2.3.1. Participants

Participants were recruited from a daycare in Mexico City, Mexico. Adult participants were all native speakers of Spanish working at the school. Children were typically-developing learners of Spanish as a first language. A total of forty-seven adults (forty-three women) and ninety-seven children (fifty-seven girls) ages 1;11 to 6;9 completed the task. One adult (n = 1) was excluded from analysis due to experimenter error (only half of the items were administered). Children were divided into three age groups for analysis: thirty-three children age three and under (≤ 3: 1;11–3;10; M = 3;3), thirty-five children age four (4: 4;0–4;11; M = 4;5), and twenty-nine children age five and older (≥5: 5;0–6;9; M = 5;8). Individual responses were excluded if the participant failed to choose one of the pictures, or if the experimenter read the prompt incorrectly. The following child responses were excluded for this reason: twenty-nine (of 1,225) age ≤3; twenty-two (of 1,308) age 4; twenty-one (of 1,081) age ≥5.

2.3.2. Design and procedure

Experimental stimuli were created by fully crossing the three cues of pronoun form (null, overt), discourse connective (temporal y después , causal porque ), and agreement number morphology (aligns with subject, aligns with nonsubject). That is, in addition to the condition in 4a, where all three cues favor the subject interpretation, and the three conditions in 4b–d, where two of three cues favor the subject, there were also three conditions where two of three cues favored the nonsubject, and one condition where all three cues favored the nonsubject.Footnote ⁷ These eight conditions were presented in both singular and plural forms, for a total of sixteen conditions.

Eight distinct experimental items were created by choosing pairs of verbs that were easily depicted and likely to be known by children under age three (e.g. sigue – sale : ‘follow’–‘go out’). In addition, as mentioned previously, we followed Johnson et al. Reference Johnson, de Villiers and Seymour2005 in masking agreement morphology information that could come from the singular -∅ or plural -s marking on the subject of the second clause (i.e. ella(s) ); we accomplished the masking by using /s/-initial predicates (e.g. sale ). This way, in overt pronoun conditions, number marking on the pronoun subject ella(s) ‘she’/‘they (f)’ would not provide participants with any additional number cues (e.g. ‘ ella sal… ’ is difficult to distinguish from ‘ ellas sal… ’). Instead, participants would need to rely on the agreement cue provided by the verbal agreement marker (e.g. sale(n) ).

Fillers were created by choosing an additional sixteen verb pairs, replacing the pronoun with the definite DP los niños ‘the boys’, and using either ‘the girls’ or ‘the teacher’ as a competing antecedent. The same sixteen fillers were presented in the same order across all versions of the experiment. See Appendix B for more details of the design and sample items.

Participants were randomly assigned to one of three versions of the experiment, with sixteen experimental trials and sixteen filler trials. Each version tested a different cue type (pronoun form, connective, morphology) by systematically aligning and pitting that cue against the other two. For example, participants assigned to the ‘morphology’ version were exposed to eight ‘congruent’ trials in which all cues favored the same antecedent (as in 4a) and eight ‘incongruent’ trials in which agreement morphology favored the opposite antecedent from that which the pronoun form and connective favored (as in 4d). The same protocol was used for participants assigned to the ‘pronoun form’ and ‘connective’ versions (pronoun form: 4a vs. 4b; connective: 4a vs. 4c). Thus, cue type was between-subject, while congruency (i.e. congruent, incongruent) was within-subject.

Within each version of the experiment, fillers and experimental items were presented in a fixed order that formed a coherent narrative arc about two consecutive days at a school with a teacher, a group of girls, and a group of boys. Each experimental item was presented twice, once in its congruent form and once in its incongruent form, which were randomly assigned to appear on either ‘day 1’ or ‘day 2’. Each pair of congruent/incongruent experimental items was randomly assigned to appear in the plural or the singular. The total experiment (three practice items, sixteen experimental items, and sixteen fillers) took about ten minutes to administer.

The task was administered on a 13-inch MacBook Air using Psychopy version 3.0.0b11 (Peirce et al. Reference Peirce, Gray, Simpson, MacAskill, Höchenberger, Sogo, Kastman and Lindeløv2019). The task began with an introductory screen introducing the characters (a teacher, three identically dressed girls, three identically dressed boys). Next, participants were given an explanation of the task and three practice trials. During each practice and experimental trial, participants saw an illustration of the first clause as it was read out loud by the experimenter (child participants) or a recording of her voice was played over headphones (adult participants), as in Screen 1 of Figure 1. Then, participants saw a blank screen and heard the second clause. Then, two illustrations appeared, one corresponding to the subject interpretation and one to the object interpretation (as in Screen 2 of Figure 1); participants chose the picture that matched their own interpretation by pointing (child participants) or by pressing the ‘4’ or ‘9’ key (adult participants).

Pictures were randomly placed on the left or right side of the screen. Upon completion, children received a piece of candy and adults received the equivalent of US$10.

2.4.1. Practice and filler trials

For the practice trials, we used a one-sided t-test to evaluate whether the correct response rate in each age group was above chance and found that indeed it was (≤3: M = 0.68, t(32) = 4.37, p < 0.001; 4: M = 0.66, t(34) = 3.60, p < 0.01; ≥5: M = 0.71, t(28) = 4.12, p < 0.001; adults: M = 0.89, t(46) = 17.09, p < 0.001). For the filler trials, we again used a one-sided t-test to evaluate whether the correct response rate in each age group was above chance and found that indeed it was (≤3: M = 0.69, t(32) = 4.45, p < 0.001; 4: M = 0.67, t(34) = 3.34, p < 0.01; ≥5: M = 0.85, t(28) = 7.727, p < 0.001; adults: M = 0.94, t(46) = 19.71, p < 0.001).

