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Maybe this old dinosaur isn't extinct: What does Bayesian modeling add to associationism?

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Maybe this old dinosaur isn't extinct: What does Bayesian modeling add to associationism?

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We agree with Jones & Love (J&L) that much of Bayesian modeling has taken a fundamentalist approach to cognition; but we do not believe in the potential of Bayesianism to provide insights into psychological processes. We discuss the advantages of associative explanations over Bayesian approaches to causal induction, and argue that Bayesian models have added little to our understanding of human causal reasoning.
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Behavioral and Brain Sciences
Manuscript Draft
Manuscript Number: BBS-D-10-00265R2
Title: Bayesian Fundamentalism or Enlightenment? On the explanatory status and theoretical
contributions of Bayesian models of cognition
Article Type: Target Article
Keywords: Bayesian modeling; cognitive processing; levels of analysis; rational analysis;
representation
Abstract: The prominence of Bayesian modeling of cognition has increased recently largely because of
mathematical advances in specifying and deriving predictions from complex probabilistic models.
Much of this research aims to demonstrate that cognitive behavior can be explained from rational
principles alone, without recourse to psychological or neurological processes and representations. We
note commonalities between this rational approach and other movements in psychology -namely,
behaviorism and evolutionary psychology- that set aside mechanistic explanations or make use of
optimality assumptions. Through these comparisons, we identify a number of challenges that limit the
rational program's potential contribution to psychological theory. Specifically, rational Bayesian
models are significantly unconstrained, both because they are uninformed by a wide range of process-
level data and because their assumptions about the environment are generally not grounded in
empirical measurement. The psychological implications of most Bayesian models are also unclear.
Bayesian inference itself is conceptually trivial, but strong assumptions are often embedded in the
hypothesis sets and the approximation algorithms used to derive model predictions, without a clear
delineation between psychological commitments and implementational details. Comparing multiple
Bayesian models of the same task is rare, as is the realization that many Bayesian models recapitulate
existing (mechanistic level) theories. Despite the expressive power of current Bayesian models, we
argue they must be developed in conjunction with mechanistic considerations to offer substantive
explanations of cognition. We lay out several means for such an integration that take into account the
representations on which Bayesian inference operates, as well as the algorithms and heuristics that
carry it out. We argue this unification will better facilitate lasting contributions to psychological theory,
avoiding the pitfalls that have plagued previous theoretical movements.
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To be published in Behavioral and Brain Sciences (in press)
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Below is the copyedited final draft of a BBS target article that has been accepted for publication.
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Bayesian Fundamentalism or Enlightenment? On the explanatory
status and theoretical contributions of Bayesian models of cognition
Matt Jones
Dept. of Psychology and Neuroscience, University of Colorado, Boulder, CO 80309
mcj@colorado.edu
http://matt.colorado.edu
Bradley C. Love
Dept. of Psychology, University of Texas, Austin, TX 78712
brad_love@mail.utexas.edu
http://love.psy.utexas.edu
Abstract: The prominence of Bayesian modeling of cognition has increased recently largely
because of mathematical advances in specifying and deriving predictions from complex
probabilistic models. Much of this research aims to demonstrate that cognitive behavior can
be explained from rational principles alone, without recourse to psychological or neurological
processes and representations. We note commonalities between this rational approach and
other movements in psychology – namely, behaviorism and evolutionary psychology – that
set aside mechanistic explanations or make use of optimality assumptions. Through these
comparisons, we identify a number of challenges that limit the rational program’s potential
contribution to psychological theory. Specifically, rational Bayesian models are significantly
unconstrained, both because they are uninformed by a wide range of process-level data and
because their assumptions about the environment are generally not grounded in empirical
measurement. The psychological implications of most Bayesian models are also unclear.
Bayesian inference itself is conceptually trivial, but strong assumptions are often embedded
in the hypothesis sets and the approximation algorithms used to derive model predictions,
without a clear delineation between psychological commitments and implementational
details. Comparing multiple Bayesian models of the same task is rare, as is the realization
that many Bayesian models recapitulate existing (mechanistic level) theories. Despite the
expressive power of current Bayesian models, we argue they must be developed in
conjunction with mechanistic considerations to offer substantive explanations of cognition.
We lay out several means for such an integration that take into account the representations on
which Bayesian inference operates, as well as the algorithms and heuristics that carry it out.
We argue this unification will better facilitate lasting contributions to psychological theory,
avoiding the pitfalls that have plagued previous theoretical movements.
Keywords: Bayesian modeling; cognitive processing; levels of analysis; rational analysis;
representation
*Target Article
Click here to download Target Article: M_Jones-BBS-D-10-00265_Copyedited_Final_Version.pdf
2
Advances in science are due not only to empirical discoveries and theoretical progress,
but also to development of new formal frameworks. Innovations in mathematics or
related fields can lead to a new class of models that enables researchers to articulate more
sophisticated theories and to address more complex empirical problems than previously
possible. This often leads to a rush of new research and a general excitement in the field.
For example in physics, the development of tensor calculus on differential
manifolds (Ricci & Levi-Civita 1900) provided the mathematical foundation for
formalizing the general theory of relativity (Einstein 1916). This formalism led to
quantitative predictions that enabled experimental verification of the theory (e.g., Dyson
et al. 1920). More recent mathematical advances have played key roles in the
development of string theory (a potential unification of general relativity and quantum
mechanics), but in this case the mathematical framework, although elegant, has yet to
make new testable predictions (Smolin 2006; Woit 2006). Therefore, it is difficult to
evaluate whether string theory represents true theoretical progress.
In the behavioral sciences, we are generally in the more fortunate position of
being able to conduct the key experiments. However, there is still a danger of confusing
technical advances with theoretical progress, and the allure of the former can lead to the
neglect of the latter. As the new framework develops, it is critical to keep the research
tied to certain basic questions such as, What theoretical issues are at stake? What are the
core assumptions of the approach? What general predictions does it make? What is being
explained and what is the explanation? How do the explanations it provides relate,
logically, to those of existing approaches? What is the domain of inquiry, and what
questions are outside its scope? This grounding is necessary for disciplined growth of the
field. Otherwise, there is a tendency to focus primarily on generating existence proofs of
what the computational framework can achieve. This comes at the expense of real
theoretical progress, in terms of deciding among competing explanations for empirical
phenomena or relating those explanations to existing proposals. By overemphasizing
computational power, we run the risk of producing a poorly grounded body of work that
is prone to collapse under more careful scrutiny.
This article explores these issues in connection with Bayesian modeling of
cognition. Bayesian methods have progressed tremendously in recent years, due largely
3
to mathematical advances in probability and estimation theory (Chater et al. 2006). These
advances have enabled theorists to express and derive predictions from far more
sophisticated models than previously possible. These models have generated excitement
for at least three reasons: First, they offer a new interpretation of the goals of cognitive
systems, in terms of inductive probabilistic inference, which has revived attempts at
rational explanation of human behavior (Oaksford & Chater 2007). Second, this rational
framing can make the assumptions of Bayesian models more transparent than in
mechanistically oriented models. Third, Bayesian models may have the potential to
explain some of the most complex aspects of human cognition, such as language
acquisition or reasoning under uncertainty, where structured information and incomplete
knowledge combine in a way that has defied previous approaches (e.g., Kemp &
Tenenbaum 2008).
Despite this promise, there is a danger that much of the research within the
Bayesian program is getting ahead of itself by placing too much emphasis on
mathematical and computational power at the expense of theoretical development. In
particular, the primary goal of much Bayesian cognitive modeling has been to
demonstrate that human behavior in some task is rational with respect to a particular
choice of Bayesian model. We refer to this school of thought as Bayesian
Fundamentalism, because it strictly adheres to the tenet that human behavior can be
explained through rational analysis – once the correct probabilistic interpretation of the
task environment has been identified – without recourse to process, representation,
resource limitations, or physiological or developmental data. Although a strong case has
been made that probabilistic inference is the appropriate framework for normative
accounts of cognition (Oaksford & Chater 2007), the fundamentalist approach primarily
aims to reinforce this position, without moving on to more substantive theoretical
development or integration with other branches of cognitive science.
We see two significant disadvantages to the fundamentalist approach. First, the
restriction to computation-level accounts (cf. Marr 1982) severely limits contact with
process-level theory and data. Rational approaches attempt to explain why cognition
produces the patterns of behavior that is does, but they offer no insight into how
cognition is carried out. Our argument is not merely that rational theories are limited in
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what they can explain (this applies to all modes of explanation), but that a complete
theory of cognition must consider both rational and mechanistic explanations as well as
their interdependencies, rather than treating them as competitors. Second, the focus on
existence proofs obfuscates that there are generally multiple rational theories of any given
task that correspond to different assumptions about the environment and the learner’s
goals. Consequently, there is insufficient acknowledgement of these assumptions and
their critical roles in determining model predictions. It is extremely rare to find a
comparison among alternative Bayesian models of the same task to determine which is
most consistent with empirical data (for a related analysis of the philosophical literature,
see Fitelson 1999). Likewise, there is little recognition when the critical assumptions of a
Bayesian model logically overlap closely with those of other theories, so that the
Bayesian model is expressing essentially the same explanation, just couched in a different
framework.
The primary aim of this article is to contrast Bayesian Fundamentalism with other
Bayesian research that explicitly compares competing rational accounts and that
considers seriously the interplay between rational and mechanistic levels of explanation.
We call this the Enlightened Bayesian approach because it goes beyond the dogma of
pure rational analysis and actively attempts to integrate with other avenues of inquiry in
cognitive science. A critical distinction between Bayesian Fundamentalism and Bayesian
Enlightenment is that the latter considers the elements of a Bayesian model as claims
regarding psychological process and representation, rather than mathematical
conveniences made by the modeler for the purpose of deriving computational-level
predictions. Bayesian Enlightenment thus treats Bayesian models as making both rational
and mechanistic commitments, and it takes as a goal the joint evaluation of both. Our aim
is to initiate a discussion of the distinctions and relative merits of Bayesian
Fundamentalism and Bayesian Enlightenment so that future research can focus effort in
the directions most likely to lead to real theoretical progress.
Before commencing, we must distinguish a third use of Bayesian methods in the
cognitive and other sciences, which we refer to as Agnostic Bayes. Agnostic Bayesian
research is concerned with inferential methods for deciding among scientific models
based on empirical data (e.g., Pitt et al. 2002; Schwarz 1978). This line of research has
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developed powerful tools for data analysis, but as with other such tools (e.g., analysis of
variance, factor analysis), they are not intended as models of cognition itself. Because it
has no position on whether the Bayesian framework is useful for describing cognition,
Agnostic Bayes is not a topic of the present article. Likewise, research in pure artificial
intelligence that uses Bayesian methods without regard for potential correspondence with
biological systems is beyond the scope of this article. There is no question that the
Bayesian framework, as a formal system, is a powerful scientific tool. The question is
how well that framework parallels the workings of human cognition, and how best to
exploit those parallels to advance cognitive science.
The remainder of this article offers what we believe is an overdue assessment of
the Bayesian approach to cognitive science, including evaluation of its theoretical
content, explanatory status, scope of inquiry, and relationship to other methods. We begin
with a discussion of the role that new metaphors play in science, and cognitive science in
particular, using connectionism as an historical example to illustrate both the potential
and the danger of rapid technical advances within a theoretical framework. An overview
of Bayesian modeling of cognition is then presented that attempts to clarify what is and is
not part of a Bayesian psychological theory. Following this, we offer a critical appraisal
of the Fundamentalist Bayesian movement. We focus on concerns arising from the
limitation to strictly computational-level accounts, by noting commonalities between the
Bayesian program and other movements – namely, Behaviorism and evolutionary
psychology – that have minimized reliance on mechanistic explanations in favor of
explaining behavior directly from the environment. Finally, we outline the Enlightened
Bayesian perspective, give examples of research in this line, and explain how this
approach leads to a more productive use of the Bayesian framework and better
integration with other methods in cognitive science. Like many others, we believe that
Bayes’s mathematical formalism has great potential to aid our understanding of
cognition. Our aim is not to undermine that potential, but to focus it by directing attention
to the important questions that will allow disciplined, principled growth and integration
with existing knowledge. Above all, our hope is that by the time the excitement has faded
over their newfound expressive power, Bayesian theories will be seen to have something
important to say.
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1. Metaphor in science
Throughout the history of science, metaphor and analogy use has helped researchers gain
insight into difficult problems and make theoretical progress (Gentner et al. 1997;
Nersessian 1986; Thagard 1989). In addition to this evidence gleaned from the personal
journals of prominent scientists, direct field observation of modern molecular biologists
finds that analogies are commonly used in laboratory discussions (Dunbar 1995).
