Mutual Information and Categorical Perception

Abstract

Categorical perception refers to the enhancement of perceptual sensitivity near category boundaries, generally along dimensions that are informative about category membership. However, it remains unclear exactly which dimensions are treated as informative and why. This article reports a series of experiments in which subjects were asked to learn statistically defined categories in a novel, unfamiliar 2D perceptual space of shapes. Perceptual discrimination was tested before and after category learning of various features in the space, each defined by its position and orientation relative to the maximally informative dimension. The results support a remarkably simple generalization: The magnitude of improvement in perceptual discrimination of each feature is proportional to the mutual information between the feature and the category variable. This finding suggests a rational basis for categorical perception in which the precision of perceptual discrimination is tuned to the statistical structure of the environment.

Categorical perception refers to the enhancement of perceptual sensitivity near category boundaries (Harnad, 1987). After learning to classify stimuli into discrete classes, subjects’ ability to make fine discriminations along perceptual dimensions that are informative about the categories can measurably improve. Categorical perception and associated changes in perceptual discrimination were first observed in phonological perception (Liberman et al., 1957) but have since been observed among a number of visual features, including orientation (Rosielle & Cooper, 2001), facial features (Rotshtein et al., 2005; Viviani et al., 2014), and shape (Folstein et al., 2014; Gauthier et al., 2003). For example, in one influential study (Goldstone, 1994), subjects trained to classify objects into two categories of objects distinguishable by their size became more sensitive to size differences (improved discrimination, d′) but not as much to brightness differences. Such findings are remarkable because they reflect a profound interaction between cognitive and perceptual mechanisms: a change to basic perceptual processes attributable to the acquisition of a new concept by an adult organism (Schyns et al., 1998).

In modern terminology, the term categorical perception is sometimes reserved for changes to categorization performance (referring to the tendency for subjects’ category judgments to change abruptly near the category boundary). Concomitant changes to discrimination performance are referred to as acquired distinctiveness (for improvements in discrimination between categories) or acquired equivalence (for degradation of discrimination within categories, which has been observed in some though not all studies; Folstein et al., 2010, 2013, 2014; Goldstone, 1994; Goldstone et al., 2001; Livingston et al., 1998; Notman et al., 2005). The neural basis of acquired distinctiveness and acquired equivalence is thought to involve modification of receptive field structure (Folstein et al., 2015; Kang et al., 2004; Li et al., 2007; Sigala & Logothetis, 2002). Although several neural-network models of categorical perception have been proposed (Casey & Sowden, 2012; Damper & Harnad, 2000), the computational mechanisms underlying these effects are still poorly understood.

Some researchers (e.g., Goldstone & Steyvers, 2001) have concluded that improvement in perceptual discrimination (acquired distinctiveness) tends to occur in proportion to each feature’s informativeness (or relevance or diagnosticity) about the categories to be learned. But it is not clear which perceptual features the system treats as informative and why. In many studies, categories are distinguished by a clear, deterministic boundary separating one class from another—often a linear boundary separating a 2D perceptual space into two clear-cut halves. In such a space, a perceptual feature that crosses the category boundary is perfectly predictive of category membership, making it informative by any reasonable metric, whereas any feature that does not cross the boundary is completely uninformative. But hard classification boundaries are not characteristic of natural categories, which have been understood for decades to have typicality gradients and correspondingly soft classification boundaries (Posner & Keele, 1968; Rosch, 1973). When categories are defined more naturalistically via statistical distributions (Huttenlocher et al., 2000), no dimension is perfectly predictive, and a variety of definitions of informativeness are possible. For example, some researchers have defined categories as bivariate Gaussian (normal) distributions in a 2D space (e.g., Lake et al., 2009; Maye et al., 2002). In this case, Lake et al. (2009) found that a particular measure of informativeness, the L² norm between the posterior distributions, predicted improvements in perceptual discrimination, but this measure was not compared with alternatives.

Statement of Relevance.

