Post-training flexibility in category learning

Abstract

Exemplar models of categorization, which assume that people make classification decisions based on item information stored in memory, typically assume that all of the exemplars are available and inform decision-making. However, in this study, we hypothesized that people may selectively emphasize subsets of exemplars, giving rise to individual differences in categorization. To verify this hypothesis, we adopted the partial-XOR category structure in Conaway and Kurtz (Psychonomic Bulletin & Review, 24, 1312–1323 2017), which has been evident to be able to induce two major response patterns in the transfer phase: the Proximity and XOR patterns. “Experiment 1” confirmed that these two patterns could be generated if participants were trained with only the exemplars of one category or the other. In “Experiment 2”, participants were asked to not only learn the category labels of all exemplars but also memorize the exemplars of only Category A (Condition A), only Category B (Condition B), or two categories (Condition AB) for a recognition test after the training phase of the categorization task. As expected, in the transfer phase, the participants tended to perform the XOR and Proximity patterns, when the exemplars of Category A and Category B were respectively targeted for the recognition test. The parameters of the SDGCM estimated by Bayesian inference for modeling the data of “Experiment 2” showed that the exemplar accessibility of Category A was larger than that of Category B for performing the XOR pattern and vice versa for performing the proximity pattern, hence verifying our hypothesis.

Appendix: Computational modeling with the SDGCM

The SDGCM (Stewart & Brown, 2005) is an exemplar model, in which categorization is based on the similarity and dissimilarity of the to-be-classified item to all exemplars. As is the same as most exemplar models, the SD-GCM also assumes that each item is represented as a dot in the psychological space and the distance between items $x_i$ and $x_j$ is computed as $d_{ij} = (\sum _m \alpha _m (|x_{mi} - x_{mj}|^p))^{1/q}$, where $\alpha _m$ is the selective attention on dimension m, $\sum _m \alpha _m = 1$. The parameters p and q determine the distance metric (e.g., $p = q = 1$ for the city-block distance and $p = q = 2$ for the Euclidean distance). We tried and found that the SDGCM could better generate our observed data in our two experiments when the Euclidean distance was used. The similarity between $x_i$ and $x_j$ is transferred from the distance between them as $S_{ij} = e^{-cd_{ij}}$, where c is the specificity of the stimulus space; the larger c, the more specific each item is. The dissimilarity between $x_i$ and $x_j$ is computed as $DS_{ij} = 1 - S_{ij}$. The probability that an item is classified as Category A is computed as $P(A) = \frac{(\beta _A\nu _A)^\gamma }{(\beta _A\nu _A)^\gamma + (\beta _B\nu _B)^\gamma }$, where $\nu _A$ and $\nu _B$ are the valences for Category A and Category B, respectively, and $\gamma$ is the decision parameter; the larger $\gamma$, the more deterministic the categorization decision is. The bias parameters $\beta _A$ and $\beta _B$ were set as equal in this study, for the exemplars of the two categories were presented equally frequently.

The valence for Category A $\nu _A$ is the sum of the total similarity of $x_i$ to all exemplars of Category A and the total dissimilarity of it to all exemplars of Category B, as $\nu _A = \sum _{j \in A}w_j S_{ij} + \sum _{j \in B}w_j DS_{ij}$, where $w_j$ ($0 \le w_j \le 1$) is the parameter to weight the contribution of the exemplar $x_j$ for categorization, which was treated as the accessibility of the exemplar $x_j$ in this study. Likewise, the valence for Category B is computed as $\nu _B = \sum _{j \in B}w_j S_{ij} + \sum _{j \in A}w_j DS_{ij}$. According to the SD-GCM, an item tends to be classified as one category if it is more similar to the exemplars of that category and more dissimilar to those of the contrasting category. In this study, it was assumed that the exemplars in the same category share the same accessibility. Thus, in “Experiment 1”, we estimated two accessibilities $w_A$ and $w_B$ for Category A and Category B, respectively. In “Experiment 2”, the accessibility of each exemplar was set up as 1, as there was only one category and no need to assess the exemplar accessibility of each category. Regardless of the experiment, the parameters of the SD-GCM were estimated by Bayesian inference, which was conducted by using the R package {rjags} with two chains each collecting 1000 posterior samples for each parameter and each response of the SDGCM, without setting up burn-in samples.

Bayesian modeling with the SDGCM for Experiment 1

In “Experiment 1”, four parameters, c, $\gamma$, $\alpha _x$, and $\lambda$ were estimated for the data of each participant in the two conditions. The first three had uniform priors, as $c\sim U(1,10)$, $\gamma \sim U(1,20)$, and $\alpha _x\sim U(0,1)$. As the data in “Experiment 2” were the probability of Category A (or 1 - Probability of Category B for Condition B) for each transfer item predicted by the participants, the likelihood for the SDGCM to generate each data point was computed by a Gaussian distribution function with the mean as the Probability of Category A computed by the SD-GCM for each transfer item and the precision as $\lambda = 1/\sigma ^2$, where $\sigma ^2$ is the variance of normal distribution. The precision parameter $\lambda$ had a Gamma prior, $\lambda \sim Gamma(0.001,0.001)$.

Bayesian modeling with the SDGCM for Experiment 2

Five parameters c, $\gamma$, $\alpha _x$, $w_A$, and $w_B$, were estimated for each participant in the XOR and Proximity groups. The first three had uniform priors as $c\sim U(1,30)$, $\gamma \sim U(3,20)$, and $\alpha _x\sim U(0,1)$. The parameters $w_A$ and $w_B$ also had uniform priors, except that their intervals depended on the latent parameter g, which had a Bernoulli prior, $g \sim Bern(0.5)$. When $g = 0$ (for the XOR group), $w_A \sim U(0.51,1)$ and $w_B\sim U(0,0.49)$, while when $g = 1$ (for the Proximity group), $w_A\sim U(0,0.49)$ and $w_B\sim U(0.51,1)$, as it was hypothesized that the XOR pattern and the Proximity pattern, respectively, result from referencing more Category A and Category B. The parameters $w_A$ and $w_B$ independently estimated, because it was not necessary to assume the total accessibility as a constant. The likelihood for the SDGCM to generate the observed response for each transfer item for each participant was computed by the binomial distribution function, given the probability of Category A computed by the SDGCM.