Metaphors and analogies provide powerful means for structuring an abstract or poorly
understood domain in terms of a more familiar domain, such as understanding the atom
in terms of the solar system (Gentner 1983). Drawing these parallels can lead to insights
and be a source of new ideas and hypotheses.
Daugman (2001) reviews historical use of metaphor for describing brain function
and concludes that current technology has consistently determined the dominant choice
of metaphor, from water technology to clocks to engines to computers. Whatever society
at large views as its most powerful device tends to become our means for thinking about
the brain, even in formal scientific settings. Despite the recurring tendency to take the
current metaphor literally, it is important to recognize that any metaphor will eventually
be supplanted. Thus, researchers should be aware of what the current metaphor
contributes to their theories, as well as what the theories’ logical content is once the
metaphor is stripped away.
One danger is mistaking metaphors for theories in themselves. In such cases,
scientific debate shifts focus from comparisons of theories within established frameworks
to comparisons among metaphors. Such debates are certainly useful in guiding future
research efforts, but it must be recognized that questions of metaphor are not scientific
questions (at best, they are metascientific). Metaphors should be viewed as tools or
languages, not theories in themselves. Conflating debates over scientific metaphors with
scientific debates per se can impede theoretical progress in a number of ways. By shifting
focus to the level of competing metaphors, the logical content of specific theories can
become neglected. Research that emphasizes existence proofs, demonstrating that a given
set of phenomena can be explained within a given framework, tends to ignore critical
comparisons among multiple, competing explanations. Likewise, the emphasis on
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differences in metaphorical frameworks can obscure that theories cast within different
frameworks can have substantial logical overlap. In both ways, basic theory loses out
because too much effort is spent debating the best way to analyze or understand the
scientific subject, at the expense of actually doing the analysis. Only by identifying
competing explanations, and distilling their differences to logical differences in
assumptions and empirically testable contrasting predictions, can true theoretical progress
be made.
1.1. The case of connectionism
One illustration of this process within cognitive science comes from the history of
connectionism. Connectionism was originally founded on a metaphor with telegraph
networks (Daugman 2001) and later on a metaphor between information-processing units
and physical neurons (in reaction to the dominant computer metaphor of the 1970s and
1980s). At multiple points in its development, research in connectionism has been
marked by technical breakthroughs that significantly advanced the computational and
representational power of existing models. These breakthroughs led to excitement that
connectionism was the best framework within which to understand the brain. However,
the initial rushes of research that followed focused primarily on demonstrations of what
could be accomplished within this framework, with little attention to the theoretical
commitments behind the models or whether their operation captured something
fundamental to human or animal cognition. Consequently, when challenges arose to
connectionism’s computational power, the field suffered major setbacks because there
was insufficient theoretical or empirical grounding to fall back on. Only after researchers
began to take connectionism seriously as a mechanistic model, to address what it could
and could not predict, and to consider what constraints it placed on psychological theory,
did the field mature to the point that it was able to make a lasting contribution. This shift
in perspective also helped to clarify the models’ scope, in terms of what questions they
should be expected to answer, and identified shortcomings that in turn spurred further
research.
There are of course numerous perspectives on the historical and current
contributions of connectionism, and it is not the purpose of the present article to debate
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these views. Instead, we merely summarize two points in the history of connectionism
that illustrate how overemphasis on computational power at the expense of theoretical
development can delay scientific progress.
Early work on artificial neurons by McCulloch and Pitts (1943) and synaptic
learning rules by Hebb (1949) showed how simple, neuronlike units could automatically
learn various prediction tasks. This new framework seemed very promising as a source of
explanations for autonomous, intelligent behavior. A rush of research followed,
culminated by Rosenblatt’s (1962) perceptron model, for which he boldly claimed,
“Given an elementary -perceptron, a stimulus world W, and any classification C(W) for
which a solution exists, . . . an error correction procedure will always yield a solution to
C(W) in finite time.” However, Minsky and Papert (1969) pointed out a fatal flaw:
Perceptrons are provably unable to solve problems requiring nonlinear solutions. This
straightforward, yet unanticipated, critique devastated the connectionist movement such
that there was little research under that framework for the ensuing 15 years.
Connectionism underwent a revival in the mid-1980s, primarily triggered by the
development of back-propagation, a learning algorithm that could be used in multilayer
networks (Rumelhart et al. 1986). This advance dramatically expanded the
representational capacity of connectionist models to the point where they were capable of
approximating any function to arbitrary precision, bolstering hopes that paired with
powerful learning rules any task could be learnable (Hornik et al. 1989). This technical
advance led to a flood of new work as researchers sought to show that neural networks
could reproduce the gamut of psychological phenomena, from perception to decision
making to language processing (e.g., McClelland et al. 1986; Rumelhart et al. 1986).
Unfortunately, the bubble was to burst, once again, following a series of attacks on
connectionism’s representational capabilities and lack of grounding. Connectionist
models were criticized for being incapable of capturing the compositionality and
productivity characteristic of language processing and other cognitive representations
(Fodor & Pylyshyn 1988); for being too opaque (e.g., in the distribution and dynamics of
their weights) to offer insight into their own operation, much less that of the brain
(Smolensky 1988); and for using learning rules that are biologically implausible and
amount to little more than a generalized regression (Crick 1989). The theoretical position
9
underlying connectionism was thus reduced to the vague claim that that the brain can
learn through feedback to predict its environment, without a psychological explanation
being offered of how it does so. As before, once the excitement over computational
power was tempered, the shortage of theoretical substance was exposed.
One reason that research in connectionism suffered such setbacks is that, although
there were undeniably important theoretical contributions made during this time, overall
there was insufficient critical evaluation of the nature and validity of the psychological
claims underlying the approach. During the initial explosions of connectionist research,
not enough effort was spent asking what it would mean for the brain to be fundamentally
governed by distributed representations and tuning of association strengths, or which
possible specific assumptions within this framework were most consistent with the data.
Consequently, when the limitations of the metaphor were brought to light, the field was
not prepared with an adequate answer. On the other hand, pointing out the shortcomings
of the approach (e.g., Marcus 1998; Pinker & Prince 1988) was productive in the long run
because it focused research on the hard problems. Over the last two decades, attempts to
answer these criticisms have led to numerous innovative approaches to computational
problems such as object binding (Hummel & Biederman 1992), structured representation
(Pollack 1990), recurrent dynamics (Elman 1990), and executive control (e.g., Miller &
Cohen 2001; Rougier et al. 2005). At the same time, integration with knowledge of
anatomy and physiology has led to much more biologically realistic networks capable of
predicting neurological, pharmacological, and lesion data (e.g., Boucher et al. 2007;
Frank et al. 2004). As a result, connectionist modeling of cognition has a much firmer
grounding than before.
1.2. Lessons for the Bayesian program?
This brief historical review serves to illustrate the dangers that can arise when a new line
of research is driven primarily by technical advances and is not subjected to the same
theoretical scrutiny as more mature approaches. We believe such a danger currently
exists in regard to Bayesian models of cognition. Principles of probabilistic inference
have been prevalent in cognitive science at least since the advent of signal detection
theory (Green & Swets 1966). However, Bayesian models have become much more
10
sophisticated in recent years, largely due to mathematical advances in specifying
hierarchical and structured probability distributions (e.g., Engelfriet & Rozenberg 1997;
Griffiths & Ghahramani 2006) and in efficient algorithms for approximate inference over
complex hypothesis spaces (e.g., Doucet et al. 2000; Hastings 1970). Some of the ideas
developed by psychologists have been sufficiently sophisticated that they have fed back
to significantly impact computer science and machine learning (e.g., Thibaux & Jordan
2007). In psychology, these technical developments have enabled application of the
Bayesian approach to a wide range of complex cognitive tasks, including language
processing and acquisition (Chater & Manning 2006), word learning (Xu & Tenenbaum
2007), concept learning (Anderson 1991), causal inference (Griffiths & Tenenbaum
2009), and deductive reasoning (Chater & Oaksford 1999a). There is a growing belief in
the field that the Bayesian framework has the potential to solve many of our most
important open questions, as evidenced by the rapid increase in the number of articles
published on Bayesian models and by optimistic assessments such as, “In the [last]
decade, probabilistic models have flourished . . . [The current wave of researchers] have
considerably extended both the technical possibilities of probabilistic models and their
range of applications in cognitive science” (Chater & Oaksford 2008, p. 25).
One attraction of the Bayesian framework is that it is part of a larger class of
models that make inferences in terms of probabilities. Like connectionist models,
probabilistic models avoid many of the challenges of symbolic models founded on
Boolean logic and classical artificial intelligence (e.g., Newell & Simon 1972). For
example, probabilistic models offer a natural account of non-monotonic reasoning,
avoiding the technical challenges that arise in the development of nonmonotonic logics
(see Gabbay et al. 1994). Oaksford and Chater (2007) make a strong case that
probabilistic models have greater computational power than propositional models, and
that the Bayesian framework is the more appropriate standard for normative analysis of
human behavior than is that of classical logic (but, for an important counterargument, see
Binmore 2009). Unfortunately, most of the literature on Bayesian modeling of cognition
has not moved past these general observations. Much current research falls into what we
have labeled Bayesian Fundamentalism, which emphasizes promotion of the Bayesian
metaphor over tackling genuine theoretical questions. As with early incarnations of
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connectionism, the Bayesian Fundamentalist movement is primarily driven by the
expressive power – both computational and representational – of its mathematical
framework. Most applications to date have been existence proofs, in that they
demonstrate a Bayesian account is possible without attempting to adjudicate among (or
even acknowledge) the multiple Bayesian models that are generally possible, or to
translate the models into psychological assumptions that can be compared with existing
approaches. Furthermore, amidst the proliferation of Bayesian models for various
psychological phenomena, there has been surprisingly little critical examination of the
theoretical tenets of the Bayesian program as a whole.
Taken as a psychological theory, the Bayesian framework does not have much to
say. Its most unambiguous claim is that much of human behavior can be explained by
appeal to what is rational or optimal. This is an old idea that has been debated for
centuries (e.g., Kant 1787/1961). More importantly, rational explanations for behavior
offer no guidance as to how that behavior is accomplished. As already mentioned, early
connectionist learning rules were subject to the same criticism, but connectionism is
naturally suited for grounding in physical brain mechanisms. The Bayesian framework is
more radical in that, unlike previous brain metaphors grounded in technology and
machines, the Bayesian metaphor is tied to a mathematical ideal and thus eschews
mechanism altogether. This makes Bayesian models more difficult to evaluate. By
locating explanations firmly at the computational level, the Bayesian Fundamentalist
program renders irrelevant many major modes of scientific inquiry, including physiology,
neuroimaging, reaction time, heuristics and biases, and much of cognitive development
(although, as we show in sect. 5, this is not a necessary consequence of the Bayesian
framework itself). All of these considerations suggest it is critical to pin Bayes down, to
bring the Bayesian movement past the demonstration phase and get to the real work of
using Bayesian models in integration with other approaches, to understand the detailed
workings of the mind and brain.
2. Bayesian inference as a psychological model
Bayesian modeling can seem complex to the outsider. The basic claims of Bayesian
modeling can be completely opaque to the non–mathematically inclined. In reality, the
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presuppositions of Bayesian modeling are fairly simple. In fact, one might wonder what
all the excitement is about once the mystery is removed. Here, by way of toy example,
we shed light on the basic components at the heart of every Bayesian model. The hope is
that this illustration will clarify what the basic claims of the Bayesian program are.
Constructing a Bayesian model involves two steps. The first step is to specify the
set of possibilities for the state of the world, which is referred to as the hypothesis space.
Each hypothesis can be thought of as a candidate prediction by the subject about what
future sensory information will be encountered. However, the term hypothesis should not
be confused with its more traditional use in psychology, connoting explicit testing of
rules or other symbolically represented propositions. In the context of Bayesian
modeling, hypotheses need have nothing to do with explicit reasoning, and indeed the
Bayesian framework makes no commitment whatsoever on this issue. For example, in
Bayesian models of visual processing, hypotheses can correspond to extremely low-level
information, such as the presence of elementary visual features (contours, etc.) at various
locations in the visual field (Geisler et al. 2001). There is also no commitment regarding
where the hypotheses come from. Hypotheses could represent innate biases or
knowledge, or they could have been learned previously by the individual. Thus the
framework has no position on nativist-empiricist debates. Furthermore, hypotheses
representing very different types of information (e.g., a contour in a particular location,
whether or not the image reminds you of your mother, whether the image is symmetrical,
whether it spells a particular word, etc.) are all lumped together in a common hypothesis
space and treated equally by the model. Thus there is no distinction between different
types of representations or knowledge systems within the brain. In general, a hypothesis
is nothing more than a probability distribution. This distribution, referred to as the
likelihood function, simply specifies how likely each possible pattern of observations is
according to the hypothesis in question.