When people classify objects into different categories, they focus on some features more than others. For example, people might classify fruits by their color but trees by the shape of their leaves. When people learn new categories, they tend to become attuned to the features that distinguish those particular categories—actually improving at discriminating small differences among those features—but it has never been clear exactly which features tend to improve and why. In the studies reported here, adult subjects learned novel categories that were differentiated by very subtle shape features. The results show that after category training, subjects became better at discriminating each feature in proportion to how objectively informative it was about the category to be learned. The results suggest that human perceptual systems tend to use perceptual features that are maximally informative about the statistical structure of the categories in the world around them.

However, classical information theory provides a more natural and well-motivated measure of informativeness: mutual information, which quantifies how much of the variation in one variable is predicted by another (Cover & Thomas, 1991). Mutual information is widely used in neuroscience (Piasini & Panzeri, 2019), animal learning (Balsam et al., 2006), and machine learning (Battiti, 1994) to quantify informational relationships among variables. The mutual information MI between a category variable C and a feature f is defined as

$$\mathrm{MI}(C,f)=H(C)-H(C|f),$$

where H(C) is the prior Shannon uncertainty about the category and H(C|f ) is the conditional uncertainty about the category once the feature is known, both measured in bits if logs are taken in base 2. The mutual information represents the degree to which learning the value of the feature reduces the observer’s uncertainty about which category the stimulus belongs to and thus constitutes a natural measure of the informativeness of the feature. Indeed, recently Bates et al. (2019) showed that features that provide mutual information about a category variable undergo more improvement in discrimination than those that do not, and Bates and Jacobs (2020) provided a comprehensive theoretical argument that the quantity of conveyed information is capped at the mutual information.

However, the manner in which the system quantifies informativeness cannot be determined using a hard category boundary, as has been used in virtually all studies (including those of Bates et al., 2019). With such a boundary, all of the information (however defined) is concentrated at the boundary, and features that do not cross the category boundary convey no information whatsoever about the category variable. This makes it impossible to test intermediate values of informativeness (again, however defined), and moreover completely confounds all reasonable measures of informativeness, because all of them are maximal at the boundary and minimal everywhere else in the feature space. The experiment below solves some of these problems by using probabilistically defined categories separated by a soft boundary in a novel 2D feature space. This allows for the evaluation of “diagonal” features through the space, in addition to the category-relevant and category-irrelevant axes to which previous studies have been restricted. The resulting experiment includes a whole range of levels of informativeness (rather than just relevant and irrelevant) and also deconfounds various potential measures of informativeness.

Moreover, most studies of categorical perception use features such as color or facial features with which the visual system has enormous prior experience and that also may have some degree of innate categorical structure (Folstein et al., 2015), making it difficult both to induce changes to perceptual sensitivity and to attribute them directly to training. To more carefully isolate the effect of learning, one should use a feature space that is as unbiased and unfamiliar to subjects as possible.

The experiments below used a space of randomized, subjectively novel perceptual features with which subjects can be assumed to have little or no prior experience (Fig. 1). Stimuli are drawn from a high-dimensional space of blob shapes created by modulating radial Fourier components (Dickinson et al., 2013; Op de Beeck et al., 2003) defining shape contours (Fig. 1a). Shape is very high-dimensional space in which most dimensions involve subtle combinations of contour geometry that are novel and difficult to verbalize (Destler et al., 2019). From the initial high-dimensional space, a 2D feature space is randomly selected by choosing three random points in the space, which define a random plane, and then randomly choosing an origin and two orthogonal-basis vectors in this plane (via the Gram-Schmidt process), which together define a coordinate frame (Fig. 1b). This results in a 2D manifold of shapes from which stimuli are chosen, any subspace of which defines a potential feature (Fig. 1c). Note that unlike in many previous studies (e.g., Dieciuc et al., 2017; Folstein et al., 2012, 2013; Viviani et al., 2014; Wallraven et al., 2014), this feature space is not a morph space constructed by weighted combinations of fixed stimuli at the poles. Rather, it is a completely novel space newly novelized (randomized) for each subject. Unlike a morph space, this feature space has no familiar or consistent stimulus shape at the poles, and indeed there are no poles, which reduces the possibility of a preexisting categorical bias present in most previous experiments.

Fig. 1.

