We commend Jones & Love's (J&L's) call for a greater integration of “Fundamentalist” Bayesian models with algorithmic- and implementational-level models in psychology. We agree that such integrations would significantly improve the empirical testability of Bayesian models, particularly in light of the currently unsatisfactory fit of Bayesian models with normative accounts of rational inference and choice. At the same time, we believe that this call for integration must be accompanied by a call for the development of new, complementary accounts of rationality for the algorithmic and implementational levels.
The target article might be read (mistakenly, in our view) as a complete surrender of the rational or computational level of modeling based on the difficulty of testing Bayesian models empirically. J&L suggest that rational Bayesian models can be tested only if the models are subject to constraints that span across the three levels – constraints which are not provided by extant Bayesian models. The problem is exacerbated by the apparent gap between standard accounts of rationality supporting the Bayesian paradigm (such as diachronic Dutch book arguments) and the empirical evidence used to confirm Bayesian models (see, e.g., Eberhardt & Danks [Reference Eberhardt and Danks2011] for a concrete example). The appropriate remedy, J&L suggest, is to tie the Bayesian models to the algorithmic- and implementational-level models. But on its own, such an integration would provide only a reduction of the computational level to the algorithmic and implementational levels, and so would relinquish the explanatory benefits the computational level provides.
Bayesian models have achieved widespread popularity in part because, as computational-level models, they describe an inference procedure that is rational or optimal given the agent's task. Bayesian models purport to provide a certain type of explanation – an optimality explanation, rather than a mechanistic one – while not being committed to a particular algorithm or implementation in the learner's brain. As already noted, however, these “explanations” are undermined by the misfit between empirical results and standard accounts of rationality. The common response in the so-called Fundamentalist Bayesian literature (to the extent that there is one) has been that rational behavior is defined on a case-by-case basis, where the appropriate principles of rationality depend on the learner's particular context (see, e.g., Oaksford & Chater Reference Oaksford and Chater2007). Obviously, from a normative perspective, such claims are unsatisfactory: We need an account of rational principles that are fixed independently of the learner's behavior.
We contend that the target article's call for a closer integration of the computational and algorithmic levels of modeling provides an ideal opportunity: Rather than attempting to develop an account of rational behavior independent of the learner's cognitive infrastructure, an integration of levels should go hand in hand with the development of an account of algorithmic rationality. That is, we require an account of rationality that is informed by the learner's constraints at the implementational and algorithmic levels. By providing such an integrated account of rationality, we can resist the purely reductionist view in which a computational-level model is ultimately assessed only on the basis of its success at the algorithmic and implementational levels, while preserving the (additional, distinctive) explanatory power provided by rational models for all levels. (J&L's discussion of a possible “Bayesian Enlightenment” is, we think, closely related.)
What would an account of algorithmic rationality look like? Often, suggestions along these lines are made under the heading of bounded rationality. In many cases, however, models of bounded rationality (e.g., Simon's “satisficing,” or Gigerenzer's “fast and frugal heuristics”) propose boundedness constraints that are not informed by particular implementational constraints of the learner, but rather are motivated by abstract limitations or efficiencies of the computational procedure. Nevertheless, they are on the right track. Accounts of algorithmic rationality would have to provide support for models that are empirically adequate, and must approximate behavior that is deemed ideally rational (e.g., maximization of expected utility, avoidance of Dutch book, etc.). At the same time, accounts of algorithmic rationality must integrate known implementational (i.e., biological) constraints. The resulting models would be neither just computational nor just algorithmic, but would instead provide an account that is both descriptively adequate and normatively supported. Confirmation of such a model would require both empirical confirmation at multiple levels (e.g., matching the right input-output relation, predicting that neurons will fire in particular places at particular times), as well as an account of why this behavior is the rational or optimal thing to do in light of the given task and the cognitive and neural infrastructures.
We thus argue that there needs to be not only a “tying down” of the rational models to algorithmic- and implementational-level constraints, but also a “pulling up” of algorithmic-level models to go beyond purely descriptive accounts (that occasionally come paired with evolutionary “how-possibly” stories) to explanations of why the particular algorithm or implementation is the right one for the task. The fundamental insight of Newell and Simon, Pylyshyn, and Marr, that there are different levels of explanation, has led to enormous productivity at each of the different levels. J&L are right to point out that these efforts have drifted apart: Models at different levels are frequently no longer informative about other levels. However, a re-integration that simply reduces one of the levels to the others would abandon one of the distinctive forms of explanation identified in that original insight. We urge instead that any re-integration must include the development of an account of algorithmic-level rationality, which then permits us to determine whether Bayesian models remain successful, both empirically and theoretically.