The second step in constructing a Bayesian model is to specify how strongly the
subject believes in each hypothesis before observing data. This initial belief is expressed
as a probability distribution over the hypothesis space and is referred to as the prior
distribution (or prior). The prior can be thought of as an initial bias in favor of some
hypotheses over others, in that it contributes extra votes (as elaborated below) that are
13
independent of any actual data. This decisional bias allows the model’s predictions to be
shifted in any direction the modeler chooses regardless of the subject’s observations. As
we discuss in section 4, the prior can be a strong point of the model if it is derived from
empirical statistics of real environments. However, more commonly the prior is chosen
ad hoc, providing substantial unconstrained flexibility to models that are advocated as
rational and assumption-free.
Together, the hypotheses and the prior fully determine a Bayesian model. The
model’s goal is to decide how strongly to believe in each hypothesis after data have been
observed. This final belief is again expressed as a probability distribution over the
hypothesis space and is referred to as the posterior distribution (or posterior). The
mathematical identity known as Bayes’s Rule is used to combine the prior with the
observed data to compute the posterior. Bayes’s Rule can be expressed in many ways, but
here we explain how it can be viewed as a simple vote-counting model. Specifically,
Bayesian inference is equivalent to tracking evidence for each hypothesis, or votes for
how strongly to believe in each hypothesis. The prior provides the initial evidence counts,
Eprior, which are essentially made-up votes that give some hypotheses a head start over
others before any actual data are observed. When data are observed, each observation
adds to the existing evidence according to how consistent it is with each hypothesis. The
evidence contributed for a hypothesis that predicted the observation will be greater than
the evidence for a hypothesis under which the observation was unlikely. The evidence
contributed by the ith observation, is simply added to the existing evidence to
update each hypothesis’s count. Therefore the final evidence, Eposterior, is nothing more
than a sum of the votes from all of the observations, plus the initial votes from the prior:1
(1)
This sum is computed for every hypothesis, H, in the hypothesis space. The vote totals
determine how strongly the model believes in each hypothesis in the end. Thus any
Bayesian model can be viewed as summing evidence for each hypothesis, with initial
evidence coming from the prior and with additional evidence coming from each new
observation. The final evidence counts are then used in whatever decision procedure is
appropriate for the task, such as determining the most likely hypothesis, predicting the
14
value of some unobserved variable (by weighting each hypothesis by its posterior
probability and averaging their predictions), or choosing an action that maximizes the
expected value of some outcome (again by weighted average over hypotheses). At its
core, this is all there is to Bayesian modeling.
To illustrate these two steps and how inference proceeds in a Bayesian model,
consider the problem of determining whether a fan entering a football stadium is rooting
for the University of Southern California (USC) Trojans or the University of Texas (UT)
Longhorns, based on three simple questions: (1) Do you live by the ocean? (2) Do you
own a cowboy hat? (3) Do you like Mexican food? The first step is to specify the space
of possibilities (i.e., hypothesis space). In this case the hypothesis space consists of two
possibilities: being a fan of either USC or UT. Both of these hypotheses entail
probabilities for the data we could observe, for example, P(ocean | USC) = .8 and
P(ocean | UT) = .3. Once these probabilities are given, the two hypotheses are fully
specified. The second step is to specify the prior. In many applications, there is no
principled way of doing this, but in this example the prior corresponds to the probability
that a randomly selected person will be a USC fan or a UT fan; that is, one’s best guess as
to the overall proportion of USC and UT fans in attendance.
With the model now specified, inference proceeds by starting with the prior and
accumulating evidence as new data are observed. For example, if the football game is
being played in Los Angeles, one might expect that most people are USC fans, and hence
the prior would provide an initial evidence count in favor of USC. If our target person
responded that he lives near the ocean, this observation would add further evidence for
USC relative to UT. The magnitudes of these evidence values will depend on the specific
numbers assumed for the prior and for the likelihood function for each hypothesis, but all
that the model does is take the evidence values and add them up. Each new observation
adds to the balance of evidence among the hypotheses, strengthening those that predicted
it relative to those under which it was unlikely.
There are several ways in which real applications of Bayesian modeling become
more complex than the foregoing simple example. However, these all have to do with the
complexity of the hypothesis space rather than the Bayesian framework itself. For
example, many models have a hierarchical structure in which hypotheses are essentially
15
grouped into higher-level overhypotheses. Overhypotheses are generally more abstract
and require more observations to discriminate among them; thus hierarchical models are
useful for modeling learning (e.g., Kemp et al. 2007). However, each overhypothesis is
just a weighted sum of elementary hypotheses, and inference among overhypotheses
comes down to exactly the same vote-counting scheme as described earlier. As a second
example, many models assume special mathematical functions for the prior, such as
conjugate priors (discussed further in sect. 5), that simplify the computations involved in
updating evidence. However, such assumptions are generally made solely for the
convenience of the modeler rather than for any psychological reason related to the likely
initial beliefs of a human subject. Finally, for models with especially complex hypothesis
spaces, computing exact predictions often becomes computationally intractable. In these
cases, sophisticated approximation schemes are used, such as Markov-chain Monte Carlo
(MCMC) or particle filtering (i.e., sequential Monte Carlo). These algorithms yield good
estimates of the model’s true predictions while requiring far less computational effort.
However, once again they are used for the convenience of the modeler and are not meant
as proposals for how human subjects might solve the same computational problems. As
we argue in section 5, all three of these issues are points where Bayesian modeling makes
potential contact with psychological theory in terms of how information is represented
and processed. Unfortunately, most of the focus to date has been on the Bayesian
framework itself, setting aside where the hypotheses and priors come from and how the
computations are performed or approximated.
The aim of this section was to clear up confusion about the nature and theoretical
claims of Bayesian models. To summarize: Hypotheses are merely probability
distributions and have no necessary connection to explicit reasoning. The model’s
predictions depend on the initial biases on the hypotheses (i.e., the prior), but the choice
of the prior does not always have a principled basis. The heart of Bayesian inference –
combining the prior with observed data to reach a final prediction – is formally
equivalent to a simple vote-counting scheme. Learning and one-off decision making both
follow this scheme and are treated identically except for timescale and specificity of
hypotheses. The elaborate mathematics that often arises in Bayesian models comes from
the complexity of their hypothesis sets or the tricks used to derive tractable predictions,
16
which generally have little to do with the psychological claims of the researchers.
Bayesian inference itself, aside from its assumption of optimality and close relation to
vote-counting models, is surprisingly devoid of psychological substance. It involves no
representations to be updated; no encoding, storage, retrieval, or search; no attention or
control; no reasoning or complex decision processes; and actually no mechanism at all,
except for a simple counting rule.
3. Bayes as the new Behaviorism
Perhaps the most radical aspect of Bayesian Fundamentalism is its rejection of
mechanism. The core assumption is that one can predict behavior by calculating what is
optimal in any given situation. Thus, the theory is cast entirely at the computational level
(in the sense of Marr 1982), without recourse to mechanistic (i.e., algorithmic or
implementational) levels of explanation. As a metascientific stance, this is a very strong
position. It asserts that a wide range of modes of inquiry and explanation are essentially
irrelevant to understanding cognition. In this regard, the Bayesian program has much in
common with Behaviorism. This section explores the parallels between these two schools
of thought in order to draw out some of the limitations of Bayesian Fundamentalism.
During much of the first half of the 20th century, American psychology was
dominated by the Behaviorist belief that one cannot draw conclusions about unobservable
mental entities (Skinner 1938; Watson 1913). Under this philosophy, theories and
experiments were limited to examination of the schedule of sensory stimuli directly
presented to the subject and the patterns of observed responses. This approach conferred
an important degree of rigor that the field previously lacked, by abolishing Dualism,
advocating rigorous Empiricism, and eliminating poorly controlled and objectively
unverifiable methods such as introspection. The strict Empiricist focus also led to
discovery of important and insightful phenomena, such as shaping (Skinner 1958) and
generalization (Guttman & Kalish 1956).
One consequence of the Behaviorist framework was that researchers limited
themselves to a very constrained set of explanatory tools, such as conditioning and
reinforcement. These tools have had an important lasting impact, for example, in
organizational behavior management (Dickinson 2000) and behavioral therapy for a wide
17
variety of psychiatric disorders (Rachman 1997). However, cognitive constructs, such as
representation and information processing (e.g., processes associated with inference and
decision making), were not considered legitimate elements of a psychological theory.
Consequently, Behaviorism eventually came under heavy criticism for its inability to
account for many aspects of cognition, especially language and other higher-level
functions (Chomsky 1959). After the so-called Cognitive Revolution, when researchers
began to focus on the mechanisms by which the brain stores and processes information,
the depth and extent of psychological theories were dramatically expanded (Miller 2003).
Relative to the state of current cognitive psychology, Behaviorist research was extremely
limited in the scientific questions that it addressed, the range of explanations it could
offer, and the empirical phenomena it could explain.
The comparison of Bayesian modeling to Behaviorism may seem surprising
considering that Bayesian models appear to contain unobservable cognitive constructs,
such as hypotheses and their subjective probabilities. However, these constructs rarely
have the status of actual psychological assumptions. Psychological theories of
representation concern more than just what information is tracked by the brain; they
include how that information is encoded, processed, and transformed. The
Fundamentalist Bayesian view takes no stance on whether or how the brain actually
computes and represents probabilities of hypotheses. All that matters is whether behavior
is consistent with optimal action with respect to such probabilities (Anderson 1990;
1991). This means of sidestepping questions of representation can be viewed as a strength
of the rational approach, but it also means that Bayesian probabilities are not necessarily
psychological beliefs. Instead, they are better thought of as tools used by the researcher to
derive behavioral predictions. The hypotheses themselves are not psychological
constructs either, but instead reflect characteristics of the environment. The set of
hypotheses, together with their prior probabilities, constitute a description of the
environment by specifying the likelihood of all possible patterns of empirical
observations (e.g., sense data). According to Bayesian Fundamentalism, this description
is an accurate one, and by virtue of its accuracy it is determined solely by the
environment. There is no room for psychological theorizing about the nature of the
hypothesis set, because such theories logically could only take the form of explaining
18
how people’s models of the environment are incorrect. According to Bayesian
Fundamentalism, by grounding the hypotheses and prior in the environment (Anderson
1990), Bayesian models make predictions directly from the environment to behavior,
with no need for psychological assumptions of any sort.
In many Bayesian models, the hypotheses are not expressed as an unstructured
set, but instead emerge from a generative model of the environment. The generative
model (which is a component of the Bayesian model) often takes the form of a causal
network in which the probabilities of observable variables depend on the values of
unobservable, latent variables. Hypotheses about observable variables correspond to
values of the latent variables. For example, in the topic model of text comprehension, the
words in a passage (the observables) are assumed to be generated by a stochastic process
parameterized by the weights of various semantic topics within the passage (Griffiths et
al. 2007). However, the model makes no claim about the psychological status of the
latent variables (i.e., the topic weights). These variables serve only to define the joint
distribution over all possible word sequences, and the model is evaluated only with
respect to whether human behavior is consistent with that distribution. Whether people
explicitly represent topic weights (or their posterior distributions) or whether they arrive
at equivalent inferences based on entirely different representations is outside the scope of
the model (Griffiths et al. 2007, p. 212). Therefore, generative models and the latent
variables they posit do not constitute psychological constructs, at least according to the
fundamentalist viewpoint. Instead, they serve as descriptions of the environment and
mathematical tools that allow the modeler to make behavioral predictions. Just as in
Behaviorist theories, the path from environmental input to behavioral prediction bypasses
any consideration of cognitive processing.