Procedure for creating a novel, randomized perceptual space in which categorization occurs. Shapes are defined (a) by four radial Fourier components (only three are depicted). Next, a random plane (b) through this four-dimensional space is chosen. In this plane, a random origin and two orthogonal-basis vectors are chosen (c), resulting in a 2D shape space through which any direction is a potential shape feature. In this space, A and B categories (d) are each circular bivariate Gaussian distributions, respectively centered at (.25, .5) and (.75, .5) of the unit square. Viewed as a contour plot (e), this category structure defines a linear optimal classification boundary (dashed vertical line) with a maximally informative dimension (solid horizontal arrow). The sample screen from a categorization task (f) shows a shape to be classified. The dotted line indicates motion and was not visible to subjects.

Within the feature space, two categories are defined by circular bivariate Gaussian distributions (Fig. 1d), which define a soft linear optimal classification boundary (shown as a dotted line in Fig. 1e). The overlap between the two Gaussian distributions can be modulated by changing their common standard deviation (σ), which determines the maximum possible proportion correct (ideal performance level, or IPL, equal to 1 minus the Bayes error). IPL was set to 95% in Experiments 1, 2, and 3 (σ = .150), 90% in Experiment 4 (σ = .188), and 99% in Experiment 5 (σ = .105).

Critically, this procedure defines a feature space that is fully rotatable (with a euclidean L² norm), meaning that any direction through this space defines a potential shape feature—including some that might be somewhat verbalizable but many others that are not (Hockema et al., 2005; Op de Beeck et al., 2003). In contrast, feature spaces in most studies (if 2D at all) consist of two separable features (e.g., size and brightness) with only the two cardinal axes as potential features, implying an L¹ (city block) norm. It is well established that diagonal dimensions that combine cardinal axes are more difficult to learn than axis-aligned features (Ashby & Maddox, 2011). But in the spaces used here, no direction is any more cardinal than any other (and thus no space is any more diagonal than any other), and moreover, the orientation of the subspace is randomized for each subject. Hence, in the experiments below, exactly which features are informative depends only on the category structure chosen. This procedure makes it possible to cleanly assess the informativeness of shape dimensions purely as a function of the category learned, without confounding from the subjects’ prior experience.

Method

Subjects

Subjects were adult members of the undergraduate community (Ns = 20, 22, 21, 21, and 21 in Experiments 1–5, respectively), recruited from introductory psychology classes and naive to the goals of the experiment.

Discrimination task

Perceptual discrimination was assessed at selected features of interest (FOIs) before and after the categorization task. Each FOI is defined as a point x = (x, y) and direction v = (u, v) in the shape space; five or six such features were evaluated in each experiment (for details, see Fig. 2 and also Table S1 in the Supplemental Material available online). Discrimination was measured by presenting pairs of shapes (size = ~4° of visual angle; white on a dark background) one at a time for 0.25 s each, separated by an interstimulus interval of 0.5 s and spatially offset by about 10° of visual angle. Subjects were asked to indicate whether the shapes were the same or different. Each pair of shapes was located at the desired location in feature space plus or minus a variable discrepancy in the given vector direction, $x\pm \mathrm{\lambda }v/2$ . The featural difference $\mathrm{\lambda }$ was then adaptively reduced on successive trials by the psi method (Kingdom & Prins, 2010) until the shapes could no longer be distinguished, resulting in an estimate of the threshold of distinguishability at each FOI. Staircases were randomly interleaved. Threshold estimates stabilized in about 15 min (about 50–100 trials per feature). Subjects performed the discrimination task before and after the categorization task, providing pretraining and posttraining estimates of discrimination threshold at each FOI. The main dependent measure is the difference in thresholds (pretraining – posttraining) at each FOI, Δthreshold.

Fig. 2.

Category structure and features of interest (FOIs) at which discrimination was evaluated in (a) Experiment 1, (b) Experiment 2, and (c) Experiments 3, 4, and 5. Green and red circles indicate A and B categories, respectively (each is a bivariate Gaussian distribution indicated by a circle of radius σ). Each FOI is a point and direction in the 2D perceptual space.

