Critique of pure Bayesian cognitive science: A view from the philosophy of science

Bayesian approaches to human cognition have been extensively advocated in the last decades, but sharp objections have been raised too within cognitive science. In this paper, we outline a diagnosis of what has gone wrong with the prevalent strand of Bayesian cognitive science (here labelled pure Bayesian cognitive science), relying on selected illustrations from the psychology of reasoning and tools from the philosophy of science. Bayesians’ reliance on so-called method of rational analysis is a key point of our discussion. We tentatively conclude on a constructive note, though: an appropriately modified variant of Bayesian cognitive science can still be coherently pursued, as some scholars have noted.


Introduction
Bayesian approaches to human cognition have been extensively advocated in the last decades, but sharp objections have been raised too within cognitive science (e.g., Baron, 1991;Bowers & Davis, 2012a, 2012b;Eberhardt & Danks, 2011;Elqayam & Evans, 2011;Glymour, 2007;Marcus & Davis, 2013, 2015;Sloman & Fernbach, 2008).In this paper, we outline a diagnosis of what has gone wrong with a prevalent strand of Bayesian cognitive science (here labelled pure Bayesian cognitive science), relying on selected illustrations from the psychology of reasoning and tools from the philosophy of science.Bayesians' reliance on so-called method of rational uncertainty or inaccuracy, as long as vowels and even numbers are assumed to be rare in the background population (larger deck), a typical illustration being P(vowel) = 0.22 and P(even) = 0.27 (see Oaksford & Chater, 2003, p. 298).You may reach the conclusion that "subjects behave as Bayesians with the rarity assumption" (Oaksford & Chater, 1994, p. 627).

Pure Bayesian cognitive science
We submit our first exhibit above as representing a cornerstone example of contemporary Bayesian cognitive science (BCS) in its most prominent form (Chater et al., 2010;Griffiths et al., 2008;Oaksford & Chater, 2007).For the purposes of the present contribution, we will have to characterize this research program more precisely, and give it a more specific label.We will thus call it pure Bayesian cognitive science (PBCS), and submit that the following claims hold for PBCS.
(1) It draws on the Bayesian framework and formal machinery to account for cognitive phenomena documented by behavioral data.
(2) It regards Bayesianism as a compelling general model of rational inference and judgment under uncertainty.
(3) It is committed to a remarkably weak notion of evidential support from the data.(4) It is bound to imply a subtle but important violation of so-called is / ought divide. 1 (5) It regards humans as essentially rational.
(1) and (2) should not be contentious, we assume.From the work of leading figures of PBCS such as Chater,Oaksford,Tenenbaum,and Griffiths,(1) is evident and (2) is also quite explicit.Concerning (5), note for instance that according to Oaksford and Chater (2007, p. 19) "to show how empirical data on human reasoning can be reconciled with the notion that people are rational" is "the central goal" of their work.The daunting theoretical latitude of the notion of rationality becomes tractable here, at least for our purposes, again due to (2): to count as rational, human behavior should comply appropriately with the benchmark of probabilistic principles of inference and judgment.We expect points (3) and (4) to look much more puzzling in comparison with the others.As they turn out to be important for our argument, we will now explain better their content and why we take it to be tenable that they in fact apply to PBCS.This will require a discussion of "the method of rational analysis". 1As we will see, the relevant violation amounts to regarding normative considerations as relevant to decide a descriptive issue, namely, the status of a cognitive model of actual behavior (see Elqayam & Evans, 2011 for a similar discussion).Such inferential move from ought to is has opposite direction as compared with the traditional is / ought fallacy, in which normative conclusions are drawn from purely descriptive considerations.

The vagaries of rational analysis
The textbook reference for "rational analysis" is Anderson (1990Anderson ( , 1991)).Rational analysis (RA from now on) of a cognitive phenomenon is normally presented as a (iterative) procedure in a series of steps (see.e.g., Anderson, 1990, p. 473;Anderson, 1991, p. 29), along the following lines.