To take a simpler example, Figure 1 shows a causal graphical model
corresponding to a simplified version of Anderson’s (1991) rational model of
categorization. The subject’s task in this example is to classify animals as birds or
mammals. The rational model assumes that these two categories are each partitioned into
subcategories, which are termed clusters. The psychological prediction is that
classification behavior corresponds (at a computational level) to Bayesian inference over
this generative model. If a subject were told that a particular animal can fly, the optimal
19
probability that it is a bird would equal the sum of the posterior probabilities of all the
clusters within the bird category (and likewise for mammal). Critically, however, the
clusters do not necessarily correspond to actual psychological representations. All that
matters for predicting behavior is the joint probability distribution over the observable
variables (i.e., the features and category labels). The clusters help the modeler to
determine this distribution, but the brain may perfom the computations in a completely
different manner. In the discussion of Bayesian Enlightenment below (sect. 5), we return
to the possibility of treating latent variables and generative models as psychological
assumptions about knowledge representation. However, the important point here is that,
according to the Fundamentalist Bayesian view, they are not. Generative models, the
hypotheses they specify, and probability distributions over those hypotheses are all
merely tools for deriving predictions from a Bayesian model. The model itself exists at a
computational level, where its predictions are defined only based on optimal inference
and decision making. The mechanisms by which those decisions are determined are
outside the model’s scope.
Figure 1.
20
3.1. Consequences of the denial of mechanism
By eschewing mechanism and aiming to explain behavior purely in terms of rational
analysis, the Fundamentalist Bayesian program raises the danger of pushing the field of
psychology back toward the sort of restrictive state experienced during the strict
Behaviorist era. Optimality and probabilistic inference are certainly powerful tools for
explaining behavior, but taken alone they are insufficient. A complete science of
cognition must draw on the myriad theoretical frameworks and sources of evidence
bearing on how cognition is carried out, as opposed to just its end product. These include
theories of knowledge representation, decision making, mental models, dynamic-system
approaches, attention, executive control, heuristics and biases, reaction time,
embodiment, development, and the entire field of cognitive neuroscience, just to name
some. Many of these lines of research would be considered meaningless within the
Behaviorist framework, and likewise they are all rendered irrelevant by the strict rational
view. Importantly, the limitation is not just on what types of explanations are considered
meaningful, but also on what is considered worthy of explanation – that is, what scientific
questions are worth pursuing and what types of evidence are viewed as informative.
An important argument in favor of rational over mechanistic modeling is that the
proliferation of mechanistic modeling approaches over the past several decades has led to
a state of disorganization, wherein models’ substantive theoretical content cannot be
disentangled from idiosyncrasies of their implementations. Distillation of models down to
their computational principles would certainly aid in making certain comparisons across
modeling frameworks. For example, both neural network (Burgess & Hitch 1999) and
production system (Anderson et al. 1998) models of serial recall have explained primacy
effects by using the same assumptions about rehearsal strategies, despite the significant
architectural differences in which this common explanation is implemented. The rational
approach is useful in this regard in that it eases comparison by emphasizing the
computational problems that models aim to solve.
However, it would be a serious overreaction simply to discard everything below
the computational level. As in nearly every other science, understanding how the subject
of study (i.e., the brain) operates is critical to explaining and predicting its behavior. As
we argue in section 4, mechanistic explanations tend to be better suited for prediction of
21
new phenomena, as opposed to post hoc explanation. Furthermore, algorithmic
explanations and neural implementations are an important focus of research in their own
right. Much can be learned from consideration of how the brain handles the
computational challenge of guiding behavior efficiently and rapidly in a complex world,
when optimal decision making (to the extent that it is even well defined) is not possible.
These mechanistic issues are at the heart of most of the questions of theoretical or
practical importance within cognitive science, including questions of representation,
timing, capacity, anatomy, and pathology.
For example, connectionist models have proven valuable in reconceptualizing
category-specific deficits in semantic memory as arising from damage to distributed
representations in the brain (for a review, see Rogers & Plaut 2002), as opposed to being
indicative of damage to localized representations (e.g., Caramazza & Shelton 1998).
Although these insights rely on statistical analyses of how semantic features are
distributed (e.g., Cree & McRae 2003), and, thus, could in principle be characterized by a
Bayesian model, the connectionist models were tremendously useful in motivating this
line of inquiry. Additionally, follow-on studies have helped characterize impaired
populations and have suggested interventions, including studies involving Alzheimer’s
patients (Devlin et al. 1998) and related work exploring reading difficulties resulting
from developmental disorders and brain injury (Joanisse & Seidenberg 1999; 2003; Plaut
et al. 1996).
Even when the goal is only to explain inference or choice behavior (setting aside
reaction time), optimal probabilistic inference is not always sufficient. This is because the
psychological mechanisms that give rise to behavior often at best only approximate the
optimal solution. These mechanisms produce signature deviations from optimality that
rational analysis has no way of anticipating. Importantly, considering how representations
are updated in these mechanisms can suggest informative experiments.
For example, Sakamoto et al. (2008) investigated learning of simple perceptual
categories that differed in the variation among items within each category. To classify
new stimuli accurately, subjects had to estimate both the means and variances of the
categories (stimuli varied along a single continuous dimension). Sakamoto et al.
considered a Bayesian model that updates its estimates optimally, given all past instances
22
of each category, and a mechanistic (cluster) model that learns incrementally in response
to prediction error. The incremental model naturally produces recency effects, whereby
more recent observations have a greater influence on its current state of knowledge (Estes
1957), in line with empirical findings in this type of task (e.g., Jones & Sieck 2003).
Simple recency effects are no challenge to Bayesian models, because one can assume
nonstationarity in the environment (e.g., Yu & Cohen 2008). However, the incremental
model predicts a more complex recency effect whereby, under certain presentation
sequences, the recency effect in the estimate of a category’s mean induces a bias in the
estimate of its variance. This bias arises purely as a by-product of the updating algorithm
and has no connection to rational, computational-level analyses of the task. Human
subjects exhibited the same estimation bias predicted by the incremental model,
illustrating the utility of mechanistic models in directing empirical investigations and
explaining behavior.
Departures from strict rational orthodoxy can lead to robust and surprising
predictions, such as in work considering the forces that mechanistic elements exert on
one another in learning and decision making (Busemeyer & Johnson 2008; Davis & Love
2010; Spencer et al. 2009). Such work often serves to identify relevant variables that
would not be deemed theoretically relevant under a Fundamentalist Bayesian view
(Clearfield et al. 2009). Even minimal departures from purely environmental
considerations, such as manipulating whether information plays the role of cue or
outcome within a learning trial, can yield surprising and robust results (Love 2002;
Markman & Ross 2003; Ramscar et al. 2010; Yamauchi & Markman 1998). The effects
of this manipulation can be seen in a common transfer task, implying that it is the
learners’ knowledge that differs and not just their present goals.
Focusing solely on computational explanations also eliminates many of the
implications of cognitive science for other disciplines. For example, without a theory of
the functional elements of cognition, little can be said about cognitive factors involved in
psychological disorders. Likewise, without a theory of the physiology of cognition, little
can be said about brain disease, trauma, or psychopharmacology. (Here the situation is
even more restrictive than in Behaviorism, which would accept neurological data as valid
and useful.) Applications of cognitive theory also tend to depend strongly on mechanistic
23
descriptions of the mind. For example, research in human factors relies on models of
timing and processing capacity, and applications to real-world decision making depend
on the heuristics underlying human judgment. Understanding these heuristics can also
lead to powerful new computational algorithms that improve the performance of
artificially intelligent systems in complex tasks (even systems built on Bayesian
architectures). Rational analysis provides essentially no insight into any of these issues.
3.2. Integration and constraints on models
One advantage of Behaviorism is that its limited range of explanatory principles led to
strong cohesion among theories of diverse phenomena. For example, Skinner (1957)
attempted to explain human verbal behavior by using the same principles previously used
in theories of elementary conditioning. It might be expected that the Bayesian program
would enjoy similar integration because of its reliance on the common principles of
rational analysis and probabilistic inference. Unfortunately, this is not the case in practice
because the process of rational analysis is not sufficiently constrained, especially as
applied to higher-level cognition.
Just as mechanistic modeling allows for alternative assumptions about process
and representation, rational modeling allows for alternative assumptions about the
environment in which the cognitive system is situated (Anderson 1990). In both cases, a
principal scientific goal is to decide which assumptions provide the best explanation.
With Bayesian models, the natural approach dictated by rational analysis is to make the
generative model faithful to empirical measurements of the environment. However, as we
observe in section 4, this empirical grounding is rarely carried out in practice.
Consequently, the rational program loses much of its principled nature, and models of
different tasks become fractionated because there is nothing but the math of Bayesian
inference to bind them together.
At the heart of every Bayesian model is a set of assumptions about the task
environment, embodied by the hypothesis space and prior distribution, or equivalently by
the generative model and prior distributions for its latent variables. The prior distribution
is the well-known and oft-criticized lack of constraint in most Bayesian models. As
explained in section 2, the prior provides the starting points for the vote-counting process
24
of Bayesian inference, thereby allowing the model to be initially biased toward some
hypotheses over others. Methods have been developed for using uninformative priors that
minimize influence on model predictions, such as Jeffreys priors (Jeffreys 1946) or
maximum-entropy priors (Jaynes 1968). However, a much more serious source of
indeterminacy comes from the choice of the hypothesis set itself or equivalently from the
choice of the generative model.
The choice of generative model often embodies a rich set of assumptions about
the causal and dynamic structure of the environment. In most interesting cases, many
alternative assumptions could be made, but only one is considered. For example, the
CrossCat model of how people learn multiple overlapping systems of categories (Shafto
et al., in press) assumes that category systems constitute different partitions of a stimulus
space, that each category belongs to exactly one system, and that each stimulus feature or
dimension is relevant to exactly one category system and is irrelevant to all others. These
assumptions are all embodied by the generative model on which CrossCat is based. There
are clearly alternatives to these assumptions, for which intuitive arguments can be made
(e.g., for clothing, the color dimension is relevant for manufacturing, laundering, and
considerations of appearance), but there is no discussion of these alternatives,
justification of the particular version of the model that was evaluated, or consideration of
the implications for model predictions. Other than the assumption of optimal inference,
all there is to a Bayesian model is the choice of generative model (or hypothesis set plus
prior), so it is a serious shortcoming when a model is developed or presented without
careful consideration of that choice. The neglected multiplicity of models is especially
striking considering the rational theorist’s goal of determining the – presumably unique –
optimal pattern of behavior.
Another consequence of insufficient scrutiny of generative models (or hypothesis
sets more generally) is a failure to recognize the psychological commitments they entail.
These assumptions often play a central role in the explanation provided by the Bayesian
model as a whole, although that role often goes unacknowledged. Furthermore, the
psychological assumptions implicitly built into a generative model can be logically
equivalent to preexisting theories of the same phenomena. For example, Kemp et al.
(2007) propose a Bayesian model of the shape bias in early word learning, whereby
25
children come to expect a novel noun to be defined by the shape of the objects it denotes
rather than other features such as color or texture. The model learns the shape bias
through observation of many other words with shape-based definitions, which shifts
evidence to an overhypothesis that most nouns in the language are shape-based. The
exposition of the model is a mathematically elegant formalization of abstract induction.
However, it is not Bayes’s Rule or even the notion of overhypotheses that drives the
prediction; rather it is the particular overhypotheses that were built into the model. In
other words, the model was endowed with the capability to recognize a particular pattern
(viz., regularity across words in which perceptual dimensions are relevant to meaning), so
the fact that it indeed recognizes that pattern when presented with it is not surprising or
theoretically informative. Furthermore, the inference made by the model is logically the
same as the notion of second-order generalization proposed previously by Smith et al.
(2002). Detailed mechanistic modeling has shown how second-order generalization can
emerge from the interplay between attentional and associative processes (Colunga &
Smith 2005), in contrast to the more tautological explanation offered by the Bayesian
model. Therefore, at the level of psychological theory, Kemp et al.’s model merely
recapitulates a previously established idea in a way that is mathematically more elegant
but psychologically less informative.
In summary, Bayesian Fundamentalism is simultaneously more restrictive and
less constrained than Behaviorism. In terms of modes of inquiry and explanation, both
schools of thought shun psychological constructs, in favor of aiming to predict behavior
directly from environmental inputs. However, under Behaviorism this restriction was
primarily a technological one. Nothing in the Behaviorist philosophy would invalidate
relatively recent tools that enable direct measurements of brain function, such as
neuroimaging, EEG, and single-unit recording (at least as targets of explanation, if not as
tools through which to develop theories of internal processes). Indeed, these techniques
would presumably have been embraced because they satisfy the criterion of direct
observation. Bayesian Fundamentalism, in contrast, rejects all measures of brain
processing out of principle because only the end product (i.e., behavior) is relevant to
rational analysis.2 At the same time, whereas Behaviorist theories were built from simple
mechanisms and minimal assumptions, Bayesian models often depend on complex
26
hypothesis spaces based on elaborate and mathematically complex assumptions about
environmental dynamics. As the emphasis is generally on rational inference (i.e., starting
with the assumptions of the generative model and deriving optimal behavior from there),
the assumptions themselves generally receive little scrutiny. The combination of these
two factors leads to a dangerously underconstrained research program in which the core
assumptions of a model (i.e., the choice of hypothesis space) can be made at the
modeler’s discretion without comparison to alternatives and without any requirement to
fit physiological or other process-level data.
