete learning process, we fit two statistical models to our data. For the incremental learning process, we used a 1-state Hidden Markov Model4. For the discrete learning process, we used a 2-state Hidden Markov Model. The results for the two models are in Table 1.
Table 1
The statistical fit of the incremental (1-state) and discrete (2-state) model in terms of maximized log likelihood, number of free parameters, Akaike Information Criterion and Bayesian Information Criterion.
Model Log(L) np AIC BIC 1-state -1868.720 4 3745.439 3768.169 2-state -1761.520 7 3537.032 3576.809
In column Log(L) is the natural logarithm of the likeli-hood of the model, which represents the relative fit of the model. Lower values represent poorer fit. However, we can compare the log likelihood only for hierarchically nested models. Hierarchically nested models are models in which one of the models is a special case of the other model. In other words, models are only hierarchically nested when they are equivalent, but with different constraints. Models with a different number of states clearly are not hierarchically nested, and models 1S and 2S can therefore not be com-pared through their log likelihood. In column np are the number of freely estimated parameters in the model, which are all parameters (initial probabilities, transition probabil-ities and response probabilprobabil-ities) that are unconstrained. A model with more freely estimated parameters is less parsi-monious. Therefore, models with fewer freely estimated pa-rameters are preferred to models with more freely estimated parameters, unless their fit is significantly poorer (Hooper, Coughlan, & Mullen, 2008).
AIC and BIC are the Akaike Information Criterion and Bayesian Information Criterion (Schwarz, 1978). Both are
relative measures of model fit. A lower AIC or BIC indicates better fit of the model. Although these measures cannot be used to assert the fit of an individual model, they can be used to compare the fit of models - even when they are not hierar-chically nested. The BIC penalizes models more heavily for having freely estimated parameters, than the AIC does.
We clearly see a better fit to the data for the discrete model as compared to the incremental model. Both AIC and BIC are lower for the discrete model, indicating a better fit. This suggests that infant categorization learning is better ex-plained as a process of discrete learning, than as a process of incremental learning. In other words, we need at least two states to adequately explain our data. Implications of these results will be tended to in the Discussion section.
Exploratory Analysis: Model Selection
The next objective was to find a learning model that suited the data well, to explain infant categorization learning in more detail. We tested 2- and 3-state models, and applied specific constraints to some of them. Models with 4 or 5 states were also fit but immediately rejected, because they theur AIC and BIC statistics were much higher than those of the simpler models; 4- and 5-state models will therefore not be discussed.
We tested 3 unconstrained models (1S, 2S, 3S) and a the-oretical All or None (AoN) model. Furthermore, we tested whether the equality constraint of a guessing state is tenable in each of these models. All models with the extension -1G are models with one guessing state. A short explana-tion for each of the fitted models is provided, and a graphical overview of the unconstrained models and the AoN model can be found in Figure 6.
Model 1S. This model is the 1-state Model. It has one state, in which participants respond at a specific level of proportion correct responses. This level may vary over trials (i.e., participants may get better and therefore develop a higher proportion of correct responses)5. Model 1S is depicted in Figure 6a.
4this model is actually equivalent to a Logistic Regression
model.
5This model has also been named the Incremental Model, see
6 ABE HUIJBERS
(a) The 1S Model. (b) The 2S Model.
(c) The AoN Model. (d) The 3S Model. Figure 6. The 4 different base-models.
Model 2S. This model is the 2-state Model. It has two states, but no constraints were applied. Each state has its own probability of a correct response, an incorrect response, or no response. Participants can start in either state, and switch freely between states. Each of the initial, transition and response probabilities are estimated freely. Model 2S is depicted in Figure 6b.
Model 2S-1G. This model is the 2-state model, but with a specific equality constraint: in one of the states, the prob-ability of a correct response is equal to the probprob-ability of an incorrect response (i.e., one of the two states is a guessing state in which participants respond at chance level). Apart from this equality constraint, each of the initial, transition and response probabilities are estimated freely.
Model 2S-2G. This model is the 2-state model, but with two specific equality constraints: in both states, the probability of a correct response is equal to the probability of an incorrect response (i.e., both states are guessing states in which participants respond at chance level). Apart from this equality constraint, each of the initial, transition and response probabilities are estimated freely.
