Rahnev & Denison (R&D) perform a useful service by reminding us that human behavior often deviates from what can be construed as optimal, once some target variable (such as expected reward) has been specified. We would like to remind readers that Bayesian statistics can be used to fit models and to contribute to our assessment of the degree of optimality. This differs from the usual Bayes rational approach (BRA). Unlike BRA, Bayesian statistics can be used to assess optimality by fitting a model to data and assessing how close the resulting parameter estimates are to optimal settings. For example, if an observer model is specified in such a way that one parameter combination results in optimal performance, one can use posterior estimates to quantify (i.e., in terms of both the parameter estimates and their uncertainty) how likely subjects were to have been performing optimally.
We have used such techniques in our own work to further support the author's conclusion that optimality is rarely observed in human decision making. The models we use are often process models that are not formulated within the likelihood function, prior, cost function, and decision rule (LPCD) framework, although variables or relationships among them within the models could be used to derive LPCD quantities. Hence, LPCD may not characterize the actual computations performed during decision making. We illustrate these points through consideration of two topics, as discussed subsequently.
Although the authors note many suboptimal results in the signal detection theory model, it may be useful to separate the question of whether people are optimal from the question of whether their behavior is best understood in terms of the elements of LPCD. Some other process models might allow the constructs of LPCD to be seen as descriptions of the outcome of a learning process that does not employ these constructs directly. Turner et al. (Reference Turner, Van Zandt and Brown2011) developed a category-learning model that encodes feature information about categories – the more general case of signal detection – and gradually forms a representation of category structure that resembles the learning process observed in several experiments. The model has explicit mechanisms that bind feature information to category structure, but also allows the degree of association through repeated presentations to affect the acquisition of new or consistent knowledge. When feature-to-category associations are first being formed, changes in the strength of associations are rapid, enabling a general sense of category structure to be established quickly. However, with continued and consistent associations, the changes in associative strengths asymptote, meaning that further stimulus exposure does not necessarily change the category representation. In this model, both prior and likelihood information are latent in the category knowledge distributions. When the model is fitted to human data, it only conforms to optimal Bayesian integration under some parameterizations (Anderson Reference Anderson1991; McClelland Reference McClelland2013; Sanborn et al. Reference Sanborn, Griffiths and Navarro2010; Shi et al. Reference Shi, Griffiths, Feldman and Sanborn2010; Turner, Reference Turnerunder review; Turner & Van Zandt Reference Turner and Van Zandt2014; Turner et al. Reference Turner, Van Zandt and Brown2011). Turner (Reference Turnerunder review) investigated the degree to which the Bayes optimal version of the model could account for data across three experiments that exploited unannounced shifts in the properties of the feature-to-category maps over time. Across all experiments, the optimal version of the model only accounted for 11 out of 135 subjects, suggesting generally weak evidence for optimal category learning across different manipulations.
We have also observed suboptimal performance in perceptual decision making when subjects are asked to integrate multiple cues to form a decision. In the context of multimodal integration, Turner et al. (Reference Turner, Gao, Koenig, Palfy and McClelland2017) investigated how subjects integrate two sources of information – auditory and visual cues – to arrive at a single perceptual representation. In their experiment, subjects completed three conditions, varying on whether subjects experienced (1) visual information alone, (2) auditory information alone, or (3) both streams of information together (the bimodal condition). Having the two unimodal conditions allowed Turner et al. to isolate the individual contribution of each modality in the decisions observed in the bimodal condition. Following the literature on optimal decision making, Turner et al. examined three different methods for integrating the two streams of information. The first two considered different forms of optimal integration, whereas the third method simply allowed the weights assigned to each cue to be freely estimated. By allowing the modality weighting parameters to freely vary, the third model could be compared to the two optimal models. If the (posterior estimate of the) weight parameter for the third model overlapped with the weights derived from either optimal model, then subjects could conceivably be combining the cue information together in an optimal fashion. However, of the six subjects in Turner et al., only two subjects showed any evidence of such an overlap, indicating that auditory and visual cues were not being integrated optimally by most of the participants.
In another multiple source integration study of perceptual decision making, Hotaling et al. (Reference Hotaling, Cohen, Shiffrin and Busemeyer2015) examined optimality of inference with an experiment in which participants categorized faces based on resemblance to a family patriarch. Each face was split into two parts, and the amount of evidence contained in the top and bottom halves of each test face was independently manipulated. In one condition, the two halves were aligned to appear as a configural whole face; in another condition, the two halves were misaligned into separate parts. These data allow us to investigate a canonical example of suboptimal information integration from the judgment and decision-making literature, the dilution effect, which refers to the finding that adding weak but positive evidence to strong positive evidence for a hypothesis decreases the estimated support participants derive for that hypothesis, rather than increasing it as required by Bayesian inference. Hotaling et al. used a multicomponent information accumulation model, a hybrid optimal/averaging model of information integration, to successfully account for key accuracy, response time, and dilution effects. The model weights represented the probability of attending to optimal or suboptimal information, which provided a parameter to assess an individual's approximation to optimal integration. Splitting the top and bottom halves of a face, a manipulation meant to encourage controlled integration of information, produced farther from optimal behavior and larger dilution effects.