Categorization task

Stimulus shapes (size = ~4° of visual angle) were drawn randomly with equal probability from either the A category (a circular bivariate Gaussian centered at x = .25, y = .5 of the unit square; see Fig. 2) or the B category (also a circular bivariate Gaussian, centered at x = .75, y = .5). The two-Gaussian category structure defines a maximally informative dimension, depicted as horizontal in figures. Stimulus shapes moved downward from the top of the screen at about 8° of visual angle per second over a starry field and were visible for maximum of 2.5 s or until response. (The motion was intended to draw subjects’ attention to the stimulus; Franconeri & Simons, 2003.) The instructions framed the task as a space-based video game (see sample screen in Fig. 1f) in which subjects had to use keyboard buttons to fire at hostile ships (category A) or to welcome friendly ships (category B); they received feedback after each response in the form of a happy face (correct classification) or frown (incorrect classification). Each subject completed 300 trials, a number that a pilot study suggested would be sufficient to induce measurable categorical-perception effects. The categorization task took about 20 min.

Design

Experiments 1, 2, and 3 all used the same two-Gaussian category structure (95% IPL), differing only in the choice of FOIs (Fig. 2). FOIs were chosen so as to broadly survey the space, including a broad range of mutual-information levels (see below), and also to target several specific comparisons. Experiment 1 (Fig. 2a) tested six FOIs, including three at the intercategory midpoint (αs = 0°, 45°, and 90° relative to the maximally informative dimension) and three at the center of category A (at the same three orientations). This comparison is potentially interesting because some researchers (e.g., Folstein et al., 2010) have suggested that mere exposure to stimuli rather than category training per se is sufficient to induce categorical perception; stimuli near a category center are more frequent but less diagnostic than those between categories. Experiment 2 (Fig. 2b) also used six FOIs, three at the midpoint (αs = 0°, 45°, and 90°) and three at a point elsewhere on the optimal classification bound (αs = 0°, 45°, and 90°). Comparing features on and off the main axis is interesting because most studies use a one-dimensional space, so all comparisons are necessarily on axis. Experiments 3, 4, and 5 investigated the effect of a more finely, using five features at the intercategory midpoint ranging from maximally to minimally informative in equal angular steps (αs = 0°, 22.5°, 45°, 67.5°, and 90°). Experiment 3 used the same category structure as Experiments 1 and 2 (95% IPL), whereas Experiment 4 used a softer category boundary (IPL = 90%) and Experiment 5 a sharper one (IPL = 99%). The manipulation of IPL does not change the optimal classification boundary, but (as discussed below) it does change the quantity of information available at the FOIs, allowing a more fine-grained evaluation of categorical perception as informativeness is varied. Table S1 provides a complete list of the FOIs used in all five experiments.

Results

Categorization task

Performance on the categorization task was highly variable, and average performance was far below the theoretical limit (IPL), presumably reflecting the extremely unfamiliar and nonverbalizable shape features over which categories were defined. Mean performance in Experiments 1 through 5 was, respectively, 85% (SD = 6%), 83% (SD = 6%), 80% (SD = 5%), 83% (SD = 7%), and 86% (SD = 7%). Results reported below include only subjects with overall performance over 70% (i.e., 80 of 105 subjects [76%]). Setting the criterion to 50% includes 97 of 105 subjects (92%), which adds noise to the results but does not affect the main conclusions.

Notwithstanding the subjects’ uneven performance, their responses showed clear evidence of the sharpening of the category boundary over the course of learning associated with categorical perception. To quantify this, I fitted subjects’ responses in the categorization task to a one-dimensional Gaussian classifier of the form $p(A|x)=N(x;{\mathrm{\mu }}_A,{\mathrm{\sigma }}^2)/[N(x;{\mathrm{\mu }}_A,{\mathrm{\sigma }}^2)+N(x;{\mathrm{\mu }}_B,{\mathrm{\sigma }}^2)]$ —that is, an ideal-observer classifier—with the single free parameter σ fitted by least squares to the data in each block of 50 trials for each subject. In this model, the parameter σ modulates the sharpness of the classification boundary, with high values of σ indicating broad category distributions and a more gradual transition between categories and low values indicating narrower category distributions and a more abrupt transition. Figure 3 shows plots of the progression of subjects’ mean estimated σs over the course of training in each experiment. In all of the plots, σs start high and progressively decrease (sharpen), gradually approaching their respective target values (i.e., those from which the stimuli were actually generated). The change from broader to narrower σs from the first block of the experiment to the last was statistically substantial (Bayes factor [BF] > 3) in all five experiments. In these plots, the classification curve is approximately linear in the first block—meaning that the classification probability changes in approximately equal increments with each step through the feature space, that is, completely noncategorically. By the last block, the fitted values of σ are such that the classification is a relatively sharp step near the boundary, in the classical pattern associated with categorical perception. Note, though, that as σs seem to be reaching asymptote near their “true” values (i.e., the values used to generate the stimuli), this increasingly categorical performance simply seems to reflect approximately optimal category learning.