4. Bayes as evolutionary psychology
In addition to the rejection of mechanistic explanation, a central principle of the
Fundamentalist Bayesian approach to cognition is that of optimality. The claim that
human behavior can be explained as adaptation to the environment is also central to
evolutionary psychology. On the surface, these two approaches to understanding behavior
seem very different, as their content and methods differ. For example, one core domain of
inquiry in evolutionary psychology is mating, which is not often studied by cognitive
psychologists, and theories in evolutionary psychology tend not to be computational in
nature, whereas rational Bayesian approaches are by definition. Thus, one advantage of
rational Bayesian accounts is that they formalize notions of optimality, which can clarify
assumptions and allow for quantitative evaluation. Despite these differences, Bayesian
Fundamentalism and evolutionary psychology share a number of motivations and
assumptions. Indeed, Geisler and Diehl (2003) propose a rational Bayesian account of
Darwin’s theory of natural selection. In this section, we highlight the commonalities and
important differences between these two approaches to understanding human behavior.
We argue below that Bayesian Fundamentalism is vulnerable to many of the
criticisms that have been leveled at evolutionary psychology. Indeed, we argue that
notions of optimality in evolutionary psychology are more complete and properly
constrained than those forwarded by Bayesian Fundamentalists, because evolutionary
psychology considers other processes than simple adaptation (e.g., Buss et al. 1998).
Bayesian Fundamentalism appropriates some concepts from evolutionary psychology
(e.g., adaptation, fitness, and optimality), but leaves behind many other key concepts
27
because of its rejection of mechanism. Because it is mechanisms that evolve, not
behaviors, Bayesian Fundamentalism’s assertions of optimality provide little theoretical
grounding and are circular in a number of cases.
Basic evolutionary theory holds that animal behavior is adapted by natural
selection, which increases inclusive fitness. High fitness indicates that an animal’s
behaviors are well suited to its environment, leading to reproductive success. On the
assumption that evolutionary pressures tune a species’ genetic code such that the
observed phenotype gives rise to optimal behaviors, one can predict an animal’s behavior
by considering the environment in which its ancestors flourished and reproduced.
According to evolutionary psychologists, this environment, referred to the environment
of evolutionary adaptedness (EEA), must be understood in order to comprehend the
functions of the brain (Bowlby 1969). Thus, evolutionary explanations of behavior tend
to focus on the environment, and this focus can on occasion occur at the expense of
careful consideration of mechanism. However, as discussed extensively below in section
4.3 and in contrast to Bayesian Fundamentalism, some key concepts in evolutionary
psychology do rely on mechanistic considerations, and these concepts are critical for
grounding notions of adaptation and optimization. These key concepts are neglected in
Bayesian Fundamentalism.
Critically, it is not any function that is optimized by natural selection, but
functions that are relevant to fitness. To use Oaksford and Chater’s (1998) example,
animals may be assumed to use optimal foraging strategies because (presumably)
gathering food efficiently is relevant to the global goal of maximizing inclusive fitness
(see Hamilton 1964). Thus, in practice, evolutionary arguments, like rational theories of
cognition, require characterizing the environment and the behaviors that increase fitness.
For example, Anderson’s (1991) rational model of category learning is intended to
maximize prediction of unknown information in the environment, which presumably
increases fitness.
Like rational approaches to cognition, evolutionary psychology draws inspiration
from evolutionary biology and views much of human behavior as resulting from
adaptations shaped by natural selection (Buss 1994; Pinker 2002; Tooby & Cosmides
2005). The core idea is that recurring challenges in our ancestral environments (i.e.,
28
EEA) shaped our mental capacities and proclivities. This environmental focus is in the
same spirit as work in ecological psychology (Gibson 1979; Michaels & Carello 1981).
Following from a focus on specific challenges and adaptations, evolutionary theories
often propose special-purpose modules. For example, evolutionary psychologists have
proposed special-purpose modules for cheater detection (Cosmides & Tooby 1992),
language acquisition (Pinker 1995), incest avoidance (Smith 2007), and snake detection
(Sperber & Hirschfeld 2003). Much like evolutionary psychology’s proliferation of
modules, rational models are developed to account for specific behaviors, such as
children’s ability to give the number of objects requested (Lee & Sarnecka 2010),
navigation when disoriented in a maze (Stankiewicz et al. 2006), and understanding a
character’s actions in an animation (Baker et al. 2009), at the expense of identifying
general mechanisms and architectural characteristics (e.g., working memory) that are
applicable across a number of tasks (in which the specific behaviors to be optimized
differ).
4.1. An illustrative example of rational analysis as evolutionary argument
Perhaps the rational program’s focus on environmental adaptation is best exemplified by
work in early vision. Early vision is a good candidate for rational investigation because
the visual environment has likely been stable for millennia and the ability to perceive the
environment accurately is clearly related to fitness. The focus on environmental statistics
is clear in Geisler et al.’s (2001) work on contour detection. In this work, Geisler and
colleagues specify how an ideal classifier detects contours and compare this ideal
classifier’s performance to human performance. To specify the ideal classifier, the
researchers gathered natural image statistics that were intended to be representative of the
environment in which our visual system evolved. Implicit in the choice of images are
assumptions about what the environment was like. Additionally, the analysis requires
assuming which measures or image statistics are relevant to the contour classification
problem. Geisler et al. selected a number of natural images of mountains, forests,
coastlines, etc., to characterize our ancestral visual environment. From these images, they
measured certain statistics they deemed relevant to contour detection. Their chosen
measures described relationships among edge segments belonging to the same contour,
29
such as the distance between the segments and their degree of colinearity. To gather these
statistics, expert raters determined whether two edge elements belonged to the same
contour in the natural images. These measures specify the likelihood and prior in the
Bayesian ideal observer. The prior for the model is simply the probability that two
randomly selected edge elements belong to the same contour. The likelihood follows
from a table of co-occurrences of various distances and angles between pairs of edge
elements indexed by whether each pair belongs to the same contour. Geisler et al.
compared human performance to the ideal observer in a laboratory task that involved
determining whether a contour was present in novel, meaningless images composed of
scattered edge elements. Human performance and the rational model closely
corresponded, supporting Geisler et al.’s account.
Notice that there is no notion of mechanism (i.e., process or representation) in this
account of contour detection. The assumptions made by the modeler include what our
ancestral environment was like and which information in this environment is relevant.
Additionally, it is assumed that the specific behavior modeled (akin to a module in
evolutionary psychology) is relevant to fitness. These assumptions, along with
demonstrating a correlation with human performance, are the intellectual contribution of
the work. Finally, rational theories assume optimal inference as reflected in the Bayesian
classification model. Specifying the Bayesian model may be technically challenging, but
is not part of the theoretical contribution (i.e., it is a math problem, not a psychology
problem). The strength of Geisler et al.’s work rests in its characterization of the
environment and the statistics of relevance.
Unfortunately, the majority of rational analyses do not include any measurements
from actual environments even though the focus of such theories is on the environment
(for a similar critique, see Murphy 1993). Instead, the vast majority of rational analysis in
cognition relies on intuitive arguments to justify key assumptions. In some cases,
psychological phenomena can be explained from environmental assumptions that are
simple and transparent enough not to require verification (e.g., McKenzie & Mikkelsen
2007; Oaksford & Chater 1994). However, more often Bayesian models incorporate
complex and detailed assumptions about the structure of the environment that are far
from obvious and are not supported by empirical data (e.g., Anderson 1991; Brown &
30
Steyvers 2009; Goodman et al. 2008; Steyvers et al. 2009; Tenenbaum & Griffiths 2001).
Cognitive work that does gather environmental measures is exceedingly rare and tends to
rely on basic statistics to explain general behavioral tendencies and judgments (e.g.,
Anderson & Schooler 1991; Griffiths & Tenenbaum 2006). This departure from true
environmental grounding can be traced back to John Anderson’s (1990; 1991) seminal
contributions in which he popularized the rational analysis of cognition. In those works,
he specified a series of steps for conducting such analyses. Step 6 of the rational method
(Anderson 1991) is to revisit assumptions about the environment and relevant statistics
when the model fails to account for human data. In practice, this step involves the
modeler’s ruminating on what the environment is like and what statistics are relevant
rather than actual study of the environment. This is not surprising given that most
cognitive scientists are not trained to characterize ancestral environments. For example,
at no point in the development of Anderson’s (1991) rational model of category learning
is anything in the environment actually measured. Although one purported advantage of
rational analysis is the development of zero-parameter, nonarbitrary models, it would
seem that the theorist has unbounded freedom to make various assumptions about the
environment and the relevant statistics (for a similar critique, see Sloman & Fernbach
2008). As discussed in the next section, similar criticisms have been made of
evolutionary psychology.
4.2. Too much flexibility in evolutionary and rational explanations?
When evaluating any theory or model, one must consider its fit to the data and its
flexibility to account for other patterns of results (Pitt et al. 2002). Models and theories
are favored that fit the data and have low complexity (i.e., are not overly flexible). One
concern we raise is whether rational approaches offer unbounded and hidden flexibility to
account for any observed data. Labeling a known behavior as rational is not theoretically
significant if it is always possible for some rational explanation to be constructed.
Likewise, evolutionary psychology is frequently derided as simply offering “just so”
stories (Buller 2005, but see Machery & Barrett 2006). Adaptationist accounts certainly
constrain explanation compared to nonadaptationist alternatives, but taken alone they still
allow significant flexibility in terms of assumptions about the environment and the extent
31
to which adaptation is possible. For example, to return to the foraging example, altering
one’s assumptions about how food rewards were distributed in ancestral environments
can determine whether an animal’s search process (i.e., the nature and balance of
exploitative and exploratory decisions) is optimal. Likewise, the target of optimization
can be changed. For example, inefficiencies in an animal’s foraging patterns for food-rich
environments can be explained after the fact as an adaptation to ensure the animal does
not become morbidly obese. On the other hand, if animals were efficient in abundant
environments and became obese, one could argue that foraging behaviors were shaped by
adaptation to environments in which food was not abundant. If, no matter the data, there
is a rational explanation for a behavior, it is not a contribution to label a behavior as
rational. Whereas previous work in the heuristics-and-biases tradition (Tversky &
Kahneman 1974) cast the bulk of cognition as irrational by using a fairly simplistic
notion of rationality, Bayesian Fundamentalism finds rationality to be ubiquitous based
on underconstrained notions of rationality.
To provide a recent example from the literature, the persistence of negative traits,
such as anxiety and insecurity, that lower an individual’s fitness has been explained by
appealing to these traits’ utility to the encompassing group in signaling dangers and
threats facing the group (Ein-Dor et al. 2010). While this ingenious explanation could be
correct, it illustrates the incredible flexibility that adaptive accounts can marshal in the
face of a challenging data point.
Similar criticisms have been leveled at work in evolutionary biology. For
example, Gould and Lewontin (1979) have criticized work that develops hypotheses
about the known functions of well-studied organs as “backward-looking.” One worry is
that this form of theorizing can lead to explanations that largely reaffirm what is currently
believed. Work in evolutionary psychology has been criticized for explaining
unsurprising behaviors (Horgan 1999), like that men are less selective about who they
will mate with than are women. Likewise, we see a tendency for rational analyses to
largely reexpress known findings in the language of Bayesian optimal behavior. The
work of Geisler et al. (2001) on contour perception is vulnerable to this criticism because
it largely recapitulates Gestalt principles (e.g., Wertheimer 1923/1938) in the language of
Bayes. In cognition, the rational rules model (Goodman et al. 2008) of category learning
32
reflects many of the intuitions of previous models, such as the rule-plus-exception
(RULEX) model (Nosofsky et al. 1994), in a more elegant and expressive Bayesian form
that does not make processing predictions. In other cases, the intuitions from previous
work are reexpressed in more general Bayesian terms in which particular choices for the
priors enable the Bayesian model to mimic the behavior of existing models. For example,
unsupervised clustering models using simplicity principles based on minimum
description length (MDL; Pothos & Chater 2002) are recapitulated by more flexible
approaches phrased in the language of Bayes (Austerweil & Griffiths 2008; Griffiths et
al. 2008). A similar path of model development has occurred in natural language
processing (Ravi & Knight 2009).
