1. What is Bayes?

In the target article, J&L worry that Bayesian inference is conceptually trivial, although its consequences may be complex. The same could be said of all mathematical science: The axioms are always “trivial”; the theorems and implications are substantive, as are the possibilities for engineering nontrivial systems using that mathematics as the base.

J&L focus their attention on Bayes’ rule, but this is just the starting point for the approach, not its core. The essence of Bayes is the commitment to representing degrees of belief with the calculus of probability. By adopting appropriate representations of a problem in terms of random variables and probabilistic dependencies between them, probability theory and its decision-theoretic extensions offer a unifying framework for understanding all aspects of cognition that can be properly understood as inference under uncertainty: perception, learning, reasoning, language comprehension and production, social cognition, action planning and motor control, as well as innumerable real-world tasks that require the integration of these capacities. The Bayesian framework provides a principled approach to solving basic inductive challenges that arise throughout cognition (Griffiths et al. Reference Griffiths, Kemp, Tenenbaum and Sun2008a; Tenenbaum et al. Reference Tenenbaum, Kemp, Griffiths and Goodman2011), such as the problem of trading off simplicity and fit to data in model evaluation, via the Bayesian Occam's razor (MacKay Reference MacKay2002) or the problem of developing appropriate domain-specific inductive biases for constraining learning and inference, via hierarchical Bayesian models (Gelman et al. Reference Gelman, Carlin, Stern and Rubin2003).

Bayes’ rule is the most familiar and most concrete form in which psychologists typically encounter Bayesian inference, so it is often where Bayesian modelers start as well. But interpreted literally as the form of a computational model – what we take to be J&L's target when they refer to Bayes’ rule as a “simple vote-counting scheme” (sect. 3, para. 9) – the form of Bayes’ rule J&L employ applies to only the simplest tasks requiring an agent to evaluate two or more mutually exclusive discrete hypotheses posited to explain observed data. Some of the earliest Bayesian models of cognition did focus on these cases; starting with the simplest and most familiar settings is often a good research strategy. But most of cognition cannot be directly cast in such a simple form, and this has been increasingly reflected in Bayesian cognitive models over the last decade. Indeed, the form of Bayes’ rule that J&L discuss hardly figures in many contemporary Bayesian cognitive models.

What does it mean in practice for a computational model of cognition to be Bayesian, if not to literally implement Bayes’ rule as a mechanism of inference? Typically, it means to adopt algorithms for generating hypotheses with high posterior probabilities based on Monte Carlo sampling, or algorithms for estimating the hypothesis with highest posterior probability (i.e., maximum a posteriori probability [MAP]) using local message-passing schemes (MacKay Reference MacKay2002). The outputs of these algorithms can be shown, under certain conditions, to give reasonable approximations to fully Bayesian inference, but can scale up to much larger and more complex problems than could be solved by exhaustively scoring all possible hypotheses according to Bayes’ rule (J&L's “simple vote-counting scheme”). A little further on we briefly discuss several examples of how these approximate inference algorithms have been explored as models of how Bayesian computations might be implemented in the mind and brain.

2. Bayesian Fundamentalism versus Bayesian cognitive science

J&L charge Bayesian Fundamentalists with a number of failings. The practice of Bayesian cognitive science is largely free of these, as we will see.

(i) J&L suggest in their Introduction that “[i]t is extremely rare to find a comparison among alternative Bayesian models of the same task to determine which is most consistent with empirical data” (sect. 1, para. 6). Yet such comparisons are commonplace (for a tiny sample, see Goodman et al. Reference Goodman, Mansinghka, Tenenbaum, McNamara and Trafton2007; Griffiths & Tenenbaum Reference Griffiths and Tenenbaum2009; Kemp & Tenenbaum Reference Kemp and Tenenbaum2009; Oaksford & Chater Reference Oaksford and Chater2003); Goo. Nonetheless, of course, Bayesian authors do sometimes press for a single model, often comparing against non-Bayesian alternative accounts (e.g., Goodman et al. Reference Goodman, Tenenbaum, Feldman and Griffiths2008b). This is entirely in line with practice in other modeling frameworks.