Model AoN. This model is the All or None Model (Wickens, 1982), as depicted in Figure 6c. It is a 2-state model with specific constrains. In the AoN model it is
as-sumed that all participants start in a guessing state in which they respond at chance level. They will remain in this state until they switch to a learned state, in which they give cor-rect responses (nearly) all the time. After they have made the switch from the guessing state to the learned state, they will not return6. The probability of going from the learned state to the guessing state is zero, as is the probability of starting in the learned state. Therefore, the probability of staying in the learned state is 1. The remaining parameters are estimated freely. Model AoN is depicted in Figure 6c.
Model AoN-1G. This model is the All or None Model, but with an extra equality constraint: in the guessing state, the probability of a correct response is equal to the probability of an incorrect response (there is a guessing state). All other properties of the model are the same as in the AoN model.
Model 3S. This model is the 3-state model. It has three states, but no constraints were applied. Each state has its own probability of a correct response, an incorrect response, or no response. Participants can start in any of the three states, and switch freely between states. Each of the initial, transition and response probabilities are estimated freely. Model 3S is depicted in Figure 6d.
Model 3S-1G. This model is the 3-state model, but with a specific equality constraint: in one of the states, the probability of a correct response is equal to the probability of an incorrect response (i.e., one of the three states is a guessing state). All other parameters are estimated freely.
Model 3S-2G. This model is the 3-state model, but with two specific equality constraints: in two of the states, the probability of a correct response is equal to the probability of an incorrect response (i.e., two of the three states are guessing states). All other parameters are estimated freely.
Model 3S-3G. This model is the 3-state model, but with three specific equality constraints: in all three states, the probability of a correct response is equal to the probability of an incorrect response (i.e., all of the three states are guessing states). All other parameters are estimated freely.
The results for these models are in Table 2. The columns Log(L), np, AIC and BIC were explained earlier. The columns Delta Log(L), Delta d.f. and p-value depict a log likelihood difference test between two hierarchically nested models (Wickens, 1982). A significant log likelihood dif-ference test suggests that the more parsimonious model has significantly poorer fit. The preferred models based on the log likelihood difference tests are printed bold.
The 1S model was rejected earlier due to its lack of fit compared to the 2S model. Both AoN models were rejected for the same reasons, because it appeared that the constraints of the AoN models were not tenable (i.e., they produced a significantly poorer fit). Models 2S and 3S were rejected as well, because the constraints of models 2S-1G and 3S-1G were tenable: models 2S-1G and 3S-1G were more parsimo-nious yet had seemingly equivalent fit. Models 2S-1G and
3S-1G were the two preferred models.
A comparison between the two models showed that model 2S-1G had the better BIC, while model 3S-1G had the better AIC. It was left undecided whether one of the two is better than the other. However, model 2S-1G was preferred by the authors because it was less complex. The model that was preferred to describe infant categorization learning, was a 2-state model with one guessing 2-state. Figure 7 depicts this model with all its fitted parameters. Model 2S-1G explains
Figure 7. The fitted 2S-1G model. P(Correct) is equal to P(Incorrect) in the first state due to the equality constraint. infant categorization learning as a process of two states, in which one is a guessing state. In the fitted model, the other state had a higher probability of correct responses than in-correct responses, but was mostly signified by its high pro-portion of non-anticipatory responses (.806). Infants tended to stay in this last state between trials with probability .974. Perhaps this state represents a learned state: the infants had Table 2
The statistical fit of the fitted models in terms of maximized log likelihood, number of free parameters, Akaike Information Criterion and Bayesian Information Criterion. For the constrained models, a log likelihood difference test can be found in the columns∆ log(L), ∆ d.f. and p-value, in which the constrained model is compared to the unconstrained version of the model. Models which are preferred to their less parsimonious versions, based on the log likelihood difference test, are printed bold.