One motivation for rational analysis was to prevent models with radically
different assumptions from making similar predictions (Anderson 1991). In reality, the
modeler has tremendous flexibility in characterizing the environment (for similar
arguments, see Buller 2005). For example, the articles by Dennis and Humphreys (1998)
and Shiffrin and Steyvers (1998) both offer rational accounts of memory (applicable to
word-list tasks) that radically differ, but both do a good job with the data and are thought-
provoking. According to the rational program, analysis of the environment and the task
should provide sufficient grounding to constrain theory development. Cognitive scientists
(especially those trained in psychology) are not expert in characterizing the environment
in which humans evolved, and it is not always clear what this environment was like. As
in experimental sciences, our understanding of past environments is constantly revised
rather than providing a bedrock from which to build rational accounts of behavior.
Adding further complexity, humans can change the environment to suit their needs rather
than adapt to it (Kurz & Tweney 1998).
One factor that provides a number of degrees of freedom to the rational modeler is
that it is not clear which environment (in terms of when and where) is evolutionarily
relevant (i.e., for which our behavior was optimized). The environment that is relevant
for determining rational action could be the local environment present in the laboratory
task, similar situations (however defined) that the person has experienced, all experiences
over the person’s life, all experiences of our species, all experiences of all ancestral
organisms traced back to single cell organisms, etc. Furthermore, once the relevant
33
environment is specified and characterized, the rational theorist has considerable
flexibility in characterizing which relevant measures or statistics from the environment
should enter into the optimality calculations. When considered in this light, the argument
that rational approaches are parameter free and follow in a straightforward manner from
the environment is tenuous at best.
4.3. Optimization occurs over biological mechanisms, not behaviors
It is noncontroversial that many aspects of our behavior are shaped by evolutionary
processes. However, evolutionary processes do not directly affect behavior, but instead
affect the mechanisms that give rise to behavior when coupled with environmental input
(McNamara & Houston 2009). Assuming one could properly characterize the
environment, focusing solely on how behavior should be optimized with respect to the
environment is insufficient because the physical reality of the brain and body is
neglected. Furthermore, certain aspects of behavior, such as the time to execute some
operation (e.g., the decision time to determine whether a person is a friend or foe), are
closely linked to mechanistic considerations.
Completely sidestepping mechanistic considerations when considering optimality
leads to absurd conclusions. To illustrate, it may not be optimal or evolutionarily
advantageous to ever age, become infertile, and die, but these outcomes are universal and
follow from biological constraints. It would be absurd to seriously propose an optimal
biological entity that is not bounded by these biological and physical realities, but this is
exactly the reasoning Bayesian Fundamentalists follow when formulating theories of
cognition. Certainly, susceptibility to disease and injury impact inclusive fitness more
than do many aspects of cognition. Therefore, it would seem strange to assume that
human cognition is fully optimized while these basic challenges, which all living
creatures past and present face, are not. Our biological reality, which is ignored by
Bayesian Fundamentalists, renders optimal solutions, defined solely in terms of choice
behavior, unrealistic and fanciful for many challenges.
Unlike evolutionary approaches, rational approaches to cognition, particularly
those in the Bayesian Fundamentalist tradition, do not address the importance of
mechanism in the adaptationist story. Certain physical limitations and realities lead to
34
certain designs prevailing. Which design prevails is determined in part by these physical
realities and the contemporaneous competing designs in the gene pool. As Marcus (2008)
reminds us, evolution is survival of the best current design, not survival of the globally
optimal design. Rather than the globally optimal design winning out, often a locally
optimal solution (i.e., a design better than similar designs) prevails (Dawkins 1987; Mayr
1982). Therefore, it is important to consider the trajectory of change of the mechanism
(i.e., current and past favored designs) rather than to focus exclusively on which design is
globally optimal.
As Marcus (2008) notes, many people are plagued with back pain because the
human spine is adapted from animals that walk on four paws, not two feet. This is clearly
not the globally optimal design, indicating that the optimization process occurs over
constraints not embodied in rational analyses. The search process for the best design is
hampered by the set of current designs available. These current designs can be adapted by
descent-with-modification, but there is no purpose or forethought to this process (i.e.,
there is no intelligent designer). It simply might not be possible for our genome to code
for shock absorbers like those in automobiles, given that the current solution is locally
optimal and distant from the globally optimal solution. In the case of the human spine,
the current solution is clearly not globally optimal, but is good enough to get the job
done. The best solution is not easily reachable and might never be reached. If evolution
settles on such a bad design for our spine, it seems unlikely that aspects of cognition are
fully optimized. Many structures in our brains share homologs with other species.
Structures more prominent in humans, such as the frontal lobes, were not anticipated, but
like the spine, resulted from descent-with-modification (Wood & Grafman 2003).
The spine example makes clear that the history of the mechanism plays a role in
determining the present solution. Aspects of the mechanism itself are often what is being
optimized rather than the resulting behavior. For example, selection pressures will
include factors such as how much energy certain designs require. The human brain
consumes 25% of a person’s energy, yet accounts for only 2% of a person’s mass (Clark
& Sokoloff 1999). Such nonbehavioral factors are enormously important to the
optimization process, but are not reflected in rational analyses, because these factors are
tied to a notion of mechanism, which is absent in rational analyses. Any discussion of
35
evolution optimizing behavior is incomplete without consideration of the mechanism that
generates the behavior. To provide an example from the study of cognition, in contrast to
Anderson’s (1991) rational analysis of concepts solely in terms of environmental
prediction, concepts might also serve other functions, such as increasing cognitive
economy in limited-capacity memory systems that would otherwise be swamped with
details (Murphy 1993; Rosch 1978).
The notion of incremental improvement of mechanisms is also important because
it is not clear that globally optimal solutions are always well defined. The optimality of
Bayesian inference is well supported in small worlds in which an observer can sensibly
assign subjective probabilities to all possible contingencies (Savage 1954). However,
Binmore (2009) argues that proponents of Bayesian rationality overextend this reasoning
when moving from laboratory tasks to the natural world. Normative support for the
Bayesian framework breaks down in the latter case because, in an unconstrained
environment, there is no clear rational basis for generating prior probabilities.
Evolutionary theory does not face this problem because it relies on incremental
adjustment rather than global optimization. Furthermore, shifting focus to the level of
mechanism allows one to study the relative performance of those mechanisms without
having to explicitly work out the optimal pattern of behavior in a complex environment
(Gigerenzer & Todd 1999).
The preceding discussion assumes that we are optimized in at least a local sense.
This assumption is likely invalid for many aspects of the mechanisms that give rise to
behavior. Optimization by natural selection is a slow process that requires consistent
selective pressure in a relatively stable environment. Many of the behaviors that are
considered uniquely human are not as evolutionarily old as basic aspects of our visual
system. It is also not clear how stable the relevant environment has been. To provide one
example, recent simulations support the notion that many syntactic properties of language
cannot be encoded in a language module, and that the genetic basis of language use and
acquisition could not coevolve with human language (Chater et al. 2009).
Finally, while rational theorists focus on adaptation in pursuit of optimality,
evolutionary theorists take a broader view of the products of evolution. Namely,
evolution yields three products: (1) adaptations, (2) by-products, and (3) noise (Buss et al.
36
1998). An adaptation results from natural selection to solve some problem, whereas a by-
product is the consequence of some adaptation. To use Bjorklund and Pelligrini’s (2000)
example, the umbilical cord is an adaptation, whereas the belly button is a by-product.
Noise includes random effects due to mutations, drift, etc. Contrary to the rational
program, one should not take all behaviors and characteristics of people to be adaptations
that increase (i.e., optimize) fitness.
4.4. Developmental psychology and notions of capacity limitation: What changes over
time?
Although rational Bayesian modeling has a large footprint in developmental psychology
(Kemp et al. 2007; Sobel et al. 2004; Xu & Tenenbaum 2007), development presents
basic challenges to the rational approach. One key question for any developmental model
is what develops. In rational models, the answer is that nothing develops. Rational
models are mechanism free, leaving only information sampled to change over time.
Although some aspects of development are driven by acquisition of more observations,
other aspects of development clearly reflect maturational changes in the mechanism (see
Xu & Tenenbaum 2007, p. 169). For example, some aspects of children’s performance
are indexed by prefrontal development (Thompson-Schill et al. 2009) rather than the
degree of experience within a domain. Likewise, teenage boys’ interest in certain stimuli
is likely attributable more to hormonal changes than to collecting examples of certain
stimuli and settling on certain hypotheses.
These observations put rational theories of development in a difficult position.
People’s mental machinery clearly changes over development, but no such change occurs
in a rational model. One response has been to posit rational theories that are collections of
discrepant causal models (i.e., hypothesis spaces). Each discrepant model is intended to
correspond to a different stage of development (Goodman et al. 2006; Lucas et al. 2009).
In effect, development is viewed as consisting of discrete stages, and a new model is
proposed for each qualitative developmental change. Model selection is used to
determine which discrepant model best accounts for an individual’s current behavior.
Although this approach may be useful in characterizing an individual’s performance and
current point in development, it does not offer any explanation for the necessity of the
37
stages or why developmental transitions occur. Indeed, rather than accounts of
developmental processes, these techniques are best viewed as methods to assess a
person’s conceptual model, akin to user modeling in tutoring systems (Conati et al.
1997). To the extent that the story of development is the story of mechanism
development, rational theories have little to say (e.g., Xu & Tenenbaum 2007).
Epigenetic approaches ease some of these tensions by addressing how experience
influences gene expression over development, allowing for bidirectional influences
between experience and genetic activity (Gottlieb 1992; Johnson 1998). One
complication for rational theories is the idea that different selection pressures are exerted
on organisms at different points in development (Oppenheim 1981). For adults, rigorous
play wastes energy and is an undue risk, but, for children, rigorous play may serve a
number of adaptive functions (Baldwin & Baldwin 1977). For example, play fighting
may prepare boys for adult hunting and fighting (Smith 1982). It would seem that
different rational accounts are needed for different periods of development.
Various mental capacities vary across development and individuals. In adult
cognition, Herbert Simon (1957) introduced the notion of bounded rationality to take into
account, among other things, limitations in memory and processing capacities. One of the
proposals that grew out of bounded rationality was optimization under constraints, which
posits that people may not perform optimally in any general sense, but, if their capacities
could be well characterized, people might be found to perform optimally, given those
limitations (e.g., Sargent 1993; Stigler 1961). For instance, objects in the environment
may be tracked optimally, given sensory and memory limitations (Vul et al. 2009).
Although the general research strategy based on bounded rationality can be
fruitful, it severely limits the meaning of labeling a behavior as rational or optimal.
Characterizing capacity limitations is essentially an exercise in characterizing the
mechanism, which represents a departure from rational principles. Once all capacity
limitations are detailed, notions of rationality lose force. To provide a perverse example,
each person can be viewed as an optimal version of himself given his own limitations,
flawed beliefs, motivational limitations, etc. At such a point, it is not clear what work the
rational analysis is doing. Murphy (1993) makes a similar argument about the circularity
of rational explanations: Animals are regarded as optimal with respect to their ecological
38
niche, but an animal’s niche is defined by its behaviors and abilities. For example, if one
assumes that a bat’s niche involves flying at night, then poor eyesight is not a
counterexample of optimality.
Although these comments may appear critical, we do believe that considering
capacity limitations is a sound approach that can facilitate the unification of rational and
mechanistic approaches. However, we have doubts as to the efficacy of current
approaches to exploring capacity limitations. For example, introducing capacity
limitations by altering sampling processes through techniques like the particle filter
(Brown & Steyvers 2009) appears to be motivated more by modeling convenience than
by examination of actual cognitive mechanisms. It would be a curious coincidence if
existing mathematical estimation techniques just happened to align with human capacity
limitations. In section 5, we consider the possibility of using (mechanistic) psychological
characterizations of one or more aspects of the cognitive system to derive bounded-
optimality characterizations of decision processes. Critically, the potential of such
approaches lies in the mutual constraint of mechanistic and rational considerations, as
opposed to rational analysis alone.
