want to draw inspiration from the theory of optimal feedback control, in which more precise control incurs greater metabolic expenses at the organismal level (Todorov and Jordan 2002). The time scale over which resources are allocated. Attention can be efficiently allocated in response to trial-to-trial variations in reward or priority (Bays 2014; Sims 2003; van den Berg and Ma 2018), in other words, on a timescale of seconds. By contrast, efficient neural codes are often assumed to be optimized with respect to natural statistics (Barlow 1961; Laughlin 1981), which vary on a much longer timescale. This distinction seems largely aligned with the one made under (1), with shorter timescales being associated with task specificity. Resource-rational models are often non-committal about the timescales over which the optimization occurs. Recent work on efficient codes in nonstationary environments (Młynarski and Hermundstad 2018) holds promise for bridging the divide. Learning to be resource-rational. A question that is not often asked is how resource-rational mechanisms are learned. The target article simply defines a constrained optimum and supposes that "evolution, cognitive development, and life-long learning" have somehow solved it, without saying how. But recognizing that a particular cognitive mechanism is optimal for one's environment requires knowledge of the statistics of the environment, which in practice can never be known with certainty from any finite body of experience. The informational requirements of the learning process may impose constraints on the degree of efficiency of cognitive mechanisms that can be learned, even asymptotically, as discussed, for example, by Robson and Whitehead (2016). The question of how well-adapted a cognitive mechanism can reasonably be assumed to be is even more important if the statistics of the environment are changing (Młynarski and Hermundstad 2018). Are finite-sampling models truly resource-rational models? In some models described in the target article, the observer simulates possible futures – technically, Markov chain Monte Carlo (MCMC) sampling from a posterior (Lieder et al. 2014; 2018; Vul et al. 2014). The high-level idea here is that samples represent computational resources, and that those are limited. More samples would correspond to a better approximation of a performance term. However, it is unclear to us if this approach falls into the framework of optimizing a linear combination of a performance term and a resource cost. Role of reasoning. An ambiguity in references to "resourcerationality" is whether "rationality" is intended to mean the outcome of a process of conscious, logical reasoning, or simply means that something is an efficient solution to a problem, however that solution may have developed (Blume and Easley 1984; Smith 2009). Theories of efficient coding in early-stage sensory processing are rather obviously not to be interpreted as hypotheses according to which sensory processing is consciously decided upon; and it seems that in general, the authors of the target article do not have intend "rationality" in this way – the distinction that they draw between the resource-rationality hypothesis and Stigler's (Stigler 1961) model of optimal information gathering indicates this. Nonetheless, this is not clear in all of the references that they cite as examples of the resource-rationality research program. In particular, the more recent economics literature that models the imperfect information of decision makers as reflecting an optimal allocation of limited attention is often written as if the decision as to what to be aware of is made quite deliberately, just as in the work of Stigler. We view these differences as challenges that need to be addressed but that do not invalidate the overall framework. Progress will require carefully distinguishing between the different formalisms, and finding ways to decide which ones are more applicable to particular settings. Can resources save rationality? "Anti-Bayesian" updating in cognition and perception Eric Mandelbauma, Isabel Wonc, Steven Grossb and Chaz Firestonec aBaruch College, CUNY Graduate Center, Department of Philosophy, New York, NY10016; bDepartment of Philosophy, Johns Hopkins University, Baltimore, MD 21218; and cDepartment of Psychological & Brain Sciences, Johns Hopkins University, Baltimore, MD 21218 emandelbaum@gc.cuny.edu iwon1@jhu.edu sgross11@jhu.edu chaz@jhu.edu http://ericmandelbaum.com http://perception.jhu.edu https://sites.google.com/site/grosssteven/ doi:10.1017/S0140525X19001717, e16 Abstract Resource rationality may explain suboptimal patterns of reasoning; but what of "anti-Bayesian" effects where the mind updates in a direction opposite the one it should? We