Model log(L) np AIC BIC ∆ log(L) ∆ d.f. p-value 1S -1868.720 4 3745.439 3768.169 2S -1761.516 7 3537.032 3576.809 2S-1G -1761.576 6 3535.153 3569.248 0.121 1 .728 2S-2G -1875.189 5 3760.378 3788.791 227.347 2 .000 AoN -1819.437 5 3648.874 3677.286 115.842 2 .000 AoN-1G -1820.809 4 3649.618 3672.347 118.586 3 .000 3S -1743.625 14 3521.720 3620.953 3S-1G -1744.126 13 3514.251 3588.123 1.001 1 .317 3S-2G -1750.307 12 3524.615 3592.805 13.365 2 .001 3S-3G -1875.189 11 3772.378 3834.886 263.128 3 .000
8 ABE HUIJBERS already mastered the task, and ceased to respond to the
stim-uli after repeated presentations. Only 2 of the 129 infants showed "unlearning" behaviour in the generalisation trials, i.e., went from state 2 to state 1 after the first 14 trials had been completed. This pattern, combined with the decreas-ing number of completed trials depicted in Figure 4 and the high proportion of non-anticipatory responses, does paint a picture of infants that ceased to respond or anticipate, while in state 2. This led us to label state 2 in the 2S-1G model the learned state.
Analysis of Conditions
Model 2S-1G was fitted for each of the three stimulus conditions in the experiment, to detect whether infants would show different learning curves in different conditions. Table 3 shows the fit of the model to each of the conditions. We then compared the log likelihood of model 2S-1G for all conditions combined with the summed log likelihood of the three conditions. The likelihood ratio test suggested that the model parameters differ between the three conditions (∆LL = 23.413, ∆ df = 12, p = .000).
Conditions and States. One interesting question how infants switched between states, and how this differed for the conditions. Infants spent most of the trials in the learned state, with percentages of 93.6% for the Human Active, 62.1% for the Human Passive, and 71.0% for the Self Propelled condition.
Human Active Condition. As is to be expected, the initial probability of state 1 is higher (.679) than the initial probability of state 2 (.321) in the Human Active condition. The states are very stable, with probabilities of staying in a state between trials of .971 for state 1 and .979 for state 2. The Human Active condition has the highest probability for a correct answer in state 2 (.236) and has high probability for no anticipation in that state as well (.606), which matches the properties of the 2S-1G model. However, there is an even higher probability for no anticipation in state 1 (.904), which was not expected for model 2S-1G. Although state 2 appears to represent the learned state (higher probability for correct than incorrect responses, and a high probability of no anticipation), the Human Active condition has a surprisingly high probability for no anticipation in the guessing state. The learned state occurred in 93.6% of the trials, indicating a high probability for learning in the Human Active condition.
Human Passive Condition. In the Human Passive con-dition, the initial probability of state 2 is higher (.710) than the initial probability of state 1 (.290). The stability of the states is more or less the same as in the Human Active Con-dition, with very stable states (.949 and .957). Although state 1 seems to produce responses as expected, state 2 only has a marginally higher probability for correct responses than
Table 3
The statistical fit of the 2S-1G model in terms of maximized log likelihood, number of free parameters, Akaike Informa-tion Criterion and Bayesian InformaInforma-tion Criterion, for the three stimulus conditions.
Condition Log Lik np AIC BIC Human Active -480.044 6 972.088 999.6711 Human Passive -591.526 6 1195.052 1222.418 Self Propelled -666.593 6 1345.186 1372.744 Table 4
The parameters of the fitted 2S-1G model, for the three stim-ulus conditions: Human Active (HA), Human Passive (HP) and Self Propelled (SP).
Parameter HA HP SP Initial Probabilities State 1 (Guessing) .679 .290 .570 State 2 (Learned) .321 .710 .430 Transistion Probabilities P(S 1t|S 1t−1) .971 .949 .906 P(S 2t|S 1t−1) .029 .051 .094 P(S 2t|S 2t−1) .979 .957 .972 P(S 1t|S 2t−1) .021 .043 .028 Response Probabilities State 1 (Guessing) P(Correct) .048 .310 .394 P(Incorrect) .048 .310 .394 P(No anticipation) .904 .380 .212 State 2 (Learned) P(Correct) .236 .082 .126 P(Incorrect) .158 .080 .092 P(No anticipation) .606 .838 .782 State Occurrence State 1 (Guessing) .064 .379 .290 State 2 (Learned) .936 .621 .710
for incorrect responses (.082 and .080). Nevertheless, the learned state occurred in 62.1% of the trials, although the learned state may not be an appropriate name: the infants hardly appeared to have learned at all in the Human Passive condition.
