The case for using quantum probability that Pothos & Busemeyer (P&B) present seems to be that quantum probability – like classical Bayesian probability – supports a probabilistic approach to cognitive modelling, but – unlike classical probability – predicts behaviour consistent with the attributes of bounded cognition such as the normal limitations in memory, processing capacity, and attentional control. However, before accepting this intriguing proposal, there are questions to consider. Is classical probability theory truly inconsistent with behavioural phenomena attributable to bounded cognition? Do we need alternatives to existing models that already incorporate insights concerning bounded cognition? Are there dangers in a unified quantum probability framework that subsumes classical probability and bounded cognition?
One feature of assigning probabilities is that uncertainty can be aleatory or epistemic. Aleatory uncertainty derives from randomness in determining outcomes, whereas epistemic uncertainty reflects incomplete knowledge of outcomes and their precipitating factors. For example, prior to flipping a coin I assign Prob(Heads) = 0.5 (aleatory uncertainty); however, I still assign Prob(Heads) = 0.5 if the coin has been flipped but is covered (epistemic uncertainty). Often, both kinds of uncertainty are implied in probability assessments: someone may be uncertain whether he or she is employed at time t both because there is some randomness in determining employment termination, and because he or she does not know what managerial decisions have occurred. Importantly, from a classical probability perspective, probabilities are always conditional on current knowledge; and, consequently, some of P&B's illustrations are perhaps not so puzzling for classical probability theory as they suggest. My state of knowledge does differ according to whether someone asks me if Gore is honest before or after asking me whether Clinton is honest. In one case my knowledge includes the information “I have just been asked to assess Clinton's honesty,” and in the other case it does not.
Moreover, as per the standard explanations of order effects that P&B acknowledge, not only does my assessment of Gore alter when comparison with Clinton is encouraged, but also my classification for “honesty” may change (e.g., the category boundaries for honesty move). If the two questions about Gore refer to different events (i.e., alternative conceptions of honesty), there is little puzzlement for classical probability theory when the probabilities of “yes” differ between them. Therefore, if classical probability theory informed by the bounded nature of cognition can accommodate order effects (and other effects described by P&B) what does it gain us to have an alternative – arguably less intuitive – account from quantum probability theory? Numerous probabilistic cognitive models (including those previously championed by P&B) already incorporate features of bounded cognition, including limitations in memory (Bush & Mostellar Reference Bush and Mostellar1955), attentional control (Birnbaum Reference Birnbaum2008), or processing capacity (Tversky Reference Tversky1972). P&B seem to be proposing models that bypass the need for instantiating definable psychological processes within cognitive models, because quantum probability can account for phenomena usually explained through bounded cognition by another route. Seemingly, in quantum probability models of cognition, all the “work” is done by a probability theory with the flexibility to account for many findings, without incorporating cognitive theory in the model. How does this advance our understanding of psychological processes?
P&B propose that: “superposition appears an intuitive way to characterize the fuzziness (the conflict, ambiguity, and ambivalence) of everyday thought” (sect. 1.1). Surely, we already have an intuitive explanation for the fuzziness of everyday thought: thinking includes difficult assessments of unpredictable events (aleatory uncertainty), under conditions of incomplete knowledge (epistemic uncertainty), where often only a subset of the potentially available information is used (bounded cognition). What does “superposition” bring to the table if we already accept that information is incomplete? How does “time evolution” differ from acknowledging that knowledge states can change rapidly? What does “interference” add to the simple observation that attention shifts between the reasons or factors that underpin assessments?
I acknowledge that it is a weak argument against quantum probability formulations of cognitive models that they offer no more than existing accounts: there is no particular reason to disfavour a framework just because other frameworks already exist, and arguably there is always room for another account that addresses a different level of explanation. In other words, if quantum probability = classical probability + bounded cognition, then quantum probability theory deserves its place. That said, it seems odd to propose a class of cognitive models that, if I understand P&B correctly, need fewer components that relate to cognitive processes because quantum probability theory already predicts phenomena that those cognitive processes explain.
Additionally, there are potential dangers associated with a framework that subsumes an earlier one. For compatible questions, quantum probability reduces to classical probability, and, therefore, P&B rightly assert that the modeler determines a priori which questions are (in) compatible (sect. 2.2). However, just how a priori is a priori? If incompatibility is determined prior to specifying any formal models but after summarising the data, we risk the following unsatisfactory state of affairs:
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1. How do we know that these questions are incompatible?
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2. Because classical probability theory was violated.
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3. Why does the model predict violations of classical probability theory?
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4. Because these questions are incompatible.
If incompatibility is determined prior to data collection, this problem does not disappear, because we already know much about predicting the kinds of situations in which order effects, and other violations of classical probability theory, occur. If the incompatibility of questions cannot be specified in some way that is fully independent of the data, then quantum probability models risk benefitting from the mother of all free parameters: explaining data consistent with classical probability, and data that violate classical probability theory. Similar concerns about the flexibility in model specification afforded by quantum probability arise in the example of modeling the Linda problem. The assumption behind placing the quantum probability subspaces was that feminism is “largely uninformative to whether a person is a bank teller or not” (sect. 3.1). However, the standard account that Linda is representative of a feminist but unrepresentative of a bank teller might imply that people perceive that feminism is informative (radical feminism being inconsistent with bank telling). Therefore, for the legitimate application of quantum probability to cognitive modeling, we may also need clear guidance on the a priori specification of subspaces. Otherwise, without such checks and balances on the specification of quantum probability models, although, seemingly, such models could account for everything, they would explain nothing (Glöckner & Betsch Reference Glöckner and Betsch2011).
