Bayesian Just-So Stories in Psychology and Neuroscience
Jeffrey S. Bowers
University of Bristol
Colin J. Davis
Royal Holloway University of London
According to Bayesian theories in psychology and neuroscience, minds and brains are (near) optimal in
solving a wide range of tasks. We challenge this view and argue that more traditional, non-Bayesian
approaches are more promising. We make 3 main arguments. First, we show that the empirical evidence
for Bayesian theories in psychology is weak. This weakness relates to the many arbitrary ways that priors,
likelihoods, and utility functions can be altered in order to account for the data that are obtained, making
the models unfalsifiable. It further relates to the fact that Bayesian theories are rarely better at predicting
data compared with alternative (and simpler) non-Bayesian theories. Second, we show that the empirical
evidence for Bayesian theories in neuroscience is weaker still. There are impressive mathematical
analyses showing how populations of neurons could compute in a Bayesian manner but little or no
evidence that they do. Third, we challenge the general scientific approach that characterizes Bayesian
theorizing in cognitive science. A common premise is that theories in psychology should largely be
constrained by a rational analysis of what the mind ought to do. We question this claim and argue that
many of the important constraints come from biological, evolutionary, and processing (algorithmic)
considerations that have no adaptive relevance to the problem per se. In our view, these factors have
contributed to the development of many Bayesian “just so” stories in psychology and neuroscience; that
is, mathematical analyses of cognition that can be used to explain almost any behavior as optimal.
Keywords: Bayes, Bayesian, optimal, heuristics, just-so stories
In recent years there has been an explosion of research directed
at a surprising claim: namely, that minds and brains are (near)
optimal in solving a wide range of tasks. This hypothesis is most
strongly associated with Bayesian theories in psychology and
neuroscience that emphasize the statistical problems confronting
all organisms. That is, cognitive, motor, and perceptual systems
are faced with noisy and ambiguous inputs (e.g., a three-
dimensional world is projected on a two-dimensional retina), and
these systems are designed to carry out or approximate Bayesian
statistics in order to make optimal decisions given the degraded
inputs. Typical conclusions include the following:
It seems increasingly plausible that . . . in core domains, human
cognition approaches an optimal level of performance. (Chater, Te-
nenbaum, & Yulle, 2006, p. 289)
These studies . . . have shown that human perception is close to the
Bayesian optimal suggesting the Bayesian process may be a funda-
mental element of sensory processing. (Körding & Wolpert, 2006, p.
321)
One striking observation from this work is the myriad ways in which
human observers behave as optimal Bayesian observers. This obser-
vation . . . has fundamental implications for neuroscience, particularly
in how we conceive of neural computations and the nature of neural
representations of perceptual and motor variables. (Knill & Pouget,
2004, p. 712)
Our results suggest that everyday cognitive judgments follow the
same optimal statistical principles as perception and memory. (Grif-
fiths & Tenenbaum, 2006, p. 767)
These conclusions are exciting, not only because they are coun-
terintuitive (who would have thought we are optimal?), but also
because they appear to constitute novel claims about mind and
brain. In the standard view, cognitive, perceptual, and motor
systems are generally good at solving important tasks, but the
limitations of the systems were always salient. For example, var-
ious heuristics are often thought to support high-level reasoning
and decision making. These heuristics are adaptive under many
conditions but not optimal (in fact, how far from optimal is a
matter of some dispute; e.g., Gigerenzer & Brighton, 2009; Gig-
erenzer, Todd, & the ABC Research Group, 1999; Kahneman,
Slovic, & Tversky, 1982; Kahneman & Tversky, 1996). In a
similar way, perception is often characterized as a “bag of tricks”
(Ramachandran, 1990). That is, the perceptual systems rely on
heuristics that generally work well enough but in no way approx-
imate Bayesian solutions. More generally, it is often assumed that
evolution produces systems that satisfice (Simon, 1956) or melior-
ize (Dawkins, 1982). That is, selective adaptation produces “good
enough” solutions, or “better than alternative” solutions, but not
optimal solutions. The Bayesian approach, by contrast, appears to
claim that evolution has endowed us with brains that are exqui-
sitely good at learning and exploiting the statistics of the environ-
ment, such that performance is close to optimal.
Jeffrey S. Bowers, School of Experimental Psychology, University of
Bristol, Bristol, England; Colin J. Davis, Department of Psychology, Royal
Holloway University of London, London, England.
We would like to thank the Steve Hammett and Michael Lee for helpful
discussion.
Correspondence concerning this article should be addressed to Jeffrey S.