Because participant performance on both the practice and filler trials was above chance, this suggested that the participants in all age groups understood the task and were capable of completing it as intended. However, because the mean score on the filler trials for children under age five seemed much lower (0.67–0.69 vs. 0.85), we report and discuss the child results only for children age five and older (≥5).

2.4.2. Results: experimental trials

Figure 2 shows the rate of subject antecedent responses (i.e. interpreting the pronoun to refer to the potential antecedent in the subject position) produced for each combination of cues (pronoun form, discourse connective, and agreement morphology) for both children and adults.

Figure 2.

Rate of subject antecedent responses by children (≥5 years old) and adults interpreting Spanish pronouns in utterances where the pronoun form favors the subject (null) or the nonsubject (overt), the discourse connective favors the subject (y después) or the nonsubject (porque), and agreement morphology favors the subject or the nonsubject. Error bars represent 95% confidence intervals.

Figure 2. Long description

The y-axis represents the Proportion subject antecedent responses from 0.00 to 1.00. The x-axis is Antecedent favored by morphology with categories subject and non-subject. A horizontal dashed line marks the 0.50 chance level. Panels are organized by age group (columns: children greater than or equal to 5 years old, N=29; adults, N=47) and pronoun type (rows: null favors subject; overt favors non-subject).

Legend: Dark teal bars represent después (favors subject) and light teal bars represent porque (favors non-subject).

* Top-Left Panel (Children, Null Pronoun): For subject morphology, both connectives are above 0.50. For non-subject morphology, después is near 0.45 and porque is lowest at approximately 0.23.

* Top-Right Panel (Adults, Null Pronoun): For subject morphology, responses are high (después approx 0.92, porque approx 0.78). For non-subject morphology, después is near 0.42 and porque is approx 0.15.

* Bottom-Left Panel (Children, Overt Pronoun): For subject morphology, responses are above 0.50. For non-subject morphology, both connectives drop below 0.30.

* Bottom-Right Panel (Adults, Overt Pronoun): For subject morphology, responses remain high (después approx 0.84, porque approx 0.69). For non-subject morphology, both connectives drop significantly to near 0.10.

Error bars indicate 95 percent C I s.

In general, we can see that children appear to interpret pronouns differently from adults, even when all three cues favor the same antecedent. For instance, children have fewer subject interpretations in conditions with subject-favoring cues (the null pronoun form, the temporal connective y después , and agreement morphology that aligns with the subject). Also, child responses seem much less extreme than adult responses: when adults strongly prefer the subject antecedent, children prefer the subject less strongly; when adults strongly prefer the nonsubject antecedent, children prefer the nonsubject less strongly.

Interestingly, adult responses indicate that agreement morphology information is perhaps not as categorical at determining adult interpretations as previously thought. That is, there are some contexts where adults seem to ignore the information coming from agreement morphology and interpret the pronoun in a way that conflicts with this information. For instance, when agreement morphology favors the nonsubject, but the pronoun form and the discourse connective favor the subject (form = null, connective = y despuès ), adults interpret the pronoun as the subject 41.2% of the time (SE: 0.06), in spite of the agreement morphology. This adult interpretation behavior underscores how adult behavior is not always accurate behavior (in this context, ‘accurate’ behavior would always have the pronoun interpreted as the nonsubject, to align with the agreement morphology). While this adult interpretation behavior may be surprising at first glance, agreement attraction errors—where agreement morphology information appears to be disregarded—are well documented in adults (e.g. Acuña-Fariña et al. Reference Acuña-Fariña, Meseguer and Carreiras2014, Lago et al. Reference Lago, Shalom, Sigman, Lau and Phillips2015, Wagers et al. Reference Wagers, Lau and Phillips2009). This (mis)interpretation behavior occurs in certain contexts that provide conflicting cues to the agreement morphology information, just as the contexts here do. The question then is why inaccurate interpretation behavior occurs—that is, (as with children) it is unknown whether this behavior is caused by inaccurate representations of pronoun information, inaccurate deployment of accurate representations during the experiment, or both inaccurate representations and inaccurate deployment of those inaccurate representations.

To identify if each cue significantly affected the probability of interpreting the pronoun as referring to the subject antecedent (whatever the underlying reason), we analyzed the data using mixed-effects logistic regressions with the lme4 package in R (Bates, Mächler, et al. Reference Bates, Mächler, Bolker and Walker2015).Footnote ⁸ Following a data-driven approach that includes both fixed-effect and random-effect terms if they improve the model fit (Bates, Kliegl, et al. Reference Bates, Kliegl, Vasishth and Baayen2015, Matuschek et al. Reference Matuschek, Kliegl, Vasishth, Baayen and Bates2017, Sonderegger Reference Sonderegger2023), we selected the model structure in 5, which includes two-way interactions with age, by-participant and by-item random intercepts, and by-participant random slopes for agreement morphology (see Appendix C for details of the model selection process). Following Sonderegger Reference Sonderegger2023, all categorical predictors were coded using Helmert contrasts, which are centered and orthogonal. This coding allows main-effect coefficients to be interpreted as average effects across the levels of the other predictors in the model.

