The target article by Barbey & Sloman (B&S) challenges the natural-frequency based theories of human judgment and decision making and, instead, supports a nested sets hypothesis. The authors conclude that judgmental errors and biases in Bayesian calculations are attenuated when problems are represented properly through nested sets of individuals. Although in general we agree with the authors' claims, in this commentary we would like to further examine, from a different perspective, why the nested sets hypothesis might provide a more adequate representational account than the natural frequency representations.

We have recently hypothesized that when assessing the likelihood of uncertain events, statistically unsophisticated people utilize a coarse scale that only has a limited number of categories (Sun et al., in press). The essence of the hypothesis is that, without adequate anchors, people's internal representations are coarse and limited, and therefore do not map one-on-one onto the continuously distributed external values, which can be either probabilities (or percentages) or frequencies.

The coarseness hypothesis is based on the large body of behavioral and neuroimaging research on mental presentations of quantity and numbers. Miller (Reference Miller1956) suggested that the number of levels of any variable that can be internalized is not only finite, but also small. In psychometric research, many researchers believe that the number of response alternatives on a scale is quite limited (for a review, see Cox Reference Cox1980). Recent studies using brain-imaging techniques have provided neurological evidence indicating the existence of a coarse scale for the internal representation of numerical values. Dehaene and colleagues (e.g., Dehaene et al. Reference Dehaene, Spelke, Pinel, Stanescu and Tsivkin1999) suggest that there is a coarse and analog mental number line, which is the foundation of a “number sense” and shared by humans and animals. Particularly, Dehaene et al. show that exact calculations involve linguistic representations of numbers and are controlled by the speech-related areas of the left frontal lobe in the brain. In contrast, approximate calculations are language-independent and rely on visuo-spatial representations of numbers controlled by the left and right parietal lobes. Therefore, it is possible that there are two different calculation processes involved in the Bayesian inference. It might be too early to link this distinction to the dual-process model (“associative” and “rule-based”) suggested in the target article, but the theoretical relevance appears to be evident. Particularly for lay persons, who do not have the ability or enough information to carry out exact calculations, it is likely that their intuitive assessment of event likelihoods is a “sense of approximation” based on a coarse internal representation.

We conducted two experiments to test the coarseness hypothesis (Sun et al., in press). In Experiment 1, participants estimated event probabilities in a free format. The experiment task was to estimate the winning probabilities of poker hands in a one-deck-and-two-player “draw poker” game. Despite the fact that the target probability has an even and nearly continuous distribution of probabilities ranging from zero to close to 100% with relatively small increments, we found that subject responses were highly clustered with approximately 5 clusters each. In Experiment 2, participants made forced comparisons using two different external response scales (a 3-level coarse scale versus a 10-level fine scale). We found that their performance did not measure up to the requirement of the finer scale. These findings indicate that besides the systematic biases, a certain portion of human errors in probabilistic judgment may be due to the low resolution of the internal representations. Further analyses of the experimental data and computer simulations implied that the number of internal categories is about 5.

This coarseness model can be used to account for physicians' overestimation of the probability of cancer, given a positive test result. Specifically, probably to a physician, what matters most is a dichotomy between a positive and a negative test result. No matter how small the probability of a breast cancer can be (due to the low base rate), a positive test result has to be taken seriously. Thus, the test result would naturally serve as an external cue to anchor two distinctive mental states. If the number of options after a test is limited (e.g., two options with one threshold), a large amount of discrepancy manifested as the overestimation of cancer probability could be accounted for by the mismatch between two different scales: one is an approximate evaluation of the seriousness of a situation (e.g., classifying it as “dangerous”), and the other is an exact numerical value in a continuous distribution. Such discrepancy may not be easily made to disappear by simply using a frequency representation, since frequencies, normalized or not, are still continuously distributed.

B&S have demonstrated that diagrammatic representations such as Euler circles, when employed to construct a nested set structure, would facilitate Bayesian inference. This is consistent with our hypothesis for a coarse scale of internal representations. Figure 1 in the target article shows a probability space nested in three levels: all possibilities, number of individuals (or chances of) testing positive, and, finally, the number of individuals (or chances of having) the disease. In this setting, computations are facilitated by external cues. Most important, these cues are represented hierarchically, so that at any moment, human subjects would only need to divide the probability space into a limited number of categories. In effect, when significant facilitation is found using natural frequency representations, it is often the case that hypotheses are pre-organized to facilitate a limited number of set operations (such as the binary hypotheses used in Gigerenzer & Hoffrage Reference Gigerenzer and Hoffrage1995). Therefore, it seems that the key factor underlying facilitation is a structure with a limited number of sets at each computation phase. The success of the nested sets model may well lie in the “chunking” mechanism (Miller Reference Miller1956) which reduces the computational demands for human subjects. Other distinctions, such as whether natural frequency is normalized to an arbitrary reference class, do not address this issue. And it would be no surprise that little or less facilitation is found when natural frequencies are used but without transparent nested set structure.