What kind of explanation does a cognitive model offer? A standard way of approaching this question is to use Marr's (Reference Marr1982) three levels of explanation. A “computational analysis” provides an abstract description of the problem that the learner must solve, along with a normative account of how that problem should be solved. Bayesian models of cognition are usually computational level explanations. An “algorithmic level” explanation describes a mechanistic process that would produce human-like behavior in some task. Most traditional information-processing models and many connectionist models lie at this level of explanation. Finally, “implementation level” explanations propose a low-level physical explanation of how the brain might perform the computations that are required. These are the kinds of models typically pursued in cognitive neuroscience.
Whereabouts in this classification scheme should we place the quantum probability (QP) framework? The implementation level is the simplest to consider. Pothos & Busemeyer (P&B) explicitly disavow any implementation level interpretation of these models; making a clear distinction between their work on the formal modeling of cognition using a quantum formalism and those researchers (e.g., Hameroff 1998) who argue that neural function should be modeled as a quantum physical system. We agree with this distinction.
Should QP models be treated as computational level analyses? Although P&B make explicit comparisons to classical probability and to Bayesian models, we do not think it makes sense to treat QP models as computational level analyses. The critical characteristic of a computational analysis is to specify what problem the learner is solving, and to present a normative account of how that problem should be solved. Bayesian models work well as computational analyses because of the fact that classical probability provides good rules for probabilistic inference in everyday life. In discussing this issue, P&B point to problems associated with statistical decision theory (e.g., that Dutch books are possible in some cases), or to well- known issues with the Kolmogorov axioms (e.g., sample spaces are hard to define in real world contexts). However, in our view their discussion misses the forest for the trees: showing that classical probability has limitations does not establish QP as a plausible alternative. There is a good reason why statistics is built on top of classical probability and not quantum probability: it is the right tool for the job of defining normative inferences in everyday data analysis. In contrast, although there are such things as “quantum t-tests” (e.g., Kumagai & Hayashi Reference Kumagai and Hayashi2011), they have yet to find a natural role within everyday statistical analysis. It is possible that such usage may emerge in time, but we think this is unlikely, simply because the situations to which such tools are applicable (e.g., data follow a quantum Gaussian distribution) do not arise very often when one is analyzing real data. Until statistical tools based on QP find a place in everyday data analysis, we remain unconvinced that QP makes sense as a normative account of everyday inference.
Regarding the algorithmic level, we think that P&B are on more solid ground: there is some justification for thinking about QP models as mechanistic accounts. Consider the model used to account for Shafir and Tversky's (Reference Shafir and Tversky1992) data on the prisoner's dilemma. It relies on an interference effect to account for the fact that participants defect whenever the opponent's action is known but cooperate when it is unknown. This interference does not emerge as part of an optimal solution to the inference problem given to the decision maker, nor is it characterized at a neural level. It is clearly intended to refer to a psychological mechanism of some kind.
In view of this, a mechanistic view of QP seems to provide the right way forward, but at times it is difficult to understand what the mechanisms actually are. To take a simple example, why are some questions incompatible and others are compatible? P&B suggest that “[a] heuristic guide of whether some questions should be considered compatible or not is whether clarifying one is expected to interfere with the evaluation of the other” (sect. 2.2). This seems sensible, but it begs the question. One is naturally led to ask why some psychological states interfere and others do not. This is difficult to answer because the QP formalism is silent on how its central constructs (e.g., interference) map onto psychological mechanisms. In our own work (Fuss & Navarro, in press) we have explored this issue in regards to the dynamic equations that describe how quantum states change over time. Specifically, we have sought to describe how these equations could arise from mechanistic processes, but our solution is specific to a particular class of models and we do not claim to have solved the problem in general. In our view, understanding how formalisms map onto mechanisms is one of the biggest open questions within the QP framework.
In short, we think that the potential in QP lies in developing sensible, interpretable psychological mechanisms that can account for the otherwise puzzling inconsistencies in human decision making. It might be that human cognition cannot be described using the standard provided by classical probability theory, but turns out to be more consistent with QP theory. That doesn't make QP a good tool for rational analysis, but it would make it an interesting psychological mechanism, particularly if it is possible to provide clear and consistent interpretations for its central constructs. Should events unfold in this way, then statistics would continue to rely on classical probability for its theoretical foundation, but cognitive modelers could use quantum probability in many instances. There is nothing incompatible about these two states.