(i) Precisely specify what are the goals of the cognitive system.(ii) Develop a formal model of the environment to which the system is (taken to be) adapted.(iii) Make the minimal assumptions about computational limitations.(iv) Derive the optimal behavioral function given items (i) to (iii).(v) Examine the empirical literature to see if the predictions of the behavioral function are confirmed.(vi) If the predictions are off, iterate, revising and refining the theory.
As a matter of fact, the connection between PBCS and RA has been consistent.RA has been often explicitly invoked (e.g., Dubey & Griffiths, 2020;Goodman et al., 2008;Oaksford & Chater, 2007) and -one could argue -it has been largely presupposed and employed as an appropriate research strategy in the PBCS literature.To illustrate, our understanding of how the Bayesian approach to the abstract Wason task instantiates the rational analysis method is roughly as follows (also see Oaksford & Chater, 1994, pp. 625-626).As concerns (i), the cognitive system is assumed to aim at minimizing uncertainty (alternatively, epistemic inaccuracy) in inquiry.As for (ii), the relevant environment is represented by a sampling model (of the four cards from a larger population).This allows for the derivation of the optimal behavioral function (iv), in fact even without substantial additional computational constraints (iii)provided, however, that some key model parameters are fixed.In fact, consideration of basic empirical findings (v) (the ranking of choice propensities for the four cards) substantially constrains parameter setting, thus leading to the rarity assumption as a specific refinement of the general Bayesian analysis (vi) (see Vindrola & Crupi, 2021 for more details).
But why should anyone follow rational analysis to understand human cognition?According to a recent and important reconstruction (Icard, 2018), the "original motivation for rational analysis" is "to help guide the search for cognitive models" when faced with "a problem of identifiability" (pp.2-3): The problem is two-fold.First, there is an obvious question of where to begin the search for high-level cognitive models.Second, there are often cases in which competing models cannot be distinguished by available measurement.
We take this passage to be revelatory.It implies that the method of RA is supposed to address two issues at once.And in fact, the "problem of identifiability" is best seen as conflating two putative problems.One has to do with theory construction, the other one with theory assessment.Accordingly, sorting out this distinction generates two distinct interpretations of rational analysis.For reasons to be clarified shortly, we will call them constructive and inferential, respectively.

Two sides of rational analysis
The constructive interpretation of RA is, we submit, the following.You start out with the working hypothesis (however motivated) that human cognition complies with Bayesian rationality.You then select a certain class of cognitive tasks as your research target (say, information search, memory retrieval, spatial reasoning, …), aiming at an account of some established phenomena in that domain.Given the breadth and generality of the Bayesian framework, a diligent and intelligent application of the prescriptions of rational analysis will essentially guarantee that you'll end up with a workable model and that the known phenomena will be recovered as implications of your model.In essence, this is nothing but a specific consequence of Duhemian underdetermination in scientific theorizing (see Laudan, 1990;Crupi, 2020).Indeed, a plurality of outcomes will typically be possible through the process, depending on several modelling choices made along the way.But in any event, this use of RA plays a key "heuristic" role in a Lakatosian sense (see Lakatos, 1978): for someone whose premises embed tenets (1), ( 2) and (5) of PBCS (see above), RA does lead to a solution of the theory-construction problem.No presentation of RA relates it to Lakatos to the best of our knowledge, but if a Lakatosian philosopher of science had to try a characterization of the "positive heuristic" of PBCS as a research program (much as Lakatos did with Newtonian physics, for instance), well, a very close approximation to RA would ensue, or so we suggest (also see Baron, 1991 on this point). 2hat we take to be the inferential interpretation of RA yields a rather different story.Here is a sketch of how it operates.Much as before, you target certain phenomena, C, in a given class of cognitive tasks.Typically, there will exist some non-Bayesian account of such phenomena, call it A. Following the steps of RA, you develop a Bayesian account, B, of the same phenomena.You then consider that "competing models", A and B, "cannot be distinguished by available measurement" (Icard, 2018, p. 3), because by hypothesis now both A and B can account for C. Note that, due to how RA proceeds, the Bayesian account of the data will typically be driven by the data themselves through tailored specifications of model and parameters (recall the high level of generality of the Bayesian framework and the crucially "iterative" nature of RA!).As a consequence, the claim of "empirical underdetermination" across