(ii) J&L are concerned that Bayesians downplay the structure of the environment. This is a particularly surprising challenge given that Anderson's path-breaking Bayesian rational analyses of cognition (e.g., Anderson Reference Anderson1990; Reference Anderson1991a; Oaksford & Chater Reference Oaksford and Chater1998b) are explicitly based on assumptions about environmental structure. Similarly, Bayesian approaches to vision essentially involve careful analysis of the structure of the visual environment – indeed, this defines the “inverse problem” that the visual system faces (e.g., Yuille & Kersten Reference Yuille and Kersten2006); and Bayesian models of reasoning are crucially dependent on environmental assumptions, such as “rarity” (Oaksford & Chater Reference Oaksford and Chater1994). Finally, in the context of language acquisition, there has been substantial theoretical and empirical progress in determining how learning depends on details of the “linguistic environment,” which determine the linguistic structures to be acquired (Chater & Vitányi Reference Chater and Vitányi2007; Foraker et al. Reference Foraker, Regier, Khetarpal, Perfors and Tenenbaum2009; Hsu & Chater Reference Hsu and Chater2010; Hsu et al., in press; Perfors et al. Reference Perfors, Tenenbaum and Wonnacott2010; Reference Perfors, Tenenbaum and Regier2011).

(iii) J&L claim (in sect. 4) that Bayesian Fundamentalism is analogous to Behaviorism, because it “eschews mechanism” (sect. 2.2, para. 3). But, as J&L note, Bayesian cognitive science, qua cognitive science, is committed to computational explanation; behaviorists believe that no such computations exist, and further that there are no internal mental states over which such computations might be defined. Assimilating such diametrically opposing viewpoints obscures, rather than illuminates, the theoretical landscape.

(iv) J&L suggest, moreover, that Bayesians are unconcerned with representation and process, and that the Bayesian approach is driven merely by technical advances in statistics and machine learning. This seems to us completely backwards: Most of the technical advances have precisely been to enrich the range of representations over which Bayesian methods can operate (e.g., Goodman et al. Reference Goodman, Ullman and Tenenbaum2011; Heller et al. Reference Heller, Sanborn, Chater, Lafferty and Williams2009; Kemp et al. Reference Kemp, Goodman and Tenenbaum2010a; Reference Kemp, Tenenbaum, Niyogi and Griffiths2010b) and/or to develop new computational methods for efficient Bayesian inference and learning. These developments have substantially expanded the range of possible hypotheses concerning representations and algorithms in human inference and learning. Moreover, some of these hypotheses have provided new mechanistic accounts. For example, Sanborn et al. (Reference Sanborn, Griffiths and Navarro2010a, p.1144) have argued that “Monte Carlo methods provide a source of ‘rational process models’ that connect optimal solutions to psychological processes”; related approaches are being explored in a range of recent work (e.g., Vul et al. Reference Vul, Goodman, Griffiths, Tenenbaum, Taatgen and van Rijn2009a; Reference Vul, Hanus and Kanwisher2009b). Moreover, there has been considerable interest in how traditional psychological mechanisms, such as exemplar models (Shi et al. Reference Shi, Griffiths, Feldman and Sanborn2010) and neural networks (e.g., McClelland Reference McClelland, Oaksford and Chater1998; Neal Reference Neal1992), may be viewed as performing approximate Bayesian inference. Such accounts have been applied to psychological data on, for example, conditional reasoning (Oaksford & Chater Reference Oaksford, Chater, Oaksford and Chater2010).

We have argued that Bayesian cognitive science as a whole is closely involved both with understanding representation and processes and with specifying environmental structure. Of course, individual Bayesian projects may not address all levels of explanation, and so forth. We believe it would be unnecessary (and pernicious) to require each project to embrace all aspects of cognition. (For instance, we would not require all connectionist models to make explicit the bridge to biological neural networks.) Indeed, according to the normal canons of scientific inference, the more that can be explained, with the fewer assumptions, the better. Thus, contra J&L, we see it as a strength, rather than weakness, of the Bayesian approach that some computational-level analyses have broad applications across cognition, independent of specific representational, processing, or environmental assumptions, as we now explore.

3. The power of Bayesian computational-level explanation: The case of explaining away

Consider the Bayesian analysis of explaining away (Pearl Reference Pearl1988). Suppose two independent causes (e.g., no petrol or dead battery) can cause a car not to start. Learning that the car did not start then raises the probability of both no petrol and dead battery: they both provide potential explanations for the car not starting. But if we then learn that the battery was dead, the probability of no petrol falls back to its original value. The battery explains the car not starting; so the apparent evidence that there might be no petrol is “explained away.”