Fig. 3.

Plots of fitted values of σ across trials in (a) Experiments 1, 2, and 3 (95% ideal performance level [IPL]), (b) Experiment 4 (90% IPL), and (c) Experiment 5 (99% IPL). Each subject’s responses in the categorization task were fitted to the ideal classification curve $p(A|x)=N(x;{\mathrm{\mu }}_A,{\mathrm{\sigma }}^2)/[N(x;{\mathrm{\mu }}_A,{\mathrm{\sigma }}^2)+N(x;{\mathrm{\mu }}_B,{\mathrm{\sigma }}^2)]$ , with μ_A and ${\mathrm{\mu }}_B$ set to their true values of .25 and .75, respectively, and σ fitted by least squares to the subject’s responses. In the main graphs, error bars represent ±1 SE. The fitted value of σ modulates how broad or narrow the subjects’ induced category is: Larger values entail a softer decision boundary, and smaller values entail a sharper one. This is visualized in the insets, which show p(A|x), probability of an A response (y-axis) as a function of x (maximally informative dimension).

Discrimination task

Discrimination improved substantially (BF₁₀ > 3) from before to after training in all 20 of the FOIs at which α was less than 90° (and thus the feature was not diagnostic at all) but not in the other seven FOIs (BF₁₀ < 3). That is, subjects became more sensitive to those features—and only those features—that were predictive of category membership. The subjects demonstrated acquired distinctiveness after only about 20 min of category training, in contrast to thousands of trials of training in many studies. This unusually rapid induction of acquired distinctiveness presumably reflects the novel feature space, whose unusual initial difficulty allowed subjects to improve rapidly with training. The finding of acquired distinctiveness with integral shape dimensions contrasts with the findings of Op de Beeck et al. (2003; though see Hockema et al., 2005). Overall thresholds decreased from a mean of 0.35 (SD = 0.007) to 0.26 (SD = 0.009) after training (recall that the category means were separated by 0.5 of the unit square). The magnitude of discrimination improvement (Δthreshold) was not correlated with performance on the categorization task (R² = .00084, BF₁₀ = 0.09). The manipulated position factors had relatively small effects in individual experiments (Fig. 4). However, a clear pattern emerges when the results of all five experiments are combined, as follows.

Fig. 4.

Results for Experiments 1 through 5, showing Δthreshold as a function of α (angular deviation from most informative dimension). For Experiments 1 and 2, results are broken down further by position in the feature space. For Experiments 3, 4, and 5, model fits (red) indicate best-fitting cosine functions. Error bars represent ±1 SE. IPL = ideal performance level.

The main analysis is the magnitude of improvement in discrimination (Δthreshold) as a function of the mutual information MI(C, f ) = H(C) – H(C|f ) between the category variable C (A or B) and a given FOI f. $H(C)=-p(A){\log }_{2}p(A)-p(B){\log }_{2}p(B)$ is the prior uncertainty about the category, which in the experiments is always 1 bit because the two categories are equally likely. H(C|f ) = -p(f )log₂(A|f ) - p(f )log₂(B|f ) - $p(\neg f){\log }_2(A|\neg f)-p(\neg f){\log }_2(B|\neg f)$ is the conditional uncertainty about the category once the feature is known. Intuitively, each feature f can be thought of as a binary division of the perceptual space into two halves (Fig. 5a); MI(C, f ) measures how much information an observer gains about C (i.e., which category a given stimulus belongs to) from learning which half of f it falls in.

Fig. 5.

Calculation of mutual information. As shown in (a), the mutual information MI(C, f) is the reduction in uncertainty about C conveyed by the binary feature f (which side of the dotted boundary does the stimulus fall on?—indicated by green shading) about the category C (which category does it belong to?). Panel (b) shows a map of MI as induced by the category structure used in Experiments 1 to 3, indicating the magnitude and direction of maximum MI at each point in the space. As in prior figures, each category is depicted by a dot at its mean and a circle of radius s.