competing models (here, A and B) relies, if implicitly, on a collapse of empirical support on plain accommodation of the data.Adopting a terminology from Worrall (2011), "empirical equivalence" is thus reduced to "data equivalence", conveying a suspiciously weak idea of evidential support which is squarely at odds, in particular, with various strands of a "predictivist" view of scientific confirmation (see Crupi, 2020).The collapse of support on accommodation is unjustified because 28 Page 6 of 17 mere data equivalence (fit with the data by each of two competing models or theories) is known to have very limited epistemological import in itself.In particular, if one model predicts a phenomenon that another model only accommodates, the former but not the latter will be supported, data equivalence notwithstanding. 3 Once the "empirical indistinguishability" claim is put forward, however, a case can be made that B is still better than A for theoretical reasons, to wit, because it reconciles human behavior with compelling standards of rationality.Sometimes this move is accompanied by the claim that the proposed Bayesian model has slightly better empirical fit or offers a more unified account as compared to the alternatives, but the emphasis is clearly put on its rational grounding.Here is a good example of this interpretative pattern as regards the analysis of inductive generalization: Our Bayesian model offers a modest but consistent quantitative advantage over the best similarity-based models of generalization, and also predicts qualitative effects of varying sample size that contradict alternative approaches.More importantly, our Bayesian approach has a principled rational foundation […].In contrast, the similarity-based approach requires arbitrary assumptions […] that have no apriori justification (Sanjana & Tenenbaum, 2003, p. 65, emphasis added).
Very similar appeals to the rational foundation of Bayesian models in a comparative assessment can be found elsewhere (e.g., Heit, 1998;Kemp & Tenenbaum, 2003;Kemp et al., 2007;Tenenbaum & Griffiths, 2001).So, in inferential RA, one goes on pointing out that B is to be favored over A because B, unlike A, is an empirically adequate descriptive model that is also supported by a powerful and general normative justification. 4 In this way, the normative status of a theory (an "ought" issue) is meant to contribute to the assessment of the theory as a candidate highlevel description of behavior (an "is" issue).

Cutting the knot
The constructive interpretation and use of RA is entirely legitimate.It serves the purposes of someone who (for whatever reason) starts out with assumptions which uncontroversially characterize PBCS -( 1), (2), and (5) above -and pursues 3 The details of the distinction between prediction and accommodation are themselves a matter of discussion and may vary in subtle ways: see Barnes (2022) for a valuable survey, and Crupi (2023) for a specific proposal and application to a major historical case. 4Some advocates of Bayesian cognitive science explicitly appeal to the normative status of Bayesian inference as supported in usual ways, such as Dutch book arguments (see, e.g., Hahn, 2014, pp. 6-10).Other scholars distinguish "normative rationality", which concerns general standards of correct performance, from "adaptive rationality", which is «defined with respect to a specific environment» (Anderson, 1990, p. 35; but see Oaksford & Chater, 2009 for an explicit attempt to collapse the two notions).In this second sense, specific Bayesian models are rationally/normatively justified based on their optimality with respect to the computational goals of the agents (e.g., Griffiths & Tenenbaum, 2009) or based on their optimal trade-off between decision accuracy and cognitive costs and limitations (e.g., Lieder & Griffiths, 2020).
the development of such research program to accommodate more and more of the empirical basis of the study of human cognition.Such variant of RA is also inferentially inconsequential, because the methodological assessment of whether and to what extent such development provides significant support to the overarching program against competing views remains out of the scope of this theory-construction process.
The inferential interpretation and use of RA, as we sketched it above, is much more intricate and contentious.It is meant to deliver an argument in favor of the rationality of humans in a Bayesian perspective.The starting point is indeed the construction of a Bayesian account of the target phenomena (say, the Wason selection task), to be compared with alternative perspectives (say, Wason's original interpretation).Then the strategy relies on a weak notion of empirical support (point 3 above) to motivate the appeal to considerations beyond "available measurement".The crucial consideration in favor of the Bayesian analysis (and thus of the rationality of behavior, point 5) turns out to be its distinctive normative status (point 2), so that an "ought" statement is taken to contribute (defeasible) support to a claim in the descriptive study of behavior and cognition (point 4).In a slogan, the Bayesian rationality of humans is a premise in constructive RA, while it is meant to be a conclusion in inferential RA.