1. (5) Model with age interactions and by-participant random slopes for agreement morphology

$$ {\displaystyle \begin{array}{c}\mathrm{subj}\hbox{-} \mathrm{antecedent}\sim \mathrm{form}\ast \mathrm{age}+\mathrm{connective}\ast \mathrm{age}+\mathrm{morph}\ast \mathrm{age}+\left(1|\mathrm{item}\right)+\left(1+\mathrm{morph}|\mathrm{participant}\right)\end{array}} $$

The analysis in Table 1 suggests that the agreement morphology cue has the strongest effect (Agreement morph: β = 0.78, p < 0.001), which increases with age (Agreement morph × Age: β = 1.04, p < 0.001). In contrast, the other two cues either have a main effect that does not increase with age (connectives: Connective: β = 0.32, p = 0.025; Connective × Age: β = 0.28, p = 0.159) or no significant main effect, but one that does change with age (pronoun form: Form: β = −0.10, p = 0.504; Form × Age: β = 0.56, p = 0.006). To interpret the significant interactions with age for pronoun form and agreement morphology, we used the emmeans package (Lenth Reference Lenth2021) to conduct simple effects analyses. For each predictor showing a significant interaction with age, we tested (i) the effect of the predictor within each age group separately (simple effects), and (ii) whether these effects differed significantly between age groups (interaction contrasts).

Table 1.

Mixed-effects logistic regression results for model subj-antecedent ~ form * age + connective * age + morph * age + (1|item) + (1 + morph|participant), including β coefficients, standard error (SE), z-values, and p-values. Significant effects are reported at alpha level p < 0.05 *, p < 0.01 **, p < 0.001 ***. Note: N = 1,181 observations from seventy-six participants across eight items. Random-effects correlation between participant intercept and morphology slope = −0.18.

Table 1. Long description

The table is divided into two main sections: Fixed effects and Random effects.

Fixed effects section columns include beta, S E, z, and p values:

* (intercept): beta -0.33, S E 0.16, z -2.10, p 0.036.

* Form: beta -0.10, S E 0.14, z -0.67, p 0.504.

* Connective: beta 0.32, S E 0.14, z 2.24, p 0.025.

* Agreement morph: beta 0.78, S E 0.21, z 3.79, p less than 0.001.

* Age group: beta 0.32, S E 0.18, z 1.80, p 0.071.

* Form times Age: beta 0.56, S E 0.20, z 2.77, p 0.006.

* Connective times Age: beta 0.28, S E 0.20, z 1.41, p 0.159.

* Agreement morph times Age: beta 1.04, S E 0.27, z 3.84, p less than 0.001.

Random effects section columns include Variance and S D:

* Participant (intercept): Variance 0.09, S D 0.31.

* Participant: Morphology: Variance 0.77, S D 0.88.

* Item (intercept): Variance 0.07, S D 0.26.

Our analysis in Table 2 suggests that age moderated the cue effects of both pronoun form (β = 1.11, p = 0.006) and agreement morphology (β = 2.07, p < 0.001). Notably, only adults were sensitive to the null/overt pronoun form distinction (≥5 children: p = 0.504; adults: OR = 2.51, p = 0.001). In contrast, both children and adults were sensitive to agreement morphology (p < 0.001), but adults were markedly more sensitive (≥5 children: OR = 4.76; adults: OR = 37.70).

Table 2.

Simple effects and interaction contrasts from estimated marginal means analysis, reporting the estimate (Est.), standard error (SE), z-values, and p-values. Significant effects are reported at alpha level p < 0.01 ** or p < 0.001 ***. Note: Estimates on log-odds scale. Negative values indicate higher log-odds when predictor favors subject antecedent.

Table 2. Long description

The table contains six columns: Predictor, Contrast/Group, Estimate (Est.), Standard Error (S E), z-value, and p-value. It is divided into two main sections.

1. Pronoun form section:

- Simple effects (Null minus Overt): For Children, Est. 0.19, S E 0.29, z 0.67, p 0.504. For Adults, Est. minus 0.92, S E 0.28, z minus 3.25, p 0.001 (significant at 0.01 level).

- Interaction (Children vs. Adults): Est. 1.11, S E 0.40, z 2.77, p 0.006 (significant at 0.01 level).

2. Agreement morphology section:

- Simple effects (Non-subject-aligned minus Aligned): For Children, Est. minus 1.56, S E 0.41, z minus 3.79, p less than 0.001 (significant at 0.001 level). For Adults, Est. minus 3.63, S E 0.37, z minus 9.88, p less than 0.001 (significant at 0.001 level).

- Interaction (Children vs. Adults): Est. 2.07, S E 0.54, z 3.84, p less than 0.001 (significant at 0.001 level).

Note: Estimates are on a log-odds scale where negative values indicate higher log-odds favoring subject antecedent.

Taken together, our analysis suggests that child pronoun interpretations at age five are influenced by agreement morphology and discourse connectives, but not by pronoun form. To become adult-like, children’s pronoun interpretations must be influenced more strongly by agreement morphology, and must additionally be influenced by pronoun form.

3.3.1. Baseline Bayesian listener incorporating antecedent type

As described above, our baseline Bayesian listener has accurate representations of both antecedent probability information and pronoun probability information (pronoun form, discourse connective, and agreement morphology) on the basis of its input. The modeled listener is also able to integrate this probability information accurately, using Bayesian inference.