Bowers, School of Experimental Psychology, University of Bristol, 12a
Priory Road, Bristol BS8 1TU, England. E-mail: j.bowers@bristol.ac.uk
Psychological Bulletin © 2012 American Psychological Association
2012, Vol. 138, No. 3, 389–414 0033-2909/12/$12.00 DOI: 10.1037/a0026450
389
In this article we challenge the Bayesian approach to studying
the mind and brain and suggest that more traditional, non-Bayesian
approaches provide a more promising way to proceed. We orga-
nize our argument as follows. In Part 1, we introduce Bayesian
statistics and summarize three ways in which these methods have
influenced theories in psychology and neuroscience. These differ-
ent approaches to Bayesian theorizing make quite different claims
regarding how the mind works. In Part 2, we highlight how there
are too many arbitrary ways that priors, likelihoods, utility func-
tions, etc., can be altered in a Bayesian model in order to account
for the data that are obtained. That is, Bayesian models are difficult
to falsify. Our concern is not just hypothetical concern; we de-
scribe a number of Bayesian models developed in a variety of
domains that were built post hoc in order to account for the data.
In Part 3, we show how the predictions of Bayesian theories are
rarely compared to alternative non-Bayesian accounts that assume
that humans are reasonably good at solving problems (e.g., heu-
ristic or adaptive theories of mind). This is problematic given that
the predictions derived from optimizing (Bayesian) and adaptive
(non-Bayesian) theories will necessarily be similar. We review a
number of cases in a variety of domains in which data taken to
support the Bayesian theories are equally consistent with non-
Bayesian accounts.
Next, in Part 4, we consider the claim that collections of neurons
perform Bayesian computations. We argue that the data in support
of this claim are weaker still. There are impressive mathematical
analyses showing how populations of neurons should compute in
order to optimize inferences given certain types of noise (variabil-
ity) in neural responding, but little or no evidence exists that
neurons actually behave in this way. Finally, in Part 5, we chal-
lenge the general scientific approach that characterizes most
Bayesian theorizing in cognitive science. A key premise that
underlies most Bayesian modeling is that the mind can be studied
by focusing on the environment and the task at hand, with little
consideration of what goes on inside the head. That is, the most
important constraints for theories of mind can be discovered
through a rational consideration of what a mind ought to do in
order to perform optimally. We question this claim and argue that
many of the important constraints come from processing (algorith-
mic), biological, and evolutionary considerations that have no
adaptive relevance to the problem per se. Not only does this
“rational” approach to cognition lead to underconstrained theories,
it dissociates theories of cognition from a wide range of empirical
findings in psychology and neuroscience.
In our view, the flexibility of Bayesian models, coupled with
the common failure to contrast Bayesian and non-Bayesian
accounts of performance, has led to a collection of Bayesian
“just so” theories in psychology and neuroscience: sophisti-
cated statistical analyses that can be used to explain almost any
behavior as (near) optimal. If the data had turned out otherwise,
a different Bayesian theory would have been carried out to
justify the same conclusion, that is, that the mind and brain
support near-optimal performance.
Part 1: What Is a Bayesian Theory in Psychology
(and Neuroscience)?
At the most general level, Bayesian theories in cognitive
psychology and neuroscience assume that the mind and brain
perform Bayesian statistics, or something functionally similar
in a given context. The premise is that cognitive and perceptual
systems need to make decisions about unique events on the
basis of noisy and ambiguous information. Bayesian statistics
are the optimal method for estimating probabilities of unique
events, and the mind and brain are assumed to apply or approx-
imate this method in order to make optimal (or near-optimal)
decisions.
Most Bayesian theories are developed at a computational
rather than an algorithmic level of description (Marr, 1982).
That is, Bayesian theories describe the goal of a computation,
why it is appropriate, and the logic of the strategy, but not the
mental representations and processes that are employed in solv-
ing a task. Bayesian theories in psychology typically adopt the
“rational analysis” methodology described by Anderson (1991).
That is, the focus is on understanding the nature of the envi-
ronment (e.g., what information is available to an organism)
and the nature of the task being performed. Together, these
factors, when combined with Bayesian probability theory, de-
termine what an optimal solution should look like. Critically,
this solution is thought to provide important constraints on
theories of mind and brain.
To avoid any confusion, it is important to note that our criticism
of the Bayesian approach has nothing to do with Bayesian statistics
or Bayesian decision theory per se. That is, we do not take issue
with the claim that Bayesian methods provide optimal methods for
determining probabilities of specific events. Nor do we dispute the
promise of Bayesian analysis methods in the evaluation of cogni-
tive models (e.g., Lee, 2008; Rouder & Lu, 2005; Wagenmakers,
Lodewyckx, Kuriyal, & Grasman, 2010). What we do question,
however, is the relevance of this statistical approach to theories of
mind and brain.