In this sense, mutual information quantifies the diagnosticity of a given stimulus property with respect to the shape’s category membership. Mutual information is maximal for horizontal (α = 0°) features lying on the classification boundary, but its value there is affected by the sharpness of the boundary, modulated in the experiments by the IPL. For example, the mutual information for such features in Experiments 1, 2, and 3 (95% IPL) is .71 bits, in Experiment 4 (90% IPL) .53 bits, and in Experiment 5 (99% IPL) .92 bits. Features perpendicular to the classification boundary (α = 90°) have a mutual-information value of 0; they are completely uninformative. Between these extremes, mutual information depends in a more complex way on both the feature’s position and orientation.

Figure 6 shows Δthreshold as a function of mutual information for all 27 FOIs aggregated across Experiments 1 through 5 (with different colors and symbols for each experiment). The plot shows a clear linear relationship: The magnitude of acquired distinctiveness (Δthreshold) rises with mutual information (R² = .5663, BF₁₀ = 3,697). The linear relationship between mutual information and Δthreshold in Experiments 1 through 5 was, respectively, R²s = .48, .71, .89, .006, and .94, suggesting that the effect is robust and replicable. As discussed above, previous studies have found that acquired distinctiveness is larger in category-relevant features than in category-irrelevant ones. The current results show that, as Bates and Jacobs (2020) suggested, the degree of acquired distinctiveness of each feature is proportional to its informativeness, as measured by the magnitude of mutual information it shares with the category variable.

Fig. 6.

Improvement in discrimination (Δthreshold) as a function of mutual information (MI) across features of interest tested. The solid line shows the best-fitting linear regression (Δthreshold = .0701 × MI + .0550). Each plotted point averages over the subjects included in that condition (N = 20 or 21), separately for each of the five experiments. Units on the y-axis are proportions of the feature space (in which categories were separated by .5). Error bars represent ±1 SE.

The changes to discrimination performance observed were all positive, indicating acquired distinctiveness rather than acquired equivalence. The regression intercept of 0.0550 gives the magnitude of acquired distinctiveness at a mutual-information value of 0—that is, an overall practice effect. On the basis of information-theoretic constraints, Bates et al. (Bates et al., 2019; Bates & Jacobs, 2020) have argued that if channel capacity is fixed, then improvements to representational precision in some dimensions need to be offset by degradations in others. No such effect is apparent in the current data, as all discrimination changes were in the same direction. It is possible that such trade-offs may have been swamped by an overall practice effect, meaning that the total channel capacity allocated to featural representation may have increased over the course of training. Unfortunately, the current data do not allow this issue to be addressed more decisively.

Given the particular categories and features used in these experiments, most of the variation in mutual information (about 85%) is due to α, whereas the rest is due to feature position and IPL. Hence, it is fair to wonder whether the effect of mutual information on acquired distinctiveness might in fact be entirely due to α rather than to mutual information per se. However, a regression of Δthreshold onto α alone is less predictive than mutual information (R² = .4257 vs. R² = .5663 for mutual information); the difference in fits is statistically substantial (BF₁₀ = 44.4). Moreover, a Bayesian analysis of variance on the entire data set favors (maximum posterior) the additive model that includes all three factors (α, feature position, and IPL) over any subset model (BF₁₀ = 11.820). Thus, the effect of mutual information appears to depend on all three component factors and in particular is not attributable to α alone, although the contribution of feature position and IPL is relatively subtle and should be more comprehensively explored in future experiments.

As mentioned, several other definitions of informativeness have been proposed, including the L² norm between the posterior distributions (Lake et al., 2009) and the (squared) derivative of the posterior, which reflects the sharpness of the category boundary (Clayards et al., 2008). The squared posterior derivative is related to the mutual information and Fisher information and plays an important role in the theoretical literature on neural population coding (Bonnasse-Gahot & Nadal, 2008; Pouget & Zemel, 2007). However, in the current data, the L² norm predicts Δthreshold less well than does mutual information (R² = .45, worse than the fit for mutual information by BF₁₀ = 24.3), as does the squared posterior derivative (R² = .25, worse than the fit for mutual information by BF₁₀ = 1,626). Hence, in addition to mutual information’s more natural axiomatic derivation as a measure of the information conveyed by one variable about another, mutual information gives a better fit to the human data.