Experiments have found that, when given reasoning problems with verbal materials, people do, indeed, follow this, and related, patterns of reasoning (e.g., Ali et al. Reference Ali, Chater and Oaksford2011), although this pattern is clearer in young children (Ali et al. Reference Ali, Schlottmann, Shaw, Chater, Oaksford, Oaksford and Chater2010), with adults imposing additional knowledge of causal structure (Walsh & Sloman Reference Walsh, Sloman, Gluck, Anderson and Kosslyn2008; Rehder & Burnett Reference Rehder and Burnett2005). Moreover, the same pattern is ubiquitous in perception: If a piece of sensory input is explained as part of one pattern, it does not provide evidence for another pattern. This principle emerges automatically from Bayesian models of perception (Yuille & Kersten Reference Yuille and Kersten2006).

Furthermore, explaining away also appears to help understand how children and adults learn about causal regularities (e.g., Gopnik et al. Reference Gopnik, Glymour, Sobel, Schulz, Kushnir and Danks2004; Griffiths & Tenenbaum Reference Griffiths and Tenenbaum2009). If a “blicket detector” is triggered whenever A and B are present, there is a prima facie case that A and/or B causes the detector to sound. But if the detector also sounds when preceded only by A, then this regularity explains away the sounding of the detector and reduces the presumed causal powers of B. In animal learning, a related pattern is known as blocking (Kamin Reference Kamin, Campbell and Church1969).

Blocking can also be explained using connectionist-style mechanistic models, such as the Rescorla-Wagner model of error-driven associative learning (Rescorla & Wagner Reference Rescorla, Wagner, Black and Prokasy1972). But such explanations fail to capture the fact that partial reinforcement (i.e., where the putative effect only sometimes follows the putative cause) extinguishes more slowly than total reinforcement. Indeed, partial reinforcement should induce a weak link which should more easily be eliminated. From a Bayesian point of view, extinction in partial reinforcement is slower, because the lack of effect must occur many times before there is good evidence that the state of the world has really changed (e.g., a causal link has been broken). This type of Bayesian analysis has led to a wide range of models of human and animal learning, which are both compared extensively with each other and with empirical data (for a review, see Courville et al. Reference Courville, Daw and Touretzky2006). Associative learning accounts of blocking also cannot explain the rapid and complex dynamics observed in adults’ and children's causal learning: the fact that causal powers may be identified from just one or a few observed events in the presence of appropriate background knowledge about possible causal mechanisms, and the strong dependence of the magnitude of causal discounting on the base rates of causes in the environment (Griffiths & Tenenbaum Reference Griffiths and Tenenbaum2009). In contrast, these phenomena are not only explained by, but were predicted by and then experimentally verified from, the dynamics of explaining away in Bayesian analyses of causal learning.

We have seen that a general qualitative principle, explaining away, which follows directly from the mathematics of probability, has broad explanatory power across different areas of cognition. This generality is possible precisely because the Bayesian analysis abstracts away from mechanism – which presumably differs in detail between verbal reasoning, perception, and human and animal learning. Thus, contra J&L, the Bayesian approach is not merely closely tied with empirical data; it provides a synthesis across apparently unconnected empirical phenomena, which might otherwise be explained by using entirely different principles.

Framing explanations of some phenomena at this high level of abstraction does not imply commitment to any kind of Bayesian Fundamentalism. Rather, Bayesian cognitive scientists are merely following the standard scientific practice of framing explanation at the level of generality appropriate to the phenomena under consideration. Thus, the details, across computational, algorithmic, and implementation levels, of accounts of animal learning, perception, or causal reasoning will differ profoundly – but the phenomenon of “explaining away” can insightfully be seen as applying across domains. This aspect of explanation is ubiquitous across the sciences: For example, an abstract principle such as the conservation of energy provides a unified insight across a wide range of physical phenomena; yet the application of such an abstract principle in no way detracts from the importance of building detailed models of individual physical systems.

4. Bayesian cognitive science as a top-down research strategy

Bayesian cognitive scientists strongly agree with J&L that it is vital to create mutually constraining accounts of cognition across each of Marr's computational levels of explanation. We stress that what is distinctive about the Bayesian approach, in distinction from many traditional process models in cognitive psychology, is a top-down, or “function-first” research strategy, as recommended by Marr (Reference Marr1982): from computational, to algorithmic, to implementational levels (see, e.g., Anderson Reference Anderson1990; Chater et al. Reference Chater, Oaksford, Nakisa and Redington2003; Griffiths et al. Reference Griffiths, Chater, Kemp, Perfors and Tenenbaum2010).