present two phenomena – belief polarization and the size-weight illusion – that are not obviously explained by performanceor resourcebased constraints, nor by the authors' brief discussion of reference repulsion. Can resource rationality accommodate them? Resource rationality takes seemingly irrational behaviors and reframes them as rational or optimal given other constraints on agents. For example, anchoring-and-adjustment and overestimating extreme events turn out be "rational" after all, by reflecting the rational allocation of cognitive resources. Thus, even for such classically irrational phenomena, "the resulting train of thought eventually converges to the Bayes-optimal inference" (p. 38). In such cases, reasoners fall short of perfectly rational updating, and it is illuminating that resourceand performance-based constraints can accommodate such suboptimal reasoning. But what about cases where we behave not merely suboptimally, but rather against the norms of Bayesian inference? Here, we explore cases where the mind is moved by prior knowledge in precisely the reverse direction of what a rational analysis would recommend. These cases are not merely suboptimal, but rather "anti-Bayesian," for actively defying Bayesian norms of inference. We consider two such phenomena: belief polarization and sensory integration (Fig. 1). Can resource rationality handle them? First, belief polarization: Receiving evidence contrary to your beliefs should soften those beliefs, even if ever-so-slightly. But, this isn't what actually happens when the beliefs in question are central to one's identity – in belief polarization, contrary or disconfirming evidence causes more extreme beliefs, not more moderate ones. A classic example was vividly documented by Festinger et al. (1956): Cult members who predict the world will end on some date – but who then see that date come and go with no Commentary/Lieder and Griffiths: Resource-rational analysis 31 : D HHH 5 4C 697 C9 5 C7 7C D : D 6 C9 0 2 2 , H 676 C : D HHH 5 4C 697 C9 5 C7 . : D D 1 7CD / C D 4 75 :7 4C 697 C7 7C D D7 4 7 cataclysm – end up strengthening their beliefs in the cult's tenets, not softening them. In other words, credible evidence against their worldview only makes them hold that worldview more strongly – directly defying Bayesian inference norms. The same phenomenon can be found under laboratory conditions. For example, one study exposed people who believe that Jesus is the Son of God to a (fake) news article reporting that archeologists had unearthed carbon-dated letters from the New-Testament authors; the letters said the Bible was fraudulent and that its authors knew Jesus was not divinely born (Batson 1975). Subjects who did not believe the article's content left their beliefs about Jesus unchanged; but, fascinatingly, subjects who did believe the article's content ended up strengthening their belief that Jesus was the Son of God. In other words, affirming new evidence against Jesus's divine birth (∼P) caused stronger beliefs in Jesus's divine birth (P). Similar "backwards" updating is also observed for beliefs about nuclear safety (Plous 1991), health (Liberman & Chaiken 1992), and affirmative action and gun control (Taber & Lodge 2006; see also Mandelbaum 2019). Why does this happen? In fact, belief polarization is not so mysterious: It has been known for decades, and it is even a predictable consequence of dissonance theory – "the psychological immune system" (Gilbert et al. 1998) – applied to one's values. What is mysterious is why this should occur in a Bayesian mind – even one constrained by "resources." Belief polarization is irrational not because people are insufficiently moved by evidence, but rather because people are moved in the direction opposite the one they should be. And, importantly, these patterns cannot be explained by biased attitudes toward the evidence's source. For example, Bayesian models of milder forms of belief polarization (e.g., Jern et al. 2014) suggest that subjects infer that contrary evidence must have come from unreliable sources (e.g., biased testimony); but this seems inapplicable to the above cases, where the sources are either nature itself (e.g., the world failing to end), or evidence the subject has actively accepted (e.g., news articles they endorsed). Indeed, "anti-Bayesian" updating is widespread, occurring even in basic perceptual processes. When we have prior expectations about new and uncertain sensory data, rational norms of inference say we should interpret such data with respect for those priors; "people should leverage their prior knowledge about the statistics of the world to resolve perceptual uncertainty" (p. 40). But, sensory integration frequently occurs the opposite way. Consider the size-weight