We have now explained why we believe that the inferential interpretation and use of RA involves a weak notion of evidential support from the data and a violation of so-called is / ought divide.Our conclusion that points (3) and ( 4) above apply to PBCS relies on the additional assumption that work in the PBCS program does often embed RA in the inferential sense.Admittedly, endorsement of the inferential variant of RA is seldom explicit and transparent. 5However, the literature suggests that the constructive and inferential variants of RA are recurrently conflated in PBCS, and indeed the inferential aspect of RA is sometimes rather overtly advocated, like in this important example (Hahn, 2014, p. 10): [T]he point of the approach is a methodological one: rational models aide the disambiguation between competing theories […].[T]his gives such models […] a special status, above and beyond degrees of 'model-fit' and so on.
To be sure, the normative status is not the only theoretical consideration that can be invoked to solve the comparative assessment in inferential RA.As a matter of fact, the unifying power of the Bayesian models in accommodating various sets of empirical data (see Colombo & Hartmann, 2017 for a critical discussion), as well as their adaptive optimality with respect to evolutionary landscapes (see Bowers &   5 In this sense, one might regard inferential RA as an idealization that may not be implemented exactly by any particular study, much as it happens with the idea of "imperial" (vs "local") Bayesianism in Mandelbaum (2019).The critical point, however, is that this notion captures a tendency which is actually present in the Bayesian cognitive science literature -indeed, several analyses closely resemble inferential RA in its pure form (e.g., Oaksford & Chater, 1994;Heit, 1998;Tenenbaum & Griffiths, 2001;Sanjana & Tenenbaum, 2003;Kemp & Tenenbaum, 2003;Kemp et al., 2007).At the same time, our reconstruction offers a coherent philosophical basis for some sparse objections raised by critics of the Bayesian approach, as we will see in Section 7.
28 Page 8 of 17 Davis, 2012a for a critical discussion) have often been often provided as reasons to prefer Bayesian over non-Bayesian accounts.This is particularly true for sub-fields of Bayesian cognitive science such as perceptual (see Rescorla, 2015) and sensorimotor psychology (see Rescorla, 2016), where comparative evaluations with respect to rationality standards are less relevant.
In the empirical study of reasoning and decision-making, however, both in the Bayesian and non-Bayesian tradition, normative considerations have proven to be pervasive and to have generated intense debates (see Elqayam & Evans, 2011;Baron, 2012;Elqayam & Over, 2016;Achourioti et al., 2014;Crupi & Girotto, 2014;Hahn, 2014;Oaksford, 2014).It has been argued that normative standards operate implicitly at many levels in the standard practice of the psychology of reasoning -for example, by constraining the choice of the a priori assumptions at the computational level of analysis (prior rule bias), or by influencing the way in which findings are reported and interpreted on the processing level (interpretation bias) (Elqayam & Evans, 2011; also see Gigerenzer, 1991).It has also been argued that the implicit appeal to normative standards, especially Bayesian norms, is an ineliminable component of our interpretative practices (Oaksford, 2014), making the departure from the is / ought divide a constitutive feature of the empirical evaluation of reasoning and decision making.Moreover, the appeal to (Bayesian) optimality/rationality has been invoked as an integral part of explanations at the computational level in David Marr's terms, where theorists must specify not only what function is computed by the cognitive system but also "why" the cognitive system operates the way it does (e.g., Griffiths et al., 2012a).