(6) $$ {\displaystyle \begin{array}{l}\hskip1.7pc p\left({\alpha}_{num},{\alpha}_{subj?}|\mathrm{form},\mathrm{con},\mathrm{mor}\right)\propto p\left({\alpha}_{num},{\alpha}_{subj?}\right)\\ {}\hskip25.7em \cdot p\left(\mathrm{form}|{\alpha}_{num},{\alpha}_{subj?}\right)\\ {}\hskip25.7em \cdot p\left(\mathrm{con}|{\alpha}_{num},{\alpha}_{subj?}\right)\\ {}\hskip25.6em \cdot p\left(\mathrm{mor}|{\alpha}_{num},{\alpha}_{subj?}\right)\end{array}} $$

Because priors and likelihoods vary depending on the number and position of the antecedent under consideration, equation 6 explicitly indicates that the precise value of the prior and each likelihood is different for different antecedent types. That is, the number of the antecedent is either singular or plural (α _num ∈ {sg, pl}), and the position of the antecedent is either the subject position or some nonsubject position, like the object (α _subj? ∈ {subj, ¬subj}). For instance, a singular subject antecedent (α _{subj. sg} ) would have α _num = sg and α _subj? = subj. A plural object antecedent (α_{¬subj. pl} ) would have α _num = pl and α _subj? = ¬subj.

Likewise, both the priors and the likelihoods depend on the values of these two variables; this accords with the probabilities from our corpus analysis in Table 3, which also varied by these properties of the antecedent. The posterior probability of a specific antecedent in a given experimental context can now be calculated by using the appropriate priors and likelihoods from Table 3. For example, to calculate the posterior for a potential singular antecedent in the subject position, the probabilities from the first row of Table 3 can be used (subj, sg).

3.3.2. Bayesian listener with inaccurate representations

Our listener implementation with inaccurate representations involves the listener calculating posterior probabilities the same way, but relying on inaccurate probability representations to do so. These probability representations correspond to the modeled listener’s priors and likelihoods: priors reflect the baseline probability of antecedents with different properties (e.g. singular antecedents in subject position); likelihoods reflect the probability of a particular pronoun cue value (e.g. a null form or singular morphology), given an antecedent with certain properties (e.g. a singular subject antecedent).

Inaccurate representations by incorporating noise. We use the same approach for implementing inaccurate representations of the probabilities corresponding to both the priors and likelihoods, which involves incorporating noise into these probability representations. The modeled listener uses the softmax function (e ^{σln(probability)} = probability ^σ), where σ serves as a ‘contrast’ parameter. When σ = 1, probabilities are kept the same (i.e. equal to the values in Table 3); when σ < 1, probabilities are made more uniform (i.e. relative contrasts are decreased, and so smoothed away); when σ > 1, probabilities are made more extreme (i.e. relative contrasts are increased, and so sharpened). Thus, our modeled learner can decrease the contrast between relative probabilities (e.g. using σ = 0.5 to smooth 0.324 vs. 0.676 into 0.409 vs. 0.591); it can also increase the contrast between relative probabilities (e.g. using σ = 2 to sharpen 0.324 vs. 0.676 into 0.187 vs. 0.813).

We note that the softmax function is a standard component of models of human decision-making tasks, including language tasks (e.g. Frank & Goodman Reference Frank and Goodman2012, Goodman & Stuhlmüller Reference Goodman and Stuhlmüller2013, and Scontras & Goodman Reference Scontras and Goodman2017). Here, we investigate contrast parameter values 0.0 ≤ σ ≤ 4.0.Footnote ¹⁴

Inaccurate priors. The prior has a single contrast parameter value (σ_α), which holds for all antecedents (i.e. σ_α is the same for all α _num and α _subj? values). This represents the same noise level for the prior on potential antecedents, irrespective of the number of the potential antecedent or its position.Footnote ¹⁵

Inaccurate likelihoods. Each pronoun cue has its own contrast parameter value (i.e. σ _form, σ _con, σ _mor), representing (potentially) different noise levels associated with each cue representation.

Implementing inaccurate representations. Equation 7 shows how the modeled listener implements potentially inaccurate priors and likelihoods, using the different σ contrast parameters.

(7) $$ {\displaystyle \begin{array}{l}\hskip1.5pc {p}_{\unicode{x03C3}}\left({\alpha}_{num},{\alpha}_{subj?}|\mathrm{form},\mathrm{con},\mathrm{mor}\right)\propto p{\left({\alpha}_{num},{\alpha}_{subj?}\right)}^{\unicode{x03C3}_{\alpha }}\\ {}\hskip26.2em \cdot p{\left(\mathrm{form}|{\alpha}_{num},{\alpha}_{subj?}\right)}^{\unicode{x03C3}_{form}}\\ {}\hskip26.2em \cdot p{\left(\mathrm{con}|{\alpha}_{num},{\alpha}_{subj?}\right)}^{\unicode{x03C3}_{con}}\\ {}\hskip26.2em \cdot p{\left(\mathrm{mor}|{\alpha}_{num},{\alpha}_{subj?}\right)}^{\unicode{x03C3}_{mor}}\end{array}} $$

We can then determine the contrast parameter values that allow the modeled Bayesian listener to best match the child and adult behavior from our experiment. By doing so, we can get a sense of if and how the true probability representations would need to be distorted in order for this modeled listener to reproduce the behavior we observe. We note that if the best-fitting value is σ = 1 for any information type, then this implies that the accurate representation for that probability information (prior or likelihood) was the best fit, rather than a distorted representation. So this implementation of the inaccurate representations listener allows for the possibility of accurate probability representations. More generally, we can also determine if a better fit occurs when relative probability differences are decreased (σ < 1), increased (σ > 1), or left undistorted (σ = 1). This would then correspond to whether listeners smooth, sharpen, or transparently use the relative probability differences from their input.

3.3.3. Bayesian listener with inaccurate deployment

Inaccurate deployment by selective dropout. We use the same approach for implementing inaccurate deployment of the represented probabilities for both the priors and likelihoods, which involves selectively dropping out the information from these probability representations. More specifically, our implementation of a listener with inaccurate deployment involves the listener having accurate probability representations, but not always using them. That is, when the listener calculates the posterior probabilities, the listener has available accurate representations of the probabilities for the priors and likelihoods, but simply ‘misses’ the information from one or more of these probability representations in the moment of a particular experimental item.