illusion, wherein subjects see two equally weighted objects – one large and one small – and then lift them both to feel their weight. Which feels heavier? We "should" resolve the ambiguous haptic evidence about which object is heavier in favor of our priors; but instead, the classic and much-replicated finding is that we experience the smaller object as heavier than the equally-weighted larger object (Buckingham 2014; Charpentier 1891). This too is "irrational" – not for falling short of Bayesian norms of inference, but for proceeding opposite to them, because we resolve the ambiguous sensory evidence – two equally weighted objects – against the larger-is-heavier prior, not in favor of it (Brayanov & Smith 2010; Buckingham & Goodale 2013). Indeed, this backwards pattern of updating is so strong that it can produce outcomes that are not merely odd or improbable, but even "impossible" (Won et al. 2019): If subjects are shown three boxes in a stack – Boxes A, B, and C – such that Box A is heavy (250 g) but Boxes B and C are light (30 g), then subjects who lift Box A alone and then Boxes A +B+C together report that Box A feels heavier than Boxes A+B+C – an "impossible" experience of weight (because a group could never weigh less than a member of that group). How can a "rational" account – even a resource-rational one – explain this? Lieder and Griffiths accommodate other sensory "repulsion" effects (Wei & Stocker 2015; 2017), but that modeling work seems inapplicable to the size-weight illusion. And whereas the original size-weight illusion could perhaps have a tortuous Bayesian explanation (Peters et al. 2016), Won et al.'s modification seemingly cannot: First, it's unclear if previous models of simultaneous lifting apply to Won et al.'s temporally-extended case; but second, there is just no logical chain of reasoning that should end with A alone being heavier than A+B+C together. More generally: What are the principles that lead to perverse "anti-Bayesian" updating? Perhaps resource rationality wasn't intended to cover all cases (in which case it is not an "Imperial Bayesian" theory; Mandelbaum 2019). But, the problem isn't merely that there are counterexamples to resource rationality, but rather that these are predictable, law-like counterexamples that do not reflect performance constraints between interacting mental processes. Indeed, when it comes to these more entrenched patterns, even "resources" may not save rationality. Figure 1. (Mandelbaum et al.) Examples of "anti-Bayesian" updating in the mind. (A) Under conditions of cognitive dissonance, acquiring – and affirming – evidence against one's beliefs can cause those beliefs to strengthen (Batson 1975), whereas Bayesian norms of inference recommend softening those beliefs. (B) In the sizeweight illusion, one is shown two objects of different sizes but equal weights; when one lifts them up, the smaller one feels illusorily heavier than the larger one (Buckingham 2014; Charpentier 1891; Won et al. 2019). In other words, ambiguous sensory data about which of two objects is heavier is resolved "against" one's prior expectations, rather than in favor of one's priors as recommended by Bayesian norms of inference. Can resource rationality accommodate such paradigmatically "irrational" phenomena? 32 Commentary/Lieder and Griffiths: Resource-rational analysis : D HHH 5 4C 697 C9 5 C7 7C D : D 6 C9 0 2 2 , H 676 C : D HHH 5 4C 697 C9 5 C7 . : D D 1 7CD / C D 4 75 :7 4C 697 C7 7C D D7 4 7 odds with one another. But there is also a tradition of pointing out the role that emotional responses can play in producing adaptive behavior, particularly in the context of interpersonal interaction (e.g., Frank 1988). Resource rationality provides a path to the resolution of this apparent contradiction, because the apparent antagonism between rationality versus emotion does not carry over into the resource rational framework. To the contrary, the computational efficiency of emotional mechanisms might make them resource rational in time-critical situations, and in certain situations, emotional mechanisms may be resource rational because they lead to better decisions than deliberation. Furthermore, emotions, such as anxiety, can guide the efficient allocation of cognitive resources to important problems, such as planning how to survive (Gagne et al. 2018). We agree with Russek et al. that emotions can be understood in terms of resource-efficient computational mechanisms, but we would like to clarify being resource rational does not require solving the meta-decision-making problem optimally – instead, a resource rational agent would select computations by a boundedly optimal heuristic. Furthermore, RRA can also illuminate how emotions and cognition interact (Krueger & Griffiths 2018). An extensive body of work that has emphasized that there are at least three distinct decision systems: the instinctive Pavlovian system that is responsible for emotional biases, a deliberative system that supports effective goal pursuit through flexible reasoning, and a model-free reinforcement learning system that leads to inflexible habits (van der Meer et al. 2012). Exactly how those systems interact is an open problem that RRA could be used to solve. One proposal is that the model-based system generates simulated data – through a kind of introspection – that is then used to refine model-free learning (Gershman et al. 2014). Another proposal is that deliberation is used to refine the valuation of past experiences in the light of new information and to update the agent's habits accordingly (Krueger & Griffiths 2018). This latter perspective instantiates the idea that our emotions teach us how to become more resource rational by allowing our regrets to improve our computationally-efficient, habitual response tendencies. Both Ross and Dingemanse point out that the formulation of RRA in the target article assumes an agent facing a problem that is generated by nature, while many of the problems that human beings have to solve require interacting with other agents. This creates a situation where the strategies adopted by one agent influence the environment experienced by another – a situation that is very familiar to any student of game theory. We do not foresee any fundamental obstacles to extending resource rationality to such situations. Indeed, we anticipate that this approach can be used to define models like those currently used in behavioral game theory (e.g., Camerer & Hua Ho 1999), but derived from the principle of optimization that underlies resource rationality. First steps in this direction have been taken by Halpern and Pass (2015). Beyond game theory, we agree with Dingemanse that language use represents a particularly rich territory for exploring this approach, including examining the extent to which speakers modify their linguistic choices based on assumptions about the cognitive load experienced by listeners. R7. Summary and Conclusion RRA is a new modeling paradigm that integrates the top-down approach that starts from the function of cognitive systems with the bottom-up approach that starts from insights into the mind's cognitive architecture and its constraints. Combining the strengths of both approaches makes RRA a promising methodology for reverse-engineering the mechanisms and representations of human cognition. RRA is an important step toward realizing David Marr's vision that theories formulated at different levels of analysis can inform and mutually constrain each other. RRA contributes to this vision by bringing insights about the function of cognitive systems (computational level) and empirical findings about the system's constraints (implementational level) to bear on models of cognitive mechanisms (algorithmic level of analysis). RRA accomplishes this in a principled way that uniquely specifies what the cognitive mechanism should be according to its function and the constraints of the available cognitive architecture. As Dimov noted, this addresses the fundamental non-identifiability problems that have been holding back progress on uncovering cognitive architectures and cognitive mechanisms for a long time. The commentaries revealed that RRA is even more broadly applicable than our target article suggested. We are looking forward to seeing RRA facilitating progress in fields ranging from cognitive development to history. We are especially excited to see RRA applied to understanding mental illness and improving people's mental health. We appreciated the commentators' suggestions for future methodological developments, including the establishment of limits on the constraints that can be postulated by RRA and the integration of insights from extant cognitive architectures and evolutionary theory. RRA is a brand-new modeling paradigm that will undoubtedly mature and develop and the dialogue started by our target article will likely accelerate this process. The commentaries also gave us the opportunity to clarify themethodological nature of the teleological and optimality assumptions of RRA. We hope that this has made it clear that we are not arguing that the human mind is (resource) rational but offering a methodology for understanding the human mind's somewhat suboptimal cognitive systems in terms of their function, mechanisms, and representations. References [The letters "a" and "r" before author's initials stand for target article and response references, respectively] Aerts, D., Gabora, L. & Sozzo, S. 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