Self Propelled Condition. In the Self Propelled condi-tion, the initial probability is a little higher for state 1 (.570) than for state 2 (.430). Both states are quite stable again, although the probability of switching from state 1 to state 2 is relatively high (.094). State 1 seems to produce responses
as expected, and so does state 2, even though the probabil-ity of a correct response (.126) is not much higher than the probability of an incorrect response (.092). Infants were in the learned state for 71.0% of the trials.
Discussion
The best variable to measure infants’ anticipatory looks was their last anticipation. The last anticipation uniquely showed a proportion of correct responses at chance level in the first series of trials, and a proportion above chance level in subsequent trials. This demonstrated the infants’ ability to learn the task at hand, albeit a moderate amount of learning. The highest percentage of correct responses was achieved in the first Generalisation phase, with a percentage of correct responses of 62.6%, compared to 37.4% incorrect responses. This percentage was lower than we had anticipated: even when only considering the set of trials in which the infants anticipated at all, the probability of an incorrect response was still over 13.
Perhaps the categorization task was too difficult for the sample. One reason for the infants’ poor results may lie in the relatively short duration of the lifting phase, leaving little time for the infants to anticipate the action. Another reason may lie in the definition of a correct response that was used. One could argue that the lifting phase should have been extended to a short period after the ball had gone in either direction7. Different conclusions might have arisen from a task where the infants had more time to anticipate.
We concluded that infant categorization learning is best described as a discrete process rather than an incremental process, because the 2-state model fit the data better than the 1-state model. The 2- and 3-state models with a guessing state did not seem to differ much from each other in terms of statistical fit. The 2-state model was preferred as it is the more parsimonious representation, although one could also argue for a 3-state model as the preferred model. Our search of a suitable model was of an exploratory nature as we did not hypothesize any particular model. The reason that the 2-state model was chosen in this study was a case of Occam’s razor: in the case of equivalently proven hypotheses, the least complex one was kept. A 3-state model may prove to be more suitable, but we found no evidence to support this notion.
Model 2S-1G was deemed the best model through exploratory model selection. The parameter values of the fitted model suggested one guessing state and one learned state. We do not pose these two states to be the true underlying dynamic of infant categorization learning, rather did we consider this to be the best explanation of the response pattern that was found. Moreover, this type of modelling is a way to account for different learning speeds between infants.
The three different stimulus conditions that were used
showed mixed results in the 2S-1G model. Infants in the Human Active condition, which was considered by Hunnius et al. (2012) to accompany categorization learning best of all, did show a high proportion of correct responses in state 2, suggesting a steeper learning curve than in the other stimulus conditions. The proportion of non-anticipations was very high in state 1; a state in which infants were expected to be guessing at random. The Human Passive condition showed a very poor learning curve altogether, with infants demonstrating only a 0.2% higher probability of a correct response than an incorrect response in the learned state. The parameters of the model did seem to validate the model for this stimulus condition, with relatively clear guessing and learned states. What does seem to hold is that the Human Active condition shows the most promising learning curve, corroborating to the claim made by van Elk et al. (2008) that infants learn about action movements more effectively when they see another human performing those actions.
The multitude of non-anticipatory trials could not be interpreted as correct or incorrect responses, because it was uncertain whether it represented a lack of understanding of the task, or a lack of interest in the task. In finding the best dependent variable for our analyses we had to ignore the trials in which infants did not anticipate altogether. In fitting the different Hidden Markov Models, we did include those trials as a separate type of response, because they do signify some type of (non)behaviour and perhaps cognition. In doing so, we complicated interpretation of the fitted models. However, this seemed the best option for interpreting the response patterns that we found. An alternative for our method lies in examining reaction times, and defining a border between anticipations and reactions between the different modes in the distributions. Future research should entertain this method as a powerful possibility for determining a border between different response types.
This study does support the notion of infants showing a form of discrete learning, indicating at least some form of rational learning in young infants. Although practice does not make absolutely perfect in this particular case, it is remarkable to see that infant categorization learning appears to involve a step by step learning process.
Acknowledgements
The author wishes to thank Sabine Hunnius for providing the dataset and Maartje Raijmakers, Caroline Junge, Ingmar Visser and Daan van Renswoude for their comments and support in writing this internship manuscript.
7For example, extending the lifting phase by 200 ms, because
the time needed to plan a saccade is estimated to be 200ms (Becker, 1988).
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