Appeals to Marr's methodology are particularly pervasive in PBCS and they deeply affect how empirical investigations are framed and conducted.Here the underlying assumption is that only rational models are suited to provide adequate explanations at the computational level, in that they specify why the supposedly computed function is normatively appropriate for the task at hand.In this sense, "rational model" and "computational model" tend to be used synonymously (see, e.g., Griffiths & Tenenbaum, 2009, pp. 665-666;Griffiths et al., 2012a, pp. 263-264). 6In turn, computational analyses are supposed to constrain the lower levels of explanation, where specific hypotheses about cognitive and neural processes are developed («[w]hatever form those cognitive and neural processes take, they need to approximate the solution to the computational problem», Griffiths et al., 2012a, p. 264).As a consequence, Bayesian cognitive hypotheses turn out to be systematically preferred over non-rational alternatives for they allegedly comply with Marr's framework in a distinctive way (e.g., Kemp & Tenenbaum, 2003;Kemp et al., 2007) and «provide a computational-level description and justification of why some phenomena occur» (Heit, 1998, p. 271, emphasis added).For the same 6 For instance, in Griffiths, Vul, and Sanborn's reading (2012a) of Marr, the computational level specifies «the ideal solution to an abstract statistical problem that people must solve: Given the decision that must be made, how should people use the limited available information?» (p.263).The appeal to rationality/optimality is meant to address the "why" component of explanations at this level: «that the solutions are optimal licenses a particular kind of explanation […] allowing us to assert that the match between the solution and human behavior may be why people act the way they do» (Griffiths et al., 2012a, p. 415).
reason, the comparative assessment is sometimes explicitly restricted to predictions generated by psychological models that have a rational foundation (e.g., Griffiths & Tenenbaum, 2009) or to cognitive algorithms that approximate ideal Bayesian solutions (e.g., Griffiths et al., 2012a). 7one of the considerations above appears to be conclusive as an epistemic justification of inferential RA anyway.For example, the appeal to the principle of charity has been severely criticized as a philosophical reason to systematically prefer normative (e.g., Bayesian) over non-normative accounts of cognitive phenomena (see Stein, 1996, Ch. 4).Similarly, it has been claimed that there is no principled reason why explanations at the computational level in Marr's terms should contain an ineliminable appeal to rationality/optimality (e.g., Elqayam & Evans, 2011).Quite on the contrary, it seems plausible to insist that «competence-level explanations [are] descriptive, "is"type theories, rather than normative, "ought"-type theories» (ivi, p. 239).

A philosophical reconstruction
A more comprehensive survey of the debates mentioned above will have to wait for another occasion.For the moment, we will claim indirect support for our diagnosis, pointing out that it contributes a coherent philosophy of science reconstruction for important but sparse complaints raised by others (see Baron, 1991;Bowers & Davis, 2012a, b;Eberhardt & Danks, 2011;Elqayam & Evans, 2011;Glymour, 2007;Jones & Love, 2011;Marcus & Davis, 2013;Sloman & Fernbach, 2008;Tauber et al., 2017). 8or instance, several scholars have complained that strong, unwarranted appeals to normativity and/or optimality are frequently invoked in the Bayesian psychological literature (e.g., Bowers & Davis, 2012a, b;Marcus & Davis, 2013) whereas such considerations are simply irrelevant within the descriptive framework of cognitive science (Tauber et al., 2017).Bayesian models have also been criticized for being too unconstrained; because «there are too many arbitrary ways that priors, likelihoods, utility functions, etc., can be altered in a Bayesian theory post hoc», as observed by Bowers and Davis (2012a, p. 394), «these models […] account for almost any pattern of results», resulting is nothing more than unfalsifiable "just-sostories".Moreover, it has been argued that Bayesian models rarely perform better than alternative, non-Bayesian models in terms of predicting human performance in reasoning tasks, and they are often preferred over alternatives just for a sort of strong confirmation bias (Bowers & Davis, 2012a).