We commend Jones & Love's (J&L's) call for a greater integration of “Fundamentalist” Bayesian models with algorithmic- and implementational-level models in psychology. We agree that such integrations would significantly improve the empirical testability of Bayesian models, particularly in light of the currently unsatisfactory fit of Bayesian models with normative accounts of rational inference and choice. At the same time, we believe that this call for integration must be accompanied by a call for the development of new, complementary accounts of rationality for the algorithmic and implementational levels.
The target article might be read (mistakenly, in our view) as a complete surrender of the rational or computational level of modeling based on the difficulty of testing Bayesian models empirically. J&L suggest that rational Bayesian models can be tested only if the models are subject to constraints that span across the three levels – constraints which are not provided by extant Bayesian models. The problem is exacerbated by the apparent gap between standard accounts of rationality supporting the Bayesian paradigm (such as diachronic Dutch book arguments) and the empirical evidence used to confirm Bayesian models (see, e.g., Eberhardt & Danks [Reference Eberhardt and Danks2011] for a concrete example). The appropriate remedy, J&L suggest, is to tie the Bayesian models to the algorithmic- and implementational-level models. But on its own, such an integration would provide only a reduction of the computational level to the algorithmic and implementational levels, and so would relinquish the explanatory benefits the computational level provides.
Bayesian models have achieved widespread popularity in part because, as computational-level models, they describe an inference procedure that is rational or optimal given the agent's task. Bayesian models purport to provide a certain type of explanation – an optimality explanation, rather than a mechanistic one – while not being committed to a particular algorithm or implementation in the learner's brain. As already noted, however, these “explanations” are undermined by the misfit between empirical results and standard accounts of rationality. The common response in the so-called Fundamentalist Bayesian literature (to the extent that there is one) has been that rational behavior is defined on a case-by-case basis, where the appropriate principles of rationality depend on the learner's particular context (see, e.g., Oaksford & Chater Reference Oaksford and Chater2007). Obviously, from a normative perspective, such claims are unsatisfactory: We need an account of rational principles that are fixed independently of the learner's behavior.
We contend that the target article's call for a closer integration of the computational and algorithmic levels of modeling provides an ideal opportunity: Rather than attempting to develop an account of rational behavior independent of the learner's cognitive infrastructure, an integration of levels should go hand in hand with the development of an account of algorithmic rationality. That is, we require an account of rationality that is informed by the learner's constraints at the implementational and algorithmic levels. By providing such an integrated account of rationality, we can resist the purely reductionist view in which a computational-level model is ultimately assessed only on the basis of its success at the algorithmic and implementational levels, while preserving the (additional, distinctive) explanatory power provided by rational models for all levels. (J&L's discussion of a possible “Bayesian Enlightenment” is, we think, closely related.)
What would an account of algorithmic rationality look like? Often, suggestions along these lines are made under the heading of bounded rationality. In many cases, however, models of bounded rationality (e.g., Simon's “satisficing,” or Gigerenzer's “fast and frugal heuristics”) propose boundedness constraints that are not informed by particular implementational constraints of the learner, but rather are motivated by abstract limitations or efficiencies of the computational procedure. Nevertheless, they are on the right track. Accounts of algorithmic rationality would have to provide support for models that are empirically adequate, and must approximate behavior that is deemed ideally rational (e.g., maximization of expected utility, avoidance of Dutch book, etc.). At the same time, accounts of algorithmic rationality must integrate known implementational (i.e., biological) constraints. The resulting models would be neither just computational nor just algorithmic, but would instead provide an account that is both descriptively adequate and normatively supported. Confirmation of such a model would require both empirical confirmation at multiple levels (e.g., matching the right input-output relation, predicting that neurons will fire in particular places at particular times), as well as an account of why this behavior is the rational or optimal thing to do in light of the given task and the cognitive and neural infrastructures.
We thus argue that there needs to be not only a “tying down” of the rational models to algorithmic- and implementational-level constraints, but also a “pulling up” of algorithmic-level models to go beyond purely descriptive accounts (that occasionally come paired with evolutionary “how-possibly” stories) to explanations of why the particular algorithm or implementation is the right one for the task. The fundamental insight of Newell and Simon, Pylyshyn, and Marr, that there are different levels of explanation, has led to enormous productivity at each of the different levels. J&L are right to point out that these efforts have drifted apart: Models at different levels are frequently no longer informative about other levels. However, a re-integration that simply reduces one of the levels to the others would abandon one of the distinctive forms of explanation identified in that original insight. We urge instead that any re-integration must include the development of an account of algorithmic-level rationality, which then permits us to determine whether Bayesian models remain successful, both empirically and theoretically.