To return to development, one interesting consideration is that reduced capacity at
certain points in development is actually seen as a benefit by many researchers. For
example, one proposal is that children’s diminished working-memory capacity may
facilitate language acquisition by encouraging children to focus on basic regularities
(Elman 1993; Newport 1990). “Less is more” theories have also been proposed in the
domain of metacognition. For example, children who overestimate their own abilities
may be more likely to explore new tasks and be less self-critical in the face of failure
(Bjorklund & Pellegrini 2000). Such findings seem to speak to the need to consider the
nature of human learners rather the nature of the environment. Human learners do not
seem to “turn off” harmful capacity to narrow the hypothesis space when it might be
prove beneficial to do so.
5. The role of Bayesian modeling in cognitive science
The observations in the preceding sections suggest that, although Bayesian modeling has
great potential to advance our understanding of cognition, several conceptual problems
39
with the Fundamentalist Bayesian program limit its potential theoretical contributions.
One possible reason is that most current work lacks a coherent underlying philosophy
regarding just what that contribution should be. In this section, we lay out three roles for
Bayesian modeling in cognitive science that potentially avoid the problems of the
fundamentalist approach and that better integrate with other modes of inquiry. We make
no strong commitment that any of the approaches proposed in this section will succeed,
but we believe these are the viable options if one wants to use Bayes’s Rule or
probabilistic inference as a component of psychological theory.
First, Bayesian inference has proven to be exceedingly valuable as an analysis
tool for deciding among scientific hypotheses or models based on empirical data. We
refer to such approaches as Bayesian Agnosticism because they take no stance on
whether Bayesian inference is itself a useful psychological model. Instead, the focus is on
using Bayesian inference to develop model-selection techniques that are sensitive to true
model complexity and that avoid many of the logical inconsistencies of frequentist
hypothesis testing (e.g., Pitt et al. 2002; Schwarz 1978).
Second, Bayesian models can offer computational-level theories of human
behavior that bypass questions of cognitive process and representation. In this light,
Bayesian analysis can serve as a useful starting point when investigating a new domain,
much like how ideal-observer analysis can be a useful starting point in understanding a
task and thus assist in characterizing human proficiency in the task. This approach is in
line with the Fundamentalist Bayesian philosophy, but, as the observations of the
previous sections make clear, several changes to current common practice would greatly
improve the theoretical impact of this research program. Foremost, rational analysis
should be grounded in empirical measurement of the environment. Otherwise, the
endeavor is almost totally unconstrained. Environmental grounding has yielded useful
results in low-level vision (Geisler et al. 2001) and basic aspects of memory (Anderson &
Schooler 1991), but the feasibility of this approach with more complex cognitive tasks
remains an open question. Furthermore, researchers are faced with the questions of what
is the relevant environment (that behavior is supposedly optimized with respect to) and
what are the relevant statistics of that environment (that behavior is optimized over).
There is also the question of the objective function that is being optimized, and how that
40
objective might vary according to developmental trajectory or individual differences
(e.g., sex or social roles). Finally, it may be impossible in cases to specify what is optimal
in any general sense without considering the nature of the mechanism. All of these
questions can have multiple possible answers, and finding which answers lead to the best
explanation of the data is part of the scientific challenge. Just as with mechanistic models,
competing alternatives need to be explicitly recognized and compared. Finally, an
unavoidable limitation of the pure rational approach is that behavior is not always
optimal, regardless of the choice of assumptions about the environment and objective
function. Evolution works locally rather than globally, and many aspects of behavior may
be by-products rather than adaptations in themselves. More importantly, evolution is
constrained by the physical system (i.e., the body and brain) that is being optimized. By
excluding the brain from psychological theory, Bayesian Fundamentalism is logically
unable to account for mechanistic constraints on behavior and unable to take advantage
of or inform us about the wealth of data from areas such as neurophysiology,
development, or timing.3
Third, rather than putting all the onus on rational analysis by attempting to explain
behavior directly from the environment, one could treat various elements of Bayesian
models as psychological assumptions subject to empirical test. This approach, which we
refer to as Bayesian Enlightenment, seems the most promising because it allows Bayesian
models to make contact with the majority of psychological research and theory, which
deals with mechanistic levels of analysis. The remainder of this section explores several
avenues within Bayesian Enlightenment. We emphasize up front that all of these
directions represent significant departures from the Fundamentalist Bayesian tenet that
behavior can be explained and understood without recourse to process or representation.
5.1. Bayesian Enlightenment: taking Bayesian models seriously as psychological
theories
The most obvious candidate within the Bayesian framework for status as a psychological
construct or assumption is the choice of hypothesis space or generative model. According
to the Fundamentalist Bayesian view, the hypotheses and their prior distribution
correspond to the true environmental probabilities within the domain of study. However,
41
as far as predicting behavior is concerned, all that should matter is what the subject
believes (either implicitly or explicitly) are the true probabilities. Decoupling information
encoded in the brain from ground truth in the environment (which cannot always be
determined) enables separation of two different tenets of the rationalist program. That is,
the question of whether people have veridical mental models of their environments can
be separated from the question of whether people reason and act optimally with respect to
whatever models they have. A similar perspective has been proposed in game theory,
whereby distinguishing between an agent’s model of the opponent(s) and rational
behavior with respect to that model can resolve paradoxes of rationality in that domain
(Jones & Zhang 2003). Likewise, Baker et al. (2009) present a model of how people
reason about the intentions of others in which the psychological assumption is made that
people view others as rational agents (given their current knowledge).
Separating Bayesian inference from the mental models it operates over opens up
those models as a fruitful topic of psychological study (e.g., Sanborn et al. 2010b).
Unfortunately, this view of Bayesian modeling is at odds with most applications, which
focus on the inferential side and take the generative model for granted, leaving that
critical aspect of the theory to be hand-coded by the researcher. Thus, the emphasis on
rationality marginalizes most of the interesting psychological issues. The choice of the
generative model or hypothesis space reflects an assumption about how the subject
imputes structure to the environment and how that structure is represented. There are
often multiple options here (i.e., there is not a unique Bayesian model of most tasks), and
these correspond to different psychological theories. Furthermore, even those cases that
ground the hypothesis space in empirical data from natural environments tend not to
address how it is learned by individual subjects. One strong potential claim of the
Bayesian framework is that the most substantial part of learning lies in constructing a
generative model of one’s environment, and that using that model to make inferences and
guide behavior is a relatively trivial (albeit computationally intensive) exercise in
conditional probability. Therefore, treating the generative model as a psychological
construct enables a shift of emphasis to this more interesting learning problem. Future
work focusing on how people develop models of their environment (e.g., Griffiths &
Tenenbaum 2006; Mozer et al. 2008; Steyvers et al. 2003) would greatly increase the
42
theoretical utility of Bayesian modeling by bringing it into closer contact with the hard
psychological questions of constructive learning, structured representations, and
induction.
Consideration of generative models as psychological constructs also highlights a
fundamental difference between a process-level interpretation of Bayesian learning and
other learning architectures such as neural networks or production systems. The Bayesian
approach suggests that learning involves working backward from sense data to compute
posterior probabilities over latent variables in the environment and then determining
optimal action with respect to those probabilities. This can be contrasted with the more
purely feed-forward nature of most extant models, which learn mappings from stimuli to
behavior and use feedback from the environment to directly alter the internal parameters
that determine those mappings (e.g., connection weights or production utilities). A
similar contrast has been proposed in the literature on reinforcement learning, between
model-based (planning) and model-free (habit) learning, with behavioral and neurological
evidence that these exist as separate systems in the brain (Daw et al. 2005). Model-based
reinforcement learning and Bayesian inference have important computational differences,
but this parallel does suggest a starting point for addressing the important question of
how Bayesian learning might fit into a more complete cognitive architecture.
Prior distributions offer another opportunity for psychological inquiry within the
Bayesian framework. In addition to the obvious connections to biases in beliefs and
expectations, the nature of the prior has potential ties to questions of representation. This
connection arises from the principle of conjugate priors (Raiffa & Schlaifer 1961). A
conjugate prior for a Bayesian model is a parametric family of probability distributions
that is closed under the evidence-updating operation of Bayesian inference, meaning that
the posterior is guaranteed also to lie in the conjugate family after any number of new
observations have been made. Conjugate priors can dramatically simplify computational
and memory demands because the learner needs only to store and update the parameters
of the conjugate family rather than the full evidence distribution. Conjugate priors are a
common assumption made by Bayesian modelers, but this assumption is generally made
solely for mathematical convenience of the modeler rather than for any psychological
reason. However, considering a conjugate prior as part of the psychological theory leads
43
to the intriguing possibility that the parameters of the conjugate family constitute the
information that is explicitly represented and updated in the brain. If probabilistic
distributions over hypotheses are indeed part of the brain’s computational currency, then
they must be encoded in some way, and it stands to reason that the encoding generally
converges on one that minimizes the computational effort of updating knowledge states
(i.e., of inferring the posterior after each new observation). Therefore, an interesting
mechanistic-level test of Bayesian theory would be to investigate whether the variables
that parameterize the relevant conjugate priors are consistent with what is known based
on more established methods about knowledge representation in various psychological
domains. Of course, it is unlikely that any extant formalism (currently adopted for
mathematical convenience) will align perfectly with human performance, but empirically
exploring and evaluating such possibilities might prove a fruitful starting point.
A final element of Bayesian models that is traditionally considered as outside the
psychological theory but that may have valuable process-level implications involves the
algorithms that are often used for approximating exact Bayesian inference. Except in
models that admit a simple conjugate prior, deriving the exact posterior from a Bayesian
model is in most practical cases exceedingly computationally intensive. Consequently,
even the articles that propose these models often resort to approximation methods such as
Markov-Chain Monte Carlo (MCMC; Hastings 1970) or specializations such as Gibbs
sampling (Geman & Geman 1984) to derive approximate predictions. To the extent that
Bayesian models capture any truth about the workings of the brain, the brain is faced with
the same estimation problems that Bayesian modelers are, so it too likely must use
approximate methods for inference and decision making. Many of the algorithms used in
current Bayesian models correspond to important recent advances in computer science
and machine learning, but until their psychological predictions and plausibility are
addressed, they cannot be considered part of cognitive theory. Therefore, instead of being
relegated to footnotes or appendices, these approximation algorithms should be a focus of
the research because this is where a significant portion of the psychology lies. Research
investigating estimation algorithms as candidate psychological models (e.g., Daw &
Courville 2007; Sanborn et al. 2010a) represents a promising step in this direction. An
alternative line of work suggests that inference is carried out by a set of simple heuristics
44
that are adapted to statistically different types of environments (Brighton & Gigerenzer
2008; Gigerenzer & Todd 1999). Deciding between these adaptive heuristics and the
aforementioned, more complex estimation algorithms is an important empirical question
for the mechanistic grounding of Bayesian psychological models.
A significant aspect of the appeal of Bayesian models is that their assumptions are
explicitly laid out in a clean and interpretable mathematical language that, in principle,
affords the researcher a transparent view of their operation. This is in contrast to other
computational approaches (e.g., connectionism), in which it can be difficult to separate
theoretically important assumptions from implementational details. Unfortunately, as we
have argued here, this is not generally the case in practice. Instead, unexamined, yet
potentially critical, assumptions are routinely built into the hypothesis sets, priors, and
estimation procedures. Treating these components of Bayesian models as elements of the
psychological theory rather than as ancillary assumptions is an important prerequisite for
realizing the transparency of the Bayesian framework. In this sense, the shift from
Bayesian Fundamentalism to Enlightenment is partly a shift of perspective, but it is one
we believe could have a significant impact on theoretical progress.
5.2. Integrating Bayesian analysis with mechanistic-level models
Viewing Bayesian models as genuine psychological theories in the ways outlined here
also allows for potential integration between rational and mechanistic approaches. The
most accurate characterization of cognitive functioning is not likely to come from
isolated considerations of what is rational or of what is a likely mechanism. More
promising is to look for synergy between the two, in the form of powerful rational
principles that are well approximated by efficient and robust mechanisms. Such an
approach would aid understanding not just of the principles behind the mechanisms
(which is the sole focus of Bayesian Fundamentalism) but also of how the mechanisms
achieve and approximate those principles and how constraints at both levels combine to
shape behavior (for one thorough example, see Oaksford & Chater 2010). We stress that
we are not advocating that every model include a complete theory at all levels of
explanation. The claim is merely that there must be contact between levels. We have
argued this point here for rational models – that they should be informed by
45
considerations of process and representation – but the same holds for mechanistic
models, as well, that they should be informed by consideration of the computational
principles they carry out (Chater et al. 2003).