Inaccurate deployment of priors. When the listener calculates the posterior probabilities, the listener has available an accurate probability representation of possible antecedents with the corresponding number and position information (i.e. the prior). However, as mentioned above, the listener simply ‘misses’ the probability information from the prior in the moment of a particular experimental item.

For example, a listener with inaccurate deployment like this might have an accurate representation of how probable it is to have a singular antecedent in subject position (i.e. an accurate prior p(α _num = sg, α _subj? = subj) = 0.362). Yet when calculating the posterior, this probability representation is unavailable; in this case, the listener defaults to an uninformative prior, which is a uniform distribution over the possible antecedents (i.e. if there are two possible antecedents, then p(α _num, α _subj?) = 0.5). So, the posterior calculation proceeds without that informative prior (which we indicate with p _unif ). An example is shown in equation 8 when the listener uses all three cues of pronoun form, discourse connective, and agreement morphology, but does not use the prior; p(unif) represents a uniform (i.e. uninformative) distribution over possible antecedents.

(8) $$ {\displaystyle \begin{array}{l}\hskip2.1pc {p}_{\mathrm{unif}}\left({\alpha}_{num},{\alpha}_{subj?}|\mathrm{form},\mathrm{con},\mathrm{mor}\right)\propto p\left(\mathrm{unif}\right)\\ {}\hskip28.5em \cdot p\left(\mathrm{form}|{\alpha}_{num},{\alpha}_{subj?}\right)\\ {}\hskip28.5em \cdot p\left(\mathrm{con}|{\alpha}_{num},{\alpha}_{subj?}\right)\\ {}\hskip28.5em \cdot p\left(\mathrm{mor}|{\alpha}_{num},{\alpha}_{subj?}\right)\end{array}} $$

In this way, the accurate prior information drops out and is effectively ignored by the listener for that particular experimental item—even though the modeled listener has an accurate representation of this prior information (and may even use it for other experimental items). Thus, inaccurate deployment of the prior is implemented via a selective dropout of the information from the prior. This means the listener’s posterior probability relies solely on the probability information coming from the pronoun form, the discourse connective, and the agreement morphology, represented by the likelihoods.

Inaccurate deployment of likelihoods. When the listener calculates the posterior probabilities, the listener has available accurate probability representations (i.e. likelihoods) for the different cues. However, as before, the listener simply ‘misses’ the probability information from one or more cues in the moment of a particular experimental item.

For example, a listener with inaccurate deployment like this might have an accurate probability representation of the agreement morphology cue (i.e. an accurate likelihood p(mor | α _num, α _subj?)), but be unable to use it when calculating the posterior. That is, the posterior calculation proceeds without that cue, as in equation 9, where only the probability information from the pronoun form and the discourse connective are used, along with the probability information from the prior.

(9) $$ {\displaystyle \begin{array}{l}\hskip1.2pc p\left({\alpha}_{num},{\alpha}_{subj?}|\mathrm{form},\mathrm{con}\right)\propto p\left({\alpha}_{num},{\alpha}_{subj?}\right)\\ {}\hskip20.8em \cdot p\left(\mathrm{form}|{\alpha}_{num},{\alpha}_{subj?}\right)\\ {}\hskip20.8em \cdot p\left(\mathrm{con}|{\alpha}_{num},{\alpha}_{subj?}\right)\end{array}} $$

In this way, the accurate probability information from the morphology cue drops out and is effectively ignored by the listener for that particular experimental item—even though the modeled listener has an accurate representation of this cue’s probability information (and may even use it for other experimental items). Thus, inaccurate deployment is again implemented via a selective dropout of probability information, this time from one or more cues.

Implementing selective deployment. We implement the frequency of this dropout process with a ‘use’ parameter β that encodes how often information for the prior or a particular cue’s likelihood is used. That is, with probability β_α, the informative prior for that antecedent will be used in the posterior calculation; with probability (1 − β_α), it will be ignored and an uninformative prior used instead. Similarly, with probability β _cue, a particular cue will be used in the posterior calculation; with probability (1 − β _cue), it will be ignored. Each cue therefore has its own β (β _form, β _con, β _mor).Footnote ¹⁶ Each β ranges between 0 and 1 (0.0 ≤ β ≤ 1.0).

Because the prior could be used (β_α) or ignored (1 − β_α), this yields two possibilities for the prior for any particular experimental item. Similarly, because each cue could be used or ignored individually, there are eight possibilities for the cues for any particular experimental item: the pronoun form’s probability is used (β _form) or ignored (1 − β _form); the discourse connective’s probability is used (β _con) or ignored (1 − β _con); the agreement morphology’s probability is used (β _mor) or ignored (1 − β _mor). So, the posterior for a particular experimental item (p _β(α | form, con, mor, α _num, α _subj?)) is a mix of these 2 × 8 possibilities.Footnote ¹⁷ We note that if all three cues are ignored, the modeled listener uses the prior over potential antecedents alone—whether informative (p(α _num, α _subj?)) or uninformative (p(unif)))—to calculate the posterior.

We can then determine the use parameter β values that allow the modeled Bayesian listener to best match the child and adult behavioral data on pronoun interpretation in our experiment. By doing so, we can get a sense of how noisy this integration process would need to be (and in what particular ways the process is noisy) in order to best account for participant behavior under this inaccurate deployment model. We note that if the best-fitting value is β = 1 for the prior or any cue’s likelihood, then this signals that accurate integration for that information (prior or likelihood) was the best fit, rather than noisy integration. That is, as with the inaccurate representation listener, the inaccurate deployment listener allows for the possibility of accuracy (in this case, for always deploying available information). More generally, we can also determine the best-fitting relative ‘use rate’ of the available information for both children and adults.