Bayesian Probability
Before considering the virtues of Bayesian theories in psychol-
ogy and neuroscience, it is perhaps worth reviewing the basics of
Bayesian statistics and their use in optimal decision making. Al-
though specific Bayesian methods can be quite complicated, it is
important to have a general understanding of Bayesian statistics,
given the claim that the mind and brain in some way implement or
approximate these methods.
Bayes’s theorem specifies the optimal way of combining new
information with old information. More specifically, if we have
some hypothesis about the world, possibly based on prior infor-
mation, Bayes’s rule tells us to how to reevaluate the probability of
this hypothesis in the light of new evidence. The rule itself is quite
straightforward, and may be written in the form
P共H兩E兲 ⫽ P共H兲 ⫻ P共E兩H兲/P共E兲. (1)
Here H is the hypothesis under investigation and E is the new
evidence. The left-hand side of the equation, P(H兩E), is called the
posterior probability. This represents the probability that the hy-
pothesis is true given the new evidence (the symbol 兩 means
“given”). The first term on the right-hand side of the equation,
P(H), is called the prior probability. This represents the prior
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probability that the hypothesis is true before the new evidence is
taken into account. The second term on the right-hand side of the
equation, P(E兩H), is called the likelihood function. This represents
how likely it is that the new evidence would have been obtained
given that the hypothesis H is true. Finally, the denominator P(E)
represents the probability of the new evidence. Because this term
is constant with respect to the hypothesis H, it can be treated as a
scaling factor, and we can write the following:
Posterior probability ␣ Likelihood function
⫻ Prior probability,
where the symbol ␣ is read “is proportional to.”
To illustrate the components of this formula and how Bayes’s
rule is used to estimate the posterior probability, consider the
following concrete example. Imagine that the hypothesis under
consideration is that a 30-year-old man has lung cancer, and the
new evidence is that he has a cough. Accordingly, the posterior
probability we want to calculate is the probability that the man has
cancer given that he has a cough, or P(cancer兩cough). Adapting
Equation 1 to the current situation, we have
P共cancer兩cough兲 ⫽ P共cancer兲 ⫻ P共cough兩cancer兲/P共cough兲.
(2)
For this example, let us assume we have exact values for the priors
and likelihoods. Namely, the prior probability of a 30-year-old
man in the relevant population having lung cancer is .005, the
probability of coughing when one has cancer is .8 (likelihood), and
the overall probability of coughing is .2. Note that the symptom of
coughing can have many different causes, most of which are not
related to lung cancer, and the probability of .2 ignores these
causes and simply represents how likely it is that someone will
have a cough for whatever reason. Plugging these numbers into
Equation 2 leads to the following:
P共cancer兩cough兲 ⫽ .005 ⫻ .8/.2 ⫽ .02.
The point to note here is that the estimated probability of cancer
has only gone up from .005 to .02, which makes intuitive sense
given that coughing is not a strong diagnostic test for cancer.
However, the resulting posterior probabilities are not always so
intuitive. Imagine that instead of relying on coughing, the relevant
new data is a patient’s positive result on a blood test that has a
much higher diagnostic accuracy. To evaluate this evidence, it is
helpful to decompose P(E) as follows:
P共E兲 ⫽ P共H兲 ⫻ P共E兩H兲 ⫹ P(⬃H) ⫻ P共E兩⬃H兲, (3)
where ⬃ means “not true.” That is, there are two possible expla-
nations of the positive result. The first possibility is that the result
is a correct detection. The probability of this is P(cancer) ⫻
P(positive test 兩 cancer), where P(positive test 兩 cancer) represents
the hit rate (or sensitivity) of the test. Suppose that this hit rate is
.995. The second possibility is that the result is a false alarm. The
probability of this is P(⬃cancer) ⫻ P(positive test 兩 ⬃cancer),
where P(positive test 兩 ⬃cancer) represents the false-alarm rate of
the test, that is, the probability of the test’s being positive when
one does not have cancer. Suppose that this false-alarm rate is .01.
We can infer from these rates that the overall probability of a
positive test result is P(cancer) ⫻ Hit rate ⫹ P(⬃cancer) ⫻
False-alarm rate ⫽ .005 ⫻ .995 ⫹ .995 ⫻ .01 ⫽ .015.
Plugging these numbers into Bayes’s rule leads to the following:
P共cancer兩positive test兲 ⫽ .005 ⫻ .995 / 共.005 ⫻ .995 ⫹ .995
⫻ .01兲 ⫽ .33.