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Rahnev & Denison (R&D) perform a useful service by reminding us that human behavior often deviates from what can be construed as optimal, once some target variable (such as expected reward) has been specified. We would like to remind readers that Bayesian statistics can be used to fit models and to contribute to our assessment of the degree of optimality. This differs from the usual Bayes rational approach (BRA). Unlike BRA, Bayesian statistics can be used to assess optimality by fitting a model to data and assessing how close the resulting parameter estimates are to optimal settings. For example, if an observer model is specified in such a way that one parameter combination results in optimal performance, one can use posterior estimates to quantify (i.e., in terms of both the parameter estimates and their uncertainty) how likely subjects were to have been performing optimally.
We have used such techniques in our own work to further support the author's conclusion that optimality is rarely observed in human decision making. The models we use are often process models that are not formulated within the likelihood function, prior, cost function, and decision rule (LPCD) framework, although variables or relationships among them within the models could be used to derive LPCD quantities. Hence, LPCD may not characterize the actual computations performed during decision making. We illustrate these points through consideration of two topics, as discussed subsequently.
Although the authors note many suboptimal results in the signal detection theory model, it may be useful to separate the question of whether people are optimal from the question of whether their behavior is best understood in terms of the elements of LPCD. Some other process models might allow the constructs of LPCD to be seen as descriptions of the outcome of a learning process that does not employ these constructs directly. Turner et al. (Reference Turner, Van Zandt and Brown2011) developed a category-learning model that encodes feature information about categories – the more general case of signal detection – and gradually forms a representation of category structure that resembles the learning process observed in several experiments. The model has explicit mechanisms that bind feature information to category structure, but also allows the degree of association through repeated presentations to affect the acquisition of new or consistent knowledge. When feature-to-category associations are first being formed, changes in the strength of associations are rapid, enabling a general sense of category structure to be established quickly. However, with continued and consistent associations, the changes in associative strengths asymptote, meaning that further stimulus exposure does not necessarily change the category representation. In this model, both prior and likelihood information are latent in the category knowledge distributions. When the model is fitted to human data, it only conforms to optimal Bayesian integration under some parameterizations (Anderson Reference Anderson1991; McClelland Reference McClelland2013; Sanborn et al. Reference Sanborn, Griffiths and Navarro2010; Shi et al. Reference Shi, Griffiths, Feldman and Sanborn2010; Turner, Reference Turnerunder review; Turner & Van Zandt Reference Turner and Van Zandt2014; Turner et al. Reference Turner, Van Zandt and Brown2011). Turner (Reference Turnerunder review) investigated the degree to which the Bayes optimal version of the model could account for data across three experiments that exploited unannounced shifts in the properties of the feature-to-category maps over time. Across all experiments, the optimal version of the model only accounted for 11 out of 135 subjects, suggesting generally weak evidence for optimal category learning across different manipulations.
We have also observed suboptimal performance in perceptual decision making when subjects are asked to integrate multiple cues to form a decision. In the context of multimodal integration, Turner et al. (Reference Turner, Gao, Koenig, Palfy and McClelland2017) investigated how subjects integrate two sources of information – auditory and visual cues – to arrive at a single perceptual representation. In their experiment, subjects completed three conditions, varying on whether subjects experienced (1) visual information alone, (2) auditory information alone, or (3) both streams of information together (the bimodal condition). Having the two unimodal conditions allowed Turner et al. to isolate the individual contribution of each modality in the decisions observed in the bimodal condition. Following the literature on optimal decision making, Turner et al. examined three different methods for integrating the two streams of information. The first two considered different forms of optimal integration, whereas the third method simply allowed the weights assigned to each cue to be freely estimated. By allowing the modality weighting parameters to freely vary, the third model could be compared to the two optimal models. If the (posterior estimate of the) weight parameter for the third model overlapped with the weights derived from either optimal model, then subjects could conceivably be combining the cue information together in an optimal fashion. However, of the six subjects in Turner et al., only two subjects showed any evidence of such an overlap, indicating that auditory and visual cues were not being integrated optimally by most of the participants.
In another multiple source integration study of perceptual decision making, Hotaling et al. (Reference Hotaling, Cohen, Shiffrin and Busemeyer2015) examined optimality of inference with an experiment in which participants categorized faces based on resemblance to a family patriarch. Each face was split into two parts, and the amount of evidence contained in the top and bottom halves of each test face was independently manipulated. In one condition, the two halves were aligned to appear as a configural whole face; in another condition, the two halves were misaligned into separate parts. These data allow us to investigate a canonical example of suboptimal information integration from the judgment and decision-making literature, the dilution effect, which refers to the finding that adding weak but positive evidence to strong positive evidence for a hypothesis decreases the estimated support participants derive for that hypothesis, rather than increasing it as required by Bayesian inference. Hotaling et al. used a multicomponent information accumulation model, a hybrid optimal/averaging model of information integration, to successfully account for key accuracy, response time, and dilution effects. The model weights represented the probability of attending to optimal or suboptimal information, which provided a parameter to assess an individual's approximation to optimal integration. Splitting the top and bottom halves of a face, a manipulation meant to encourage controlled integration of information, produced farther from optimal behavior and larger dilution effects.