3.3.4. Bayesian listener with both inaccurate representations and inaccurate deployment

Our implementation of a listener with both inaccurate probability representations and inaccurate deployment of those representations combines the implementations of the listeners with only inaccurate probability representations and the listeners with only inaccurate deployment. More specifically, the softmax function with contrast parameter σ is used for the representations of the prior and likelihood probabilities, as shown in p _σ in equation 7, using σ_σ, σ _form, σ _con, and σ _mor; the selective dropout function with use parameter β is used for the deployment of the prior and likelihood probabilities, as shown in p _β in equation A1 in Appendix F, using β_α, β _form, β _con, and β _mor. More concretely, the both-inaccurate listener calculates the posterior p _σ,β, which uses the inaccurate representation posterior calculation p _σ within the mixture model defined for the inaccurate deployment posterior calculation p _β. So, p _σ,β looks identical to p _β, except that p _σ(α | cues, α _num, α _subj?) is used in place of p(α | cues, α _num, α _subj?) in equation A1 in Appendix F; similarly, p _unif,σ(α _num, α _subj? | cues) is used in place of p _unif (α _num, α _subj? | cues), with equation 10 showing this implementation when all three cues are used.

(10) $$ {\displaystyle \begin{array}{l}\hskip1.7pc {p}_{\mathrm{UNIF},\unicode{x03C3}}\left({\alpha}_{num},{\alpha}_{subj?}|\mathrm{form},\mathrm{con},\mathrm{mor}\right)\propto p\left(\mathrm{unif}\right)\\ {}\hskip29.3em \cdot p{\left(\mathrm{form}|{\alpha}_{num},{\alpha}_{subj?}\right)}^{\unicode{x03C3}_{form}}\\ {}\hskip29.3em \cdot p{\left(\mathrm{con}|{\alpha}_{num},{\alpha}_{subj?}\right)}^{\unicode{x03C3}_{con}}\\ {}\hskip29.3em \cdot p{\left(\mathrm{mor}|{\alpha}_{num},{\alpha}_{subj?}\right)}^{\unicode{x03C3}_{mor}}\end{array}} $$

3.3.5. Comparing Bayesian listeners

To sum up, we defined two ways that modeled Bayesian listeners can be inaccurate when it comes to the posterior calculation they use to decide how a pronoun is interpreted in a particular context. First, listeners can have inaccurate representations of the probabilities for the priors and likelihoods that go into the posterior calculation. Second, listeners can inaccurately deploy their represented probabilities when doing that posterior calculation, specifically by using only some (or none) of the probabilities. These two options lead to four modeled Bayesian listener types: (i) the baseline listener with accurate representations of probability information about potential antecedents and cues, who deploys those representations accurately; (ii) the listener with inaccurate representations of probability information about potential antecedents and cues, who deploys those probability representations accurately; (iii) the listener with inaccurate deployment of accurate probability representations; and (iv) the listener with both inaccurate probability representations and inaccurate deployment.

We can now compare these modeled listeners on their ability to best capture the child and adult data from the pronoun interpretation task (in particular, the responses summarized in Figure 2 that correspond to the subject and nonsubject responses provided by participants in each condition). More specifically, a modeled listener can generate the probability of a data point being produced (e.g. a subject or nonsubject response in a particular experimental context), given that listener’s implemented hypotheses about how judgments are generated in context; here, the context is a certain combination of pronoun form, discourse connective, and agreement morphology. The probabilities of all participant responses can then be aggregated together, representing the probability that the modeled listener would generate the entire set of responses. This will allow us to evaluate which hypothesis, as implemented by the different modeled listeners, provides the best explanation for children’s observed behavior and how this compares to the best explanation for adults’ observed behavior.

We can also look within the best-fitting modeled listener to find the parameter values that yield the best fit. This will give us an estimate of exactly how inaccurate participants’ probability representations are (i.e. how much σ differs from 1) and/or how inaccurate their deployment is (i.e. how much β differs from 1), and whether this changes as children become adults.

3.4.1. Which modeled listener?

To fit each modeled listener type, we varied the value of the contrast parameters (0.0 ≤ σ ≤ 4.0) and/or the use parameters (0.0 ≤ β ≤ 1.0) in increments of 0.01 and chose the combination with the best log probability of the participant responses (i.e. log likelihood scoreFootnote ¹⁸) in the pronoun interpretation experiment.

To determine which of the fitted modeled listener types best matches participant pronoun interpretation behavior, we used the Bayesian information criterion (BIC) (Schwarz Reference Schwarz1978), shown in equation 11; the BIC balances a model’s ability to fit the data against the number of parameters m that it uses. Since models with more parameters have a natural advantage, the BIC rewards models with more parameters only if they provide a substantially better fit to the data than models with fewer parameters. In our case, this means that to have a better BIC score, the both-inaccurate listener (with four σ parameters and four β parameters, for a total of eight parameters) has to have a substantially better fit than the inaccurate representation listener (with four σ parameters) and the inaccurate deployment listener (with four β parameters). Similarly, all three inaccurate listeners have to have a substantially better fit than the baseline listener (with zero parameters) to have better BIC scores.