28 Page 10 of 17 Note that none of the criticisms just mentioned is decisive in itself, as several scholars have noted.For instance, normative considerations might be indeed relevant in descriptive psychology if the aim is suggesting new testable hypotheses (Zednik & Jäkel, 2016) or measuring human performance against given standards of rationality (Crupi & Girotto, 2014;Tauber et al., 2017).Moreover, a certain degree of freedom in accommodating the observed behavioural data, as well as the tendency to be influenced by some kind of confirmation bias (Lakatosian "dogmatism"), are not epistemic peculiarities of Bayesian models but characterize any scientific account of the human mind (Griffiths et al., 2012b;Zednik & Jäkel, 2016).As observed by Griffiths and colleagues (2012b, p. 416), for instance, «Bayesian models are […] falsifiable as any empirical hypothesis -any hypothesis can be "saved" by suitable ad hoc adjustments to other aspects of the theory».Similarly, according to the same authors (ivi): [b]eing careful about the degrees of freedom and using appropriate procedures for comparing and testing models are important things to keep in mind for all forms of computational modelling; they do not constitute a problem that is specific to Bayesian models.
What is specifically problematic to certain applications of Bayesian modelling, as we have seen, is rather the peculiar combination of theoretical moves that characterizes the inferential interpretation and use of RA, a combination which has not been fully captured by standard criticisms of Bayesian models in cognitive science. 9 On the other hand, some remarks put forward by advocates of Bayesian cognitive models are coherently vindicated in the light of what we called the constructive interpretation of RA.For instance, against the claim that Bayesian models make unwarranted claims about human rationality, it has been argued that the «Bayesian framework is a means of generating empirical hypotheses, rather than an assertion that people are optimal» (Griffiths et al., 2012b, p. 416).Similarly, some scholars have argued that «most of the heuristics that contribute to Bayesian reverse-engineering serve to formulate testable hypotheses, but not to directly support the claim that one of these hypotheses is actually true» (Zednik & Jäkel, 2016, p. 3973-74).In this sense, Bayesian models are just «catalysts for inspiration» (ivi.p. 3974), allowing «merely to navigate the space of computational-level hypotheses» (ivi, p. 3980) and thus making «an invaluable contribution to scientific discovery» (ivi, p. 3974).These remarks fit well with the spirit of the positive heuristic as characterized by Lakatos, of which the constructive variant of RA is but an instance, as we suggest. 9For instance, as we have claimed, the questionable move in inferential RA is not accommodation per se but the tacit equation of data accommodation with empirical support.This move is only loosely captured by saying that «theorists take the successful predictions of a Bayesian model as support to their approach and ignore the fact that alternative non-Bayesian theories might account for the data just as well, and sometimes better» (Bowers & Davis, 2012a, p. 403; see also Bowers & Davis, 2012b), as critics of Bayesian models have sometimes argued.In addition, as we have seen, in inferential RA rationality considerations are invoked with the specific purpose of providing support for empirical models, leading to a specific confusion between normative and descriptive considerations.Again, this confusion is only loosely captured by saying that «it is not always obvious when a Bayesian model is intended to imply a normative claim and when it is not» (Tauber et al., 2017, p. 411), as some scholars have observed.

Resource-RA
As a possible objection to the considerations that we have provided so far, one might argue that the inferential interpretation of RA might have been common only in the early phase of Bayesian cognitive science.In recent years -so the objection goes -normative considerations are progressively less relevant and theorists in this field are increasingly moving towards more realistic and descriptively adequate models of cognitive phenomena.
As a representative example, Lieder and Griffiths (2020) have recently formulated a modification of the methodology proposed by Anderson called resourcerational analysis (resource-RA), which gives more emphasis to computational limitations and the trade-off between optimality and psychological constraints.Here is a sketch of how it operates.To begin with, (i) the theorist formulates a computational-level theory of a cognitive problem and (ii) explicitly considers the class of algorithms that the mind might use to solve it as well as the cognitive costs of these algorithms.Then, (iii) an algorithm in this class that optimally trades off resources and approximation accuracy is individuated, and (iv) predictions made by the model are evaluated in light of behavioural data.If the predictions are off, (v) the theorist is expected to reiterate the analysis by refining either the computational-level theory or the assumed constraints.Alternatively, if the proposed cognitive model is already sufficiently realistic, the theorist can stop the analysis.