The motivation for this approach is tactical, rather than ideological. Consider attempting to understand an alien being's pocket calculator that uses input and output symbols we don’t understand. If we realize that an object is doing addition (computational level), we have some chance of discerning which type of representations and algorithms (algorithmic level) might be in play; it is hard to see how any amount of study of the algorithmic level alone might lead to inferences in the opposite direction. Indeed, it is difficult to imagine how much progress could be made in understanding an algorithm, without an understanding of what that algorithm is computing.

Thus, the problem of reverse engineering a computational system, including the human mind, seems to inevitably move primarily from function to mechanism. Of course, constraints between levels will flow in both directions (Chater & Oaksford Reference Chater and Oaksford1990). The hardware of the brain will place strong constraints on what algorithms can be computed (e.g., Feldman & Ballard Reference Feldman and Ballard1982), and the possible algorithms will place strong constraints on what computational-level problems can be solved or approximated (Garey & Johnson Reference Garey and Johnson1979). Yet, from this reverse-engineering perspective, the first task of the cognitive scientist is to specify the nature of the computational problem that the cognitive system faces, and how such problems might, in principle, be solved. This specification typically requires, moreover, describing the structured environment, the goal of the cognitive system, and, frequently, computational constraints or representational commitments (Anderson Reference Anderson1990; Oaksford & Chater Reference Oaksford and Chater1998b).

The appropriate mathematical frameworks used for this description cannot, of course, be determined a priori, and will depend on the nature of the problem to be solved. Analyzing the problem of moving a multi-jointed motor system might, for example, require invoking, among other things, tensor calculus and differential geometry (which J&L mention as important to developments in physics). A rational analysis of aspects of early auditory and visual signal processing might invoke Fourier analysis or wavelet transforms. A computational-level analysis of language use might involve the application of symbolic grammatical and computational formalism. In each case, the appropriate formalism is also open to challenge: For example, researchers differ widely concerning the appropriate grammatical or logical formalism required to represent language and thought; or, indeed, as to whether symbolic formalism is even required at all (e.g., McClelland Reference McClelland2010).

Within this diversity, there is an important common mathematical thread. A wide range of cognitive problems, from motor control to perception, language processing, and commonsense reasoning, involve (among other things) making inferences with uncertain information, for which probability theory is a natural mathematical framework. For example, the problem of finding an underlying pattern in a mass of sensory data – whether that pattern be the layout of the environment, a set of causal dependencies, the words, syntactic structure, or meaning of a sentence, or even the grammatical structure of a language – is naturally framed in terms of probabilistic (or Bayesian) inference. This explains why probability is a common theme in Bayesian modeling, and why engineering approaches to solving these and many other problems often take a Bayesian approach (though there are important alternatives) (Bishop Reference Bishop1996; Manning & Schütze, Reference Manning and Schütze1999; Russell & Norvig Reference Russell and Norvig2011). Indeed, Bayesian cognitive scientists have themselves contributed to extending the boundaries of engineering applications in some domains (e.g., Griffiths & Ghahramani Reference Griffiths, Ghahramani, Weiss, Schölkopf and Platt2006; Johnson et al. Reference Johnson, Griffiths, Goldwater, Schölkopf, Platt and Hofmann2007; Kemp & Tenenbaum Reference Kemp and Tenenbaum2008; Kemp et al. Reference Kemp, Tenenbaum, Griffiths, Yamada, Ueda and Cohn2006; Goodman et al. Reference Goodman, Mansinghka, Roy, Bonawitz, Tenenbaum, McAllester and Myllymaki2008a).

J&L are concerned that a close relationship between hypotheses in Bayesian cognitive science and technical/mathematical developments in engineering (broadly construed to include statistics and computer science) may amount to a confusion of “technical advances with theoretical progress” (sect. 1, para. 3). We suggest, by contrast, that theoretical approaches in cognitive science that are not tied to rich technical developments have little chance of success. Indeed, given that the human mind/brain is the most complex mechanism known, and that its information-processing capacities far outstrip current artificial intelligence, it is surely inevitable that, in the long term, successful reverse engineering will be possible only in the light of spectacular technical developments, alongside careful use of empirical data. We suggest that Bayesian cognitive science promises to be a small forward step along this path.