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The case for using quantum probability that Pothos & Busemeyer (P&B) present seems to be that quantum probability – like classical Bayesian probability – supports a probabilistic approach to cognitive modelling, but – unlike classical probability – predicts behaviour consistent with the attributes of bounded cognition such as the normal limitations in memory, processing capacity, and attentional control. However, before accepting this intriguing proposal, there are questions to consider. Is classical probability theory truly inconsistent with behavioural phenomena attributable to bounded cognition? Do we need alternatives to existing models that already incorporate insights concerning bounded cognition? Are there dangers in a unified quantum probability framework that subsumes classical probability and bounded cognition?
One feature of assigning probabilities is that uncertainty can be aleatory or epistemic. Aleatory uncertainty derives from randomness in determining outcomes, whereas epistemic uncertainty reflects incomplete knowledge of outcomes and their precipitating factors. For example, prior to flipping a coin I assign Prob(Heads) = 0.5 (aleatory uncertainty); however, I still assign Prob(Heads) = 0.5 if the coin has been flipped but is covered (epistemic uncertainty). Often, both kinds of uncertainty are implied in probability assessments: someone may be uncertain whether he or she is employed at time t both because there is some randomness in determining employment termination, and because he or she does not know what managerial decisions have occurred. Importantly, from a classical probability perspective, probabilities are always conditional on current knowledge; and, consequently, some of P&B's illustrations are perhaps not so puzzling for classical probability theory as they suggest. My state of knowledge does differ according to whether someone asks me if Gore is honest before or after asking me whether Clinton is honest. In one case my knowledge includes the information “I have just been asked to assess Clinton's honesty,” and in the other case it does not.
Moreover, as per the standard explanations of order effects that P&B acknowledge, not only does my assessment of Gore alter when comparison with Clinton is encouraged, but also my classification for “honesty” may change (e.g., the category boundaries for honesty move). If the two questions about Gore refer to different events (i.e., alternative conceptions of honesty), there is little puzzlement for classical probability theory when the probabilities of “yes” differ between them. Therefore, if classical probability theory informed by the bounded nature of cognition can accommodate order effects (and other effects described by P&B) what does it gain us to have an alternative – arguably less intuitive – account from quantum probability theory? Numerous probabilistic cognitive models (including those previously championed by P&B) already incorporate features of bounded cognition, including limitations in memory (Bush & Mostellar Reference Bush and Mostellar1955), attentional control (Birnbaum Reference Birnbaum2008), or processing capacity (Tversky Reference Tversky1972). P&B seem to be proposing models that bypass the need for instantiating definable psychological processes within cognitive models, because quantum probability can account for phenomena usually explained through bounded cognition by another route. Seemingly, in quantum probability models of cognition, all the “work” is done by a probability theory with the flexibility to account for many findings, without incorporating cognitive theory in the model. How does this advance our understanding of psychological processes?
P&B propose that: “superposition appears an intuitive way to characterize the fuzziness (the conflict, ambiguity, and ambivalence) of everyday thought” (sect. 1.1). Surely, we already have an intuitive explanation for the fuzziness of everyday thought: thinking includes difficult assessments of unpredictable events (aleatory uncertainty), under conditions of incomplete knowledge (epistemic uncertainty), where often only a subset of the potentially available information is used (bounded cognition). What does “superposition” bring to the table if we already accept that information is incomplete? How does “time evolution” differ from acknowledging that knowledge states can change rapidly? What does “interference” add to the simple observation that attention shifts between the reasons or factors that underpin assessments?
I acknowledge that it is a weak argument against quantum probability formulations of cognitive models that they offer no more than existing accounts: there is no particular reason to disfavour a framework just because other frameworks already exist, and arguably there is always room for another account that addresses a different level of explanation. In other words, if quantum probability = classical probability + bounded cognition, then quantum probability theory deserves its place. That said, it seems odd to propose a class of cognitive models that, if I understand P&B correctly, need fewer components that relate to cognitive processes because quantum probability theory already predicts phenomena that those cognitive processes explain.
Additionally, there are potential dangers associated with a framework that subsumes an earlier one. For compatible questions, quantum probability reduces to classical probability, and, therefore, P&B rightly assert that the modeler determines a priori which questions are (in) compatible (sect. 2.2). However, just how a priori is a priori? If incompatibility is determined prior to specifying any formal models but after summarising the data, we risk the following unsatisfactory state of affairs:
1. How do we know that these questions are incompatible?
2. Because classical probability theory was violated.
3. Why does the model predict violations of classical probability theory?
4. Because these questions are incompatible.
If incompatibility is determined prior to data collection, this problem does not disappear, because we already know much about predicting the kinds of situations in which order effects, and other violations of classical probability theory, occur. If the incompatibility of questions cannot be specified in some way that is fully independent of the data, then quantum probability models risk benefitting from the mother of all free parameters: explaining data consistent with classical probability, and data that violate classical probability theory. Similar concerns about the flexibility in model specification afforded by quantum probability arise in the example of modeling the Linda problem. The assumption behind placing the quantum probability subspaces was that feminism is “largely uninformative to whether a person is a bank teller or not” (sect. 3.1). However, the standard account that Linda is representative of a feminist but unrepresentative of a bank teller might imply that people perceive that feminism is informative (radical feminism being inconsistent with bank telling). Therefore, for the legitimate application of quantum probability to cognitive modeling, we may also need clear guidance on the a priori specification of subspaces. Otherwise, without such checks and balances on the specification of quantum probability models, although, seemingly, such models could account for everything, they would explain nothing (Glöckner & Betsch Reference Glöckner and Betsch2011).