Discussion

Several studies have found that feature discrimination tends to improve more for features that are informative about learned categories than for those that are not (e.g., Bates et al., 2019; Folstein et al., 2013, 2014; Goldstone & Steyvers, 2001). The results of Experiments 1 through 5 show that informativeness can be quantified by mutual information: The more information a feature conveys about the category, in a classical Shannon sense, the more subjects (on average) tend to gain in sensitivity for that feature. This improvement in discrimination (acquired distinctiveness) is directly attributable to category training and is associated with the progressive development of sharper category boundaries over the course of training (categorical perception). The effect is better predicted by mutual information than it is by other measures of informativeness, such as the posterior slope, the posterior L² norm, or the orientation of the feature with respect to the maximally informative dimension. Overall, these results corroborate the role of information theory in quantifying how the brain allocates representational resources (Balsam et al., 2006; Nelson et al., 2010; Sims, 2018), and they suggest that such allocation is rationally tuned to the category structure of the world (Bates & Jacobs, 2020; Feldman et al., 2009; Lake et al., 2009; Maye et al., 2002; Soto & Ashby, 2015).

One notable consequence of these results is to deemphasize the division between features that cross category boundaries and those that do not, which is often highlighted in definitions of categorical perception. In the mutual-information account, features that cross the category boundary are the most informative but are not qualitatively different from other features in the space that convey information about the category, albeit to lesser degrees. This observation helps explain a variety of previous results, such as those of Goldstone (1994), who found that discrimination improvement was not limited to the category boundary but was distributed throughout the space in a somewhat complex pattern. Note that this pattern is not consistent with traditional attention-weighting models (e.g., Kruschke, 1992) that elevate or attenuate entire perceptual dimensions rather than specific feature values. As mentioned, this pattern cannot be clearly established using a hard category boundary, where boundary-crossing features are the only informative ones, nor in a one-dimensional perceptual space, where all features lie along the (sole) informative dimension. The clear relationship between mutual information and acquired distinctiveness becomes apparent only with statistically defined category structure over at least two dimensions.

Dieciuc et al. (2017) have suggested that some feature learning can be explained by relatively short-term reallocation of attention. The current experiments cannot address the time course of the observed changes to perceptual discrimination, because discrimination was tested only in the immediate aftermath of category training. Note, however, that allocation of spatial attention cannot explain these results, because the shape features tested were all global aspects of each stimulus shape and could not be localized to any one location within it. The results might, however, reflect the reallocation of feature-based attention (Maunsell & Treue, 2006). But note that these shape features represented novel, complex combinations of shape contour features and thus could not be evaluated simply by reallocating resources within an existing feature space. Hence, although these data do not directly address the role of attention, it seems difficult to explain the observed improvements in discrimination by reallocation of feature-based attention alone. Future experiments evaluating the durability of these discrimination changes would be very valuable.

Conclusion

The results reported here suggest that feature learning is rationally tuned to the statistical structure of the environment (Bates & Jacobs, 2020; Feldman et al., 2009; Lake et al., 2009; Maye et al., 2002; Soto & Ashby, 2015) and support a principled information-theoretic quantification of the way representational resources are allocated. More specifically, the new finding supports previous arguments (Harnad, 1993; Schyns et al., 1998) that categorical perception reflects the process by which the brain constructs a vocabulary of features suitable for representing the world.

Important questions for future studies include how to apply the mutual-information measure to unsupervised categorization, in which the category variable C is not directly available to the subject. In unsupervised learning, which is ubiquitous in everyday cognition, mutual information might be computed between features and an estimated latent category variable (Lake et al., 2015). Another important question is whether the relationship between acquired distinctiveness and mutual information extends to more complex conceptual structures, such as multimodal categories (Briscoe & Feldman, 2011), in which the mutual-information map can become much more complex.

Transparency

Action Editor: Krishnankutty Sathian

Editor: Patricia J. Bauer

Author Contributions

J. Feldman is the sole author of this article and is responsible for its content.