(11) $$ \mathrm{BIC}=m\cdot log\left(| data|\right)-2\cdot log\left( model\; fit\right) $$

BIC evaluates a model’s fit to the data using the log likelihood of the data, and the best-fitting parameter values for that model.Footnote ¹⁹

The data in this case are the number of antecedent responses provided by participants in each condition (e.g. adults provided 736 responses); so, we calculate P (all participant responses across all conditions | best-fitting parameter values) for each modeled listener type and take the log—this is the log likelihood score. A better BIC score, which incorporates the log likelihood score, is closer to 0. BIC scores for all modeled listener types are shown in Table 4 for children ≥5 years old and adults.

Table 4.

BIC scores for modeled listeners with accurate representation and deployment (baseline), potentially inaccurate representations only (σ ≠ 1), potentially inaccurate deployment only (β ≠ 1), or both (σ ≠ 1, β ≠ 1). The best-performing modeled listener(s) for each age group are shown in bold. Scores within 2.0 of each other are considered equivalent, following Kass & Raftery Reference Kass and Raftery1995.

Table 4. Long description

The table consists of five columns and three rows.

Column headers from left to right are:

1. Age

2. Baseline

3. Inaccurate representation sigma is not equal to 1

4. Inaccurate deployment beta is not equal to 1

5. Both inaccurate sigma is not equal to 1, beta is not equal to 1

Row 1 (Age group greater than or equal to 5):

- Baseline: 1931.33

- Inaccurate representation: 590.04 (bold)

- Inaccurate deployment: 590.70 (bold)

- Both inaccurate: 607.12

Row 2 (Adults):

- Baseline: 1323.82

- Inaccurate representation: 642.48 (bold)

- Inaccurate deployment: 646.96

- Both inaccurate: 668.58

We see that the baseline modeled listener, with both accurate probability representations and accurate deployment, fares the worst (children ≥5 BIC: 1931.33 vs. < 607.12; adults BIC: 1323.82 vs. < 668.58); this result suggests that both children and adults are being inaccurate somewhere. That is, they are using inaccurate probability representations, inaccurately deploying accurate probability representations, or both using inaccurate probability representations and deploying those representations inaccurately. Importantly for the target state of acquisition, adult behavior is not best captured by a totally accurate listener. That is, becoming adult-like in pronoun interpretation does not mean that children should represent and deploy available pronoun probability information with total accuracy. Rather, children have to become more adult-like in how they inaccurately do these things.

We can see in Table 4 that children’s behavior is best captured by the modeled listener types that use either inaccurate probability representations (BIC: 590.04) or inaccurate deployment (BIC: 590.70), but not both (BIC: 607.12). These results suggest that children either (i) have inaccurate probability representations, but deploy them accurately, or (ii) have accurate probability representations, but deploy them inaccurately. Table 4 also suggests that adult behavior is best captured by the modeled listener type using inaccurate probability representations, but deploying those representations accurately (BIC: 642.48 vs. 646.96 and 668.58).Footnote ²⁰

It could be that children are basically doing the same thing as adults: accurately deploying inaccurate probability representations. In this case, children would not need to qualitatively shift their approach to interpreting pronouns in order to become adult-like. However, our results are also compatible with children doing something fundamentally different from adults, by inaccurately deploying accurate probability representations. In this case, children would need to qualitatively shift their pronoun interpretation approach as part of development; more specifically, children would need to learn how to (perhaps strategically) distort their probability representations of available pronoun information, but accurately deploy those distorted representations in the moment.

3.4.2. What’s changing?

To better understand the potential change between children and adults, we can examine the best-fitting model parameters that yielded the BIC scores in Table 4. In particular, we can look at the best-fitting σ contrast values for the inaccurate-representations listener to understand what children and adults may be doing, as well as the best-fitting β use values for the inaccurate-deployment listener to understand what else children may be doing. These parameter values are shown in Table 5.Footnote ²¹

Table 5.

Best-fitting parameter values for the modeled listeners that best capture the pronoun interpretation behavior of children and adults, based on BIC scores. σ values are used for inaccurate probability representations, while β values are used for inaccurate deployment of representations. Accurate deployment in the inaccurate-representations listener uses β values of 1; accurate representations in the inaccurate-deployment listener use σ values of 1.

Table 5. Long description

The table consists of nine columns. The first column is unlabeled and identifies the age group. The subsequent columns are sigma sub form, sigma sub con, sigma sub mor, sigma sub alpha, beta sub form, beta sub con, beta sub mor, and beta sub alpha.

* Children: Inaccurate representations (Age greater than or equal to 5): sigma sub form 0.02, sigma sub con 0.28, sigma sub mor 0.11, sigma sub alpha 0.00. All beta values are 1.

* Children: Inaccurate deployment (Age greater than or equal to 5): All sigma values are 1. beta sub form 0.00, beta sub con 0.43, beta sub mor 0.30, beta sub alpha 0.00.

* Adults: Inaccurate representations: sigma sub form 0.25, sigma sub con 0.33, sigma sub mor 0.28, sigma sub alpha 0.00. All beta values are 1.

Inaccurate representations in adults and possibly in children. Adult behavior is best captured by a modeled listener with inaccurate probability representations, and this is one of the two modeled listener types that children’s behavior is also best captured by. In Table 5, we see that the way in which these probability representations are inaccurate is qualitatively the same between adults and children. In particular, the best-fitting σ values for both adults and children are all less than 1 (adults: 0.00–0.33, children: 0.00–0.28). These values mean that both adults and children tend to smooth away any relative probability differences that the pronoun information provides—and in some cases, to do so quite dramatically. For instance, a σ value of 0.33 (the highest among the adults’ σ values) would take a probability distribution of 0.75 vs. 0.25 and transform it into 0.59 vs. 0.41. Lower σ values have even stronger smoothing effects: a σ value of 0.28 (the highest among the children’s σ values) transforms 0.75 vs. 0.25 into 0.58 vs. 0.42; σ = 0.11 transforms 0.75 vs. 0.25 into 0.53 vs. 0.47; σ = 0.02 transforms 0.75 vs. 0.25 into 0.505 vs. 0.495 (i.e. nearly a uniform distribution). These results suggest that even the least-distorted representation information for both adults and children has smoothed away a great deal of the probability contrast in the input.