Resource-RA is explicitly presented as a constructive procedure for Bayesian cognitive science: «it provides a tool for replacing the traditional method of developing cognitive process models […] with a means of automatically deriving hypotheses about cognitive processes from the problem people have to solve and the resources they have available to do so» (Lieder & Griffiths, 2020, p. 6).Nevertheless, worries remain about the ability of resource-RA to overcome some limitations of the traditional approach, such as the reliance on a weak notion of evidential support from the data (point 4 above).As noted by Lieder and Griffiths themselves (p. 14), [e]ncouraging modellers to revise their assumptions about cognitive constraints in the face of data […] could also produce bad models that overfit observations of idiosyncratic or genuinely irrational behaviours with wrong assumptions.
The ability of resource-RA to avoid subtle violations of the is / ought divide is also yet to be assessed, provided that the appeal to rationality (under constraints) is still one of the avowed virtues and the central guiding principles of the approach.
Perhaps the most pressing worry of resource-RA is that it remains somehow anchored to the claim that humans are essentially rational (point 5).After all, the spirit of the project is to demonstrate that the reasoning mechanisms that are commonly interpreted as evidence against human rationality can be reinterpreted as reflecting the optimal use of finite time and limited computational resources (Lieder & Griffiths, 2020, p. 7).This claim appears to be inconsistent, however, with the established body of evidence supporting the idea that «computational limits don't fully explain human cognitive limitations» (Davis & Marcus, 2020, p. 21), given the existence of «cases [that] are not only suboptimal, but rather "anti-Bayesian", for actively defying Bayesian norms of inference» (Mandelbaum et al., 2020, p. 31).In light of the amount of data showing that human cognition is not rational or optimal, and not even boundedly rational under constraints, some Bayesian cognitive scientists have started to suggest «setting optimality aside and letting data drive psychological theory» (Tauber et al., 2017, p. 410).

Second exhibit: The conjunction fallacy and Bayesian confirmation theory
In order to figure out one way for a Bayesian perspective on cognition to possibly survive without being tainted by the methodologically dubious implications of inferential rational analysis, we will now refer to another widely know example from the psychology of reasoning.In certain circumstances, people have a systematic tendency to assess a conjunctive statement as more likely than one of its conjuncts, contrary to the principles of probability theory.A number of studies have documented this phenomenon, so-called "conjunction fallacy" (see Wedell & Moro, 2008;Tentori & Crupi, 2012 for an assessment).The most widely known illustration is of course the Linda scenario, taken from the seminal works of Tversky andKahneman (1982, 1983).When faced with the description of a character, Linda (who is 31 years old, single, outspoken, and very bright, with a major in philosophy, concerns about discrimination and social justice, and an involvement in anti-nuclear demonstrations as a university student), most people ranked the statement "Linda is a bank teller and is active in the feminist movement" as more probable than "Linda is a bank teller".The conjunction fallacy does not occur indiscriminately.When does it occur then?Pretty simple question to ask, not so easy to answer, it turns out.Tversky and Kahneman drew from their general framework for the study of judgment under uncertainty (Tversky & Kahneman, 1974), but they did not achieve (nor they claimed) full success in explaining the rich variety of their findings.Virtually all later approaches posited that the extent of the conjunction fallacy effect should simply increase with the judged probability of the added conjunct ("feminist", in the Linda scenario; see Tentori et al., 2013 for an overview).
If you are a Bayesian, you may wonder if there's a better interpretation of this phenomenon.Maybe some latent subtlety is implied when people face conjunction fallacy tasks.You notice that the "feminist" conjunct is not just relatively probable as concerned Linda; perhaps more interestingly, it clearly seems to be confirmed by the evidence initially provided (Linda's description) in the specific sense of Bayesian confirmation theory (Crupi et al., 2008).Indeed, one can account for all main extant variants of the phenomenon on the basis of confirmation-theoretic connections among the key elements in the scenario, the idea being -roughly -that people tend to make a conjunction fallacy judgment to the extent the conjunction is more strongly confirmed than the target isolated conjunct by available evidence that is explicitly given or psychologically salient (see Tentori et al., 2013;Crupi & Tentori, 2016).In this view, subtle (and sound) assessments of evidential relevance can guide judgments of probability, thus generating systematic biases (also see Tentori et al., 2016).