What kind of explanation does a cognitive model offer? A standard way of approaching this question is to use Marr's (Reference Marr1982) three levels of explanation. A “computational analysis” provides an abstract description of the problem that the learner must solve, along with a normative account of how that problem should be solved. Bayesian models of cognition are usually computational level explanations. An “algorithmic level” explanation describes a mechanistic process that would produce human-like behavior in some task. Most traditional information-processing models and many connectionist models lie at this level of explanation. Finally, “implementation level” explanations propose a low-level physical explanation of how the brain might perform the computations that are required. These are the kinds of models typically pursued in cognitive neuroscience.
Whereabouts in this classification scheme should we place the quantum probability (QP) framework? The implementation level is the simplest to consider. Pothos & Busemeyer (P&B) explicitly disavow any implementation level interpretation of these models; making a clear distinction between their work on the formal modeling of cognition using a quantum formalism and those researchers (e.g., Hameroff 1998) who argue that neural function should be modeled as a quantum physical system. We agree with this distinction.
Should QP models be treated as computational level analyses? Although P&B make explicit comparisons to classical probability and to Bayesian models, we do not think it makes sense to treat QP models as computational level analyses. The critical characteristic of a computational analysis is to specify what problem the learner is solving, and to present a normative account of how that problem should be solved. Bayesian models work well as computational analyses because of the fact that classical probability provides good rules for probabilistic inference in everyday life. In discussing this issue, P&B point to problems associated with statistical decision theory (e.g., that Dutch books are possible in some cases), or to well- known issues with the Kolmogorov axioms (e.g., sample spaces are hard to define in real world contexts). However, in our view their discussion misses the forest for the trees: showing that classical probability has limitations does not establish QP as a plausible alternative. There is a good reason why statistics is built on top of classical probability and not quantum probability: it is the right tool for the job of defining normative inferences in everyday data analysis. In contrast, although there are such things as “quantum t-tests” (e.g., Kumagai & Hayashi Reference Kumagai and Hayashi2011), they have yet to find a natural role within everyday statistical analysis. It is possible that such usage may emerge in time, but we think this is unlikely, simply because the situations to which such tools are applicable (e.g., data follow a quantum Gaussian distribution) do not arise very often when one is analyzing real data. Until statistical tools based on QP find a place in everyday data analysis, we remain unconvinced that QP makes sense as a normative account of everyday inference.
Regarding the algorithmic level, we think that P&B are on more solid ground: there is some justification for thinking about QP models as mechanistic accounts. Consider the model used to account for Shafir and Tversky's (Reference Shafir and Tversky1992) data on the prisoner's dilemma. It relies on an interference effect to account for the fact that participants defect whenever the opponent's action is known but cooperate when it is unknown. This interference does not emerge as part of an optimal solution to the inference problem given to the decision maker, nor is it characterized at a neural level. It is clearly intended to refer to a psychological mechanism of some kind.
In view of this, a mechanistic view of QP seems to provide the right way forward, but at times it is difficult to understand what the mechanisms actually are. To take a simple example, why are some questions incompatible and others are compatible? P&B suggest that “[a] heuristic guide of whether some questions should be considered compatible or not is whether clarifying one is expected to interfere with the evaluation of the other” (sect. 2.2). This seems sensible, but it begs the question. One is naturally led to ask why some psychological states interfere and others do not. This is difficult to answer because the QP formalism is silent on how its central constructs (e.g., interference) map onto psychological mechanisms. In our own work (Fuss & Navarro, in press) we have explored this issue in regards to the dynamic equations that describe how quantum states change over time. Specifically, we have sought to describe how these equations could arise from mechanistic processes, but our solution is specific to a particular class of models and we do not claim to have solved the problem in general. In our view, understanding how formalisms map onto mechanisms is one of the biggest open questions within the QP framework.
In short, we think that the potential in QP lies in developing sensible, interpretable psychological mechanisms that can account for the otherwise puzzling inconsistencies in human decision making. It might be that human cognition cannot be described using the standard provided by classical probability theory, but turns out to be more consistent with QP theory. That doesn't make QP a good tool for rational analysis, but it would make it an interesting psychological mechanism, particularly if it is possible to provide clear and consistent interpretations for its central constructs. Should events unfold in this way, then statistics would continue to rely on classical probability for its theoretical foundation, but cognitive modelers could use quantum probability in many instances. There is nothing incompatible about these two states.