With reference to the problem of model fractionation discussed earlier, one way
to unite Bayesian models of different phenomena is to consider their rational
characterizations in conjunction with mechanistic implementations of belief updating and
knowledge representation, with the parsimony-derived goal of explaining multiple
computational principles with a common set of processing mechanisms. In this way the
two levels of analysis serve to constrain each other and to facilitate broader and more
integrated theories. From the perspective of theories as metaphors, the rationality
metaphor is unique in that is has no physical target, which makes it compatible with
essentially any mechanistic metaphor and suggests that synthesis between the two levels
of explanation will often be natural and straightforward (as compared to the challenge of
integrating two distinct mechanistic architectures). In the context of conditioning, Daw et
al. (2008) offer an excellent example of this approach by mapping out the relationships
between learning algorithms and the rational principles they approximate and by showing
how one can distinguish behavioral phenomena reflecting rational principles from
mechanistic signatures of the approximation schemes.
Examples of work that integrates across levels of explanation can also be found in
computational neuroscience. Although the focus is not on explaining behavior, models in
computational neuroscience relate abstract probabilistic calculations to operations in
mechanistic neural network models (Denève 2008; Denève et al. 1999). Other work
directly relates and evaluates aspects of Bayesian models to brain areas proposed to
perform the computation (Doll et al. 2009; Soltani & Wang 2010). For example, Köver
and Bao (2010) relate the prior in a Bayesian model to the number of cells devoted to
representing possible hypotheses. This work makes contact with all three of Marr’s
(1982) levels of analysis by making representational commitments and relating these
aspects of the Bayesian model to brain regions.
An alternative to the view of mechanisms as approximations comes from the
research of Gigerenzer and colleagues on adaptive heuristics (e.g., Gigerenzer & Todd
1999). Numerous studies have found that simple heuristics can actually outperform more
46
complex inference algorithms in naturalistic prediction tasks. For example, with certain
datasets, linear regression can be outperformed in cross-validation (i.e., transfer to new
observations) by a simple tallying heuristic that gives all predictors equal weight
(Czerlinski et al. 1999; Dawes & Corrigan 1974). Brighton and Gigerenzer (2008)
explain how the advantage of simple heuristics is rooted in the bias-variance dilemma
from statistical estimation theory, specifically that more constrained inference algorithms
can perform better on small datasets because they are less prone to overfitting (e.g.,
Geman et al. 1992). Although this conclusion has been used to argue against
computational-level theories of rationality in favor of ecological rationality based on
mechanisms adapted to specific environments (Gigerenzer & Brighton 2009), we believe
the two approaches are highly compatible. The connection lies in that any inference
algorithm implicitly embodies a prior expectation about the environment, corresponding
to the limitations in what patterns of data it can fit and hence the classes of environments
in which it will tend to succeed (cf. Wolpert 1996). For example, the tallying heuristic is
most successful in environments with little variation in true cue validities and in cases
where the validities cannot be precisely estimated (Hogarth & Karelaia 2005). This
suggests that tallying should be matched or even outperformed by Bayesian regression
with a prior giving more probability to more homogeneous regression weights. The point
here is that the ecological success of alternative algorithms (tallying versus traditional
regression) can inform a rational analysis of the task and hence lead to more accurate
normative theories. This sort of approach could alleviate the insufficient environmental
grounding and excessive flexibility of Bayesian models discussed in section 4.
Formalizing the relationship between algorithms and implicit priors – or between
statistical regularities in particular environments and algorithms that embody those
regularities – is therefore a potentially powerful route to integrating mechanistic and
rational approaches to cognition.
Another perspective on the relationship between Bayesian and mechanistic
accounts of cognition comes from the recognition that, at its core, Bayes’s Rule is a
model of the decision process. This is consistent with (and partly justifies) the
observation that most work in the Bayesian Fundamentalist line avoids commitments
regarding representation. However, the thesis that inference and decision making are
47
optimal is meaningful only in the context of the knowledge (i.e., beliefs about the
environment) with respect to which optimality is being defined. In other words, a
complete psychological theory must address both how knowledge is acquired and
represented and how it is acted upon. As argued earlier in section 3.1, questions of the
structure of people’s models of their environments, and of how those models are learned,
are better addressed by traditional, mechanistic psychological methods than by rational
analysis. Taken together, these observations suggest a natural synthesis in which
psychological mechanisms are used to model the learner’s state of knowledge, and
rational analysis is used to predict how that knowledge is used to determine behavior.
The line between knowledge and decision making, or representation and process,
is of course not so well defined as this simple proposal suggests, but the general idea is
that rational analysis can be performed not in the environment but instead within a
mechanistic model, thus taking into account whatever biases and assumptions the
mechanisms introduce. This approach allows the modeler to postulate decision rules that
are optimal with respect to the representations and dynamics of the rest of the model. The
result is a way of enforcing good design while still making use of what is known about
mental representations. It can improve a mechanistic model by replacing what might
otherwise be an arbitrary decision rule with something principled, and it also offers an
improvement over rational analysis that starts and ends with the environment and is not
informed by how information is actually represented. This approach has been used
successfully to explain, for example, aspects of memory as optimal retrieval, given the
nature of the encoding (Shiffrin & Steyvers 1998), patterns of short-term priming as
optimal inference with unknown sources of feature activation (Huber et al. 2001), and
sequential effects in speeded detection tasks as optimal prediction with respect to a
particular psychological representation of binary sequences (Wilder et al. 2009). A
similar approach has been applied at the neural level, for example, to model activity of
lateral intraparietal (LIP) neurons as computing a Bayesian posterior from activity of
middle temporal (MT) cells (Beck et al. 2008). One advantage of bringing rational
analysis inside cognitive or neural models is that it facilitates empirical comparison
among multiple Bayesian models that make different assumptions about knowledge
representation (e.g., Wilder et al. 2009). These lines of research illustrate that the
48
traditional identification of rational analysis with computational-level theories is an
artificial one, and that rational analysis is in fact applicable at all levels of explanation
(Danks 2008).
A complementary benefit of moving rational analysis inside psychological models
is that the assumption of optimal inference can allow the researcher to decide among
multiple candidate representations, through comparison to empirical data. The
assumption of optimal inference allows for more unambiguous testing of representation
because representation becomes the only unknown in the model. This approach has been
used successfully in the domain of category induction by Tenenbaum et al. (2006).
However, such conclusions depend on a strong assumption of rational inference. The
question of rational versus biased or heuristic inference has been a primary focus of much
of the judgment and decision-making literature for several decades, and a large body of
work argues for the latter position (e.g., Tversky & Kahneman 1974). On the other hand,
some of these classic findings have been given rational reinterpretations under new
assumptions about the learner’s knowledge and goals (e.g., Oaksford & Chater 1994).
This debate illustrates how the integration of rational and mechanistic approaches brings
probabilistic inference under the purview of psychological models where it can be more
readily empirically tested.
Ultimately, transcending the distinction between rational and mechanistic
explanations should enable significant advances of both and for cognitive science as a
whole. Much of how the brain operates reflects characteristics of the environment to
which it is adapted, and therefore an organism and its environment can be thought of as a
joint system, with behavior depending on aspects of both subsystems. There is of course a
fairly clear line between organism and environment, but that line has no more
epistemological significance than the distinctions between different sources of
explanation within either category. In other words, the gap between an explanation rooted
in some aspect of the environment and one rooted in a mechanism of neural or cognitive
processing should not be qualitatively wider than the gap between explanations rooted in
different brain regions, different processing stages or modules, or uncertainty in one
latent variable versus another. The joint system of organism and environment is a
complex one, with a large number of constituent processes, and a given empirical
49
phenomenon (of behavior, brain activity, etc.) can potentially be ascribed to any of them.
Just as in other fields, the scientific challenge is to determine which explanation is best in
each case, and for most interesting phenomena the answer will most likely involve an
interaction of multiple, disparate causes.
6. Conclusions
The recent advances in Bayesian modeling of cognition clearly warrant excitement.
Nevertheless, many aspects of current research practice act to severely limit the
contributions to psychological theory. This article traces these concerns to a particular
philosophy that we have labeled Bayesian Fundamentalism, which is characterized by the
goal of explaining human behavior solely in terms of optimal probabilistic inference
without recourse to mechanism. It is motivated by the thesis that, once a given task is
correctly characterized in terms of environmental statistics and goals of the learner,
human behavior in that task will be found to be rational. As the numerous citations
throughout this article demonstrate, Bayesian Fundamentalism constitutes a significant
portion (arguably the majority) of current research on Bayesian modeling of cognition.
Establishing the utility of the Bayesian framework, and the rational metaphor
more generally, is an important first step, and convincing arguments have been made for
this position (e.g., Oaksford & Chater 2007). However, excessive focus on this
metascientific issue severely limits the scope and impact of the research. Focusing on
existence proofs distracts from the more critical work of deciding among competing
explanations and identifying the critical assumptions behind models. In the context of
rational Bayesian modeling, existence proofs hide that there are generally many Bayesian
models of any task, corresponding to different assumptions about the learner’s goals and
model of the environment. Comparison among alternative models would potentially
reveal a great deal about what people’s goals and mental models actually are. Such an
approach would also facilitate comparison to models within other frameworks by
separating the critical assumptions of any Bayesian model (e.g., those that specify the
learner’s generative model) from the contribution of Bayes’s Rule itself. This separation
should ease recognition of the logical relationships between assumptions of Bayesian
models and of models cast within other frameworks, so that theoretical development is
50
not duplicated and so that the core differences between competing theories can be
identified and tested.
The total focus on rational inference that characterizes Bayesian Fundamentalism
is especially unfortunate from a psychological standpoint because the belief updating of
Bayes’s Rule is psychologically trivial, amounting to nothing more than vote counting.
Much more interesting are other aspects of Bayesian models, including the algorithms
and approximations by which inference is carried out, the representations on which those
algorithms operate (e.g., the parameters of conjugate priors), and the structured beliefs
(i.e., generative models) that drive them. The Enlightened Bayesian view takes these
seriously as psychological constructs and evaluates them according to theoretical merit
rather than mathematical convenience. This important shift away from Bayesian
Fundamentalism opens up a rich base for psychological theorizing, as well as contact
with process-level modes of inquiry.
It is interesting to note that economics, the field of study with the richest history
of rational modeling of behavior and the domain in which rational theories might be
expected to be most accurate, has increasingly questioned the value of rational models of
human decision making (Krugman 2009). Economics is thus moving away from purely
rational models toward theories that consider psychological mechanisms and biases
(Thaler & Sunstein 2008). Therefore it is surprising to observe a segment of the
psychological community moving in the opposite direction. Bayesian modeling certainly
has much to contribute, but its potential impact will be much greater if developed in a
way that does not eliminate the psychology from psychological models. We believe this
will be best achieved by treating Bayesian methods as a complement to mechanistic
approaches rather than as an alternative.
ACKNOWLEDGMENTS
This research was supported in part by Air Force Office of Scientific Research (AFOSR)
Grant FA9550-07-1-0178 and Army Research Laboratory (ARL) Grant W911NF-09-2-
0038 to Bradley C. Love. We thank John Anderson, Colin Bannard, Lera Boroditsky,
David Buss, Matt Keller, Mike Mozer, Mike Oaksford, Randy O’Reilly, Michael
51
Ramscar, Vladimir Sloutsky, and Alan Yuille for helpful comments on an earlier draft of
this article.
NOTES
1. Formally, Eposterior equals the logarithm of the posterior distribution, Eprior is the
logarithm of the prior, and Edata(H) is the logarithm of the likelihood of the data under
hypothesis H. The model’s prediction for the probability that hypothesis H is correct,
after data have been observed, is proportional to exp[Eposterior(H)] (cf. Luce 1963).
2. Bayesian analysis has been used to interpret neural spike recordings (e.g., Gold
& Shadlen 2001), but this falls outside Bayesian Fundamentalism, which is concerned
only with behavioral explanations of cognitive phenomena.
3. Note that we refer here to Bayesian models that address behavior, not those that
solely aim to explain brain data without linking to behavior, such as Mortimer et al.’s
(2009) model of axon wiring.
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Book synopsis: This book explores a new approach to understanding the human mind - rational analysis - that regards thinking as a facility adapted to the structure of the world. This approach is most closely associated with the work of John R Anderson, who published the original book on rational analysis in 1990. Since then, a great deal of work has been carried out in a number of laboratories around the world, and the aim of this book is to bring this work together for the benefit of the general psychological audience. The book contains chapters by some of the world's leading researchers in memory, categorisation, reasoning, and search, who show how the power of rational analysis can be applied to the central question of how humans think. It will be of interest to students and researchers in cognitive psychology, cognitive science, and animal behaviour.
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