Here, one is letting the data "drive psychological theory" (or better, theory assessment) not because Bayesian theorizing is avoided, but because empirical support is not claimed from inferential RA.Indeed, some novel, independent, and successful predictions arise from this approach.For instance, in the Linda scenario, "Linda is a bank teller and a feminist activist" will be the target of more (not less) conjunction fallacy judgments than "Linda is a bank teller and owns a pair of black shoes", even if people appreciate that the "black shoes" conjunct is more probable than "feminist", because they also perceive that the former gets less evidential support than the latter from the information given (see Tentori et al., 2013).Even more surprisingly, perhaps, the confirmation-theoretic analysis implies that a substantial amount of double conjunction fallacy judgments can be obtained in case a conjunction is confirmed by the evidence while none of the conjuncts is -a pattern that cannot be anticipated by any competing account (see Crupi et al., 2018a).

Discussion
Let us recap.Our outline of the inferential form of RA indicates that (1)-( 5) are not independent features of PBCS.Roughly, (1)-( 4) generate premises by which the claim of human rationality can be derived (if informally).To repeat, a Bayesian model is developed (1), which is assumed to fulfil compelling rationality requirements (2); a weak notion of evidential support is invoked to claim empirical indistinguishability of Bayesian and alternative accounts (3), and the normative status of the Bayesian analysis is employed as a theoretical virtue to solve the problem of comparative assessment (4).Of course, in such context, the final acceptance of a Bayesian analysis entails the rationality of behavior (5).
But the claim of rationality is untenable.Despite the sustained effort, the epic task of rationalizing human cognition through PBCS has met only limited success."Given the reams of evidence that cognition is fallible, [pure] Bayesians are fighting an uphill battle" (Fernbach & Sloman, 2011, p. 199).This raises the question of whether a Bayesian approach to cognition is possible at all, without the debatable features represented by (3), (4), and (5).Here, we take our second exhibit to provide an existence proof.For the sake of convenience, we will label this (by now admittedly minor) variant critical Bayesian cognitive science (CBCS), and characterize it as follows.
(1) It draws on the Bayesian framework and formal machinery to account for cognitive phenomena documented by behavioral data.(2) It regards Bayesianism as a compelling general model of rational inference and judgment under uncertainty.(3) *It meets the standards of a robust, predictivist view of evidential support from the data.(4) *It does not imply any violation of so-called is / ought divide.
(5) *It acknowledges that some systematic violations of rational constraints in humans are real.
28 Page 14 of 17 Note that (1) still distinguishes CBCS from the heuristic and biases approach in its traditional outlook / variation (Gilovich et al., 2002), while both (1) and (2) distinguish CBCS from the "fast and frugal heuristics" program by Gigerenzer and colleagues (Gigerenzer et al., 2011).We happen to consider (3*)-(5*) much more attractive than their counterparts (3)-( 5) for both philosophical and scientific reasons.Someone who shares such inclinations may find comfort in the conclusion that we do not have to "throw out the Bayes with the bathwater" (Fernbach & Sloman, 2011).
It is not our aim to reconstruct the philosophy of science foundations of CBCS in detail in the present context.Note, however, that within cognitive science a similar perspective has been recently suggested by Tauber and colleagues (2017) under the name of descriptive Bayesian approach. 10Within the descriptive approach outlined by Tauber and colleagues, a Bayesian model needs not to imply any claim that the underlying cognition is optimal or rational (even under constraints), and is used solely as a tool for building a psychological theory.This is a «vision that explicitly, as part of its structure, rejects assumptions or interpretations of optimality» (p.413).
In particular, in one of the case studies presented in the article (case 1), the authors formulate a Bayesian model that squarely incorporates some assumptions that are problematic from a normative point of view (such as inferential "conservatism", see pp. .Of course, as also Tauber and colleagues suggest, more work is needed to make this methodological approach take root in actual experimental practice.Nevertheless, we take this as a sign that the time is ripe for promoting a substantial change in perspective within (Bayesian) cognitive science, and we hope that the philosophical considerations provided here might serve the cause.

Financial or non-financial interests None.
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