Interestingly, for some information, any probability contrasts present in the input are completely smoothed away, with σ = 0.00 (σ_α for adults and all children). This can be interpreted as participants completely ignoring this probability information, as a σ of 0.00 yields a uniform distribution. For both adults and children, the information in the prior is completely smoothed away, yielding a uniform distribution over potential antecedents. So, children would not need to change at all to become adult-like in this respect: it is adult-like to ignore how often a pronoun refers to a potential antecedent in general, irrespective of the cues available in the current context.

However, for the cue of pronoun form (represented by σ _form), children would need to smooth away less of the probability contrast available from the input (adults: 0.25; children: 0.02). Similarly, children would need to smooth away less of the probability contrasts available for the information about discourse connectives and agreement morphology (adult σ _con: 0.33, child σ _con: 0.28; adult σ _mor: 0.28, child σ _mor: 0.11). More generally, under this view, acquisition involves tuning how much probability information is smoothed away, lessening the amount smoothed away to adult-like levels.

Inaccurate deployment possibly in children. Another possibility for children’s observable pronoun interpretation behavior is that they have accurate representations of pronoun probability information, but are inaccurately deploying them. In particular, children would completely ignore information about the prior (i.e. children use only probability information from the current pronoun context) and completely ignore probability information about the pronoun form, as β_α = β _form = 0.00. The other two pronoun information types, discourse connectives and agreement morphology, are ignored less (β _con = 0.43, β _mor = 0.30). In contrast, adults would be deploying their representations accurately (all βs = 1), rather than selectively deploying them.

So, to become adult-like in deployment, children would need to make a qualitative shift in how they interpret pronouns by accurately, rather than inaccurately, deploying those probability representations. We do note that adult-like smoothing of probability representations involves completely smoothing away information about the prior distribution over antecedents (σ_α = 0.00); this total smoothing is behaviorally equivalent to never using that probability information in the moment (β_α = 0.00), because it means children completely ignore the probability information from the input. So for probability information about prior antecedents, children would not necessarily need to qualitatively change what they end up doing, which is to completely ignore that probability information.

Becoming adult-like: summary. Taken together, our modeling results suggest one of two options for children: they are either always using highly smoothed representations of the pronoun probability information available (inaccurate representations), or they are selectively using accurate representations of the pronoun probability information available (inaccurate deployment). Notably, the best-fitting inaccurate-representations listener for children completely smooths away all information in the prior distribution over antecedents (σ_α = 0.00) and nearly does so for pronoun form (σ _form = 0.02); smoothing away all information in a listener relying on inaccurate probability representations yields a result for these information types that is equivalent to that of the best-fitting inaccurate-deployment listener that never deploys them (β_α = β _form = 0.00). That is, children’s behavior is best captured by ignoring information about the prior antecedent distribution and pronoun form (either by smoothing away all probability contrasts available in the input, or by never deploying that available information in the moment).

In contrast, adults are best fit by an inaccurate-representations listener that ignores only the information about the prior antecedent distribution (σ_α = 0.00), and thus completely smooths away any probability contrasts available from the input. So, to become adult-like, children need to pay attention to pronoun form—whether by smoothing an inaccurate probability representation less (to achieve adult-like σ _form = 0.25), or by making a qualitative shift to always deploy (i.e. β _form = 1) a probability representation that is itself inaccurate because it is smoothed (i.e. σ _form = 0.25). Children also need to pay attention to information about the discourse connective and agreement morphology more than they currently do, again either by smoothing these probability representations less or by making a qualitative shift to always use a smoothed probability representation.

3.4.3. Comparison to behavioral findings

Recall that we observed some specific behaviors in children and adults (discussed in Section 2.4). In particular, the logistic regression analyses suggested that both discourse connectives and agreement morphology mattered for children age five and older, and all three cues (pronoun form, discourse connectives, and agreement morphology) mattered for adults. Our modeling results align with these findings, while additionally uncovering more specifically how these information types may matter for each group.

In particular, if we look at the modeling results for children, the best-fitting modeled listeners either (i) rely on inaccurate probability representations that are smoothed or (ii) inaccurately deploy accurate probability representations. Either option results in children ignoring specific types of probability information, in line with the logistic regression analysis from before. For children age five and older, both discourse-connective probabilities and agreement-morphology probabilities were either less smoothed (σ _con = 0.28, σ _mor = 0.11 vs. σ _form = 0.02) or deployed more often (β _con = 0.43, β _mor = 0.30 vs. β _form = 0.00). So, the information types whose representations were either less smoothed (thus more adult-like) or deployed more often (thus more adult-like) are the same ones our behavioral analysis identified as mattering for explaining children’s behavior.

If we look at the modeling results for adults, the best-fitting modeled listener relied on inaccurate representations that smoothed away some—but not all—of the probability contrasts available in the input for the three pronoun cues. So the modeled listener aligns with the behavioral results by showing that, for adults, all cue information mattered.

For both children and adults, what the modeling results additionally uncover is exactly how the probability information that mattered might actually matter. For children, either the representations of probability information perceived as relevant were relatively sharper (i.e. less smoothed), or they were deployed relatively more often than information that did not seem to matter to them. For adults, the representations of the probability information perceived as relevant were equally relevant (though smoothed) and were always deployed to interpret pronouns in context.