That is, even though the test has 99.5% detection accuracy when
cancer is present, and has 99% accuracy when it is not, the
probability that the patient has cancer is only 33%. Why is this? It
reflects the fact that posterior probabilities are sensitive to both
likelihoods and prior probabilities. The reason that the estimate of
cancer given the blood test is higher than the estimate of cancer
given a cough is that the likelihood has changed. And the reason
that P(cancer 兩 positive test) is much lower than 99% is that the
prior probability of cancer is so low. To see the influence of the
prior, consider the same test result when the prior probability of
cancer is much higher, say, 1 in 3 (e.g., imagine that the person
being tested is a 90-year-old lifetime smoker). Then the computed
probability is
P共cancer兩positive test兲 ⫽ .333 ⫻ .995/共.333 ⫻ .995 ⫹ .667 ⫻ .01兲
⫽ .98.
The important insight here is that the very same test producing the
same result is associated with a very different posterior probability
because the prior probability has changed. The necessity of con-
sidering prior probabilities (base rates) is one that most people find
quite counterintuitive. Indeed, doctors tend to perform very poorly
at determining the correct posterior probability in real-world prob-
lems like the above. For example, Steurer, Fischer, Bachmann,
Koller, and ter Riet (2002) found that only 22% of general prac-
titioners were able to use information about base rate and test
accuracy to correctly estimate the (low) probability of a patient
having a disease following a positive test result. Indeed, the
authors suggested that their findings might overestimate the aver-
age performance of general practitioners, as their participants were
recruited from doctors attending courses on evidence-based med-
icine. Similarly, Eddy (1982) reported that 95 out of 100 doctors
estimated the posterior probability to be approximately equal to the
likelihood P(E兩H), apparently ignoring the prior probability P(H).
This assumption would be equivalent to reporting a 99.5% prob-
ability instead of a 33% probability (see Gigerenzer, Gaissmaier,
Kurz-Milcke, Schwartz, & Woloshin, 2007, for discussion of the
conditions that facilitate Bayesian reasoning).
In the Bayesian models discussed in this article it is often the
case that there are multiple hypotheses under evaluation. For
example, in a model of word identification, there may be separate
hypotheses corresponding to each possible word. We can then
make use of the law of total probability, which says that if the set
of hypotheses H
1
, H
2
, ... , H
n
is exhaustive, and each of these
hypotheses is mutually exclusive, then
P共E兲 ⫽ 兺
i⫽1
n
P共H
i
兲 ⫻ P共EⱍH
i
兲. (4)
That is, the probability of the evidence can be found by summing
the joint probabilities over all hypotheses (where the joint proba-
bility is the prior probability times the likelihood), a process
referred to as marginalization. In practice, it may not be possible to
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BAYESIAN JUST-SO STORIES
guarantee the exclusivity and exhaustivity of a set of hypotheses.
However, for many practical purposes this does not matter, as we
are typically interested in the relative probability of different
hypotheses (i.e., is this new evidence best explained by H
1
or H
2
?),
meaning that the absolute value of the scaling factor P(E), which
is independent of any specific hypothesis H
i
, is not critical.
The value of using Bayes’s rule to make rational judgments
about probability is clear in the case of clinical examples like
cancer diagnosis. The strong claim made by proponents of Bayes-
ian inference models of cognition and perception is that humans
make use of the same rule (or close approximations to it) when
perceiving their environment, making decisions, and performing
actions. For example, while walking across campus, suppose you
see someone who looks like your friend John. In the Bayesian
formulation, “looks like John” might correspond to a specific
likelihood, such as P(E兩John) ⫽ .8. Is it actually John? A rational
answer must take into account your prior knowledge about John’s
whereabouts. If John is your colleague, and you often see him on
campus, the chances are quite high that the person you have seen
is in fact John. If John is your next-door neighbor, and you have
never seen him on campus, the prior probability is lower, and you
should be less confident that the person you have seen is John. If
your friend John died 20 years ago, the rational prior probability is
0, and thus you should be perfectly confident that the person you
saw is not John, however much he may resemble him (i.e., what-
ever the likelihood). Of course, the idea that previous knowledge
influences perception is not novel. For example, there are abundant
examples of top-down influences on perception. What is unique
about the Bayesian hypothesis is the claim about how prior knowl-
edge is combined with evidence from the world, that is, the claim
that humans combine these sources of information in an optimal
(or near-optimal) way, following Bayes’s rule.
There are a number of additional complications that should
briefly be mentioned. First, unlike the simple examples above, the
priors and likelihoods in Bayesian theories of the mind generally
take the form of probability distributions rather than unique esti-
mates, given noise in the estimates. For example, the estimate of
the likelihood of a given hypothesis might be a distribution cen-
tered around a probability of .8, rather than consist of a single point
estimate of .8. This is illustrated in Figure 1, in which there is
uncertainty associated with both the likelihood and the prior,
resulting in a distribution for the posterior probability; as can be
seen, the posterior distribution is pulled in the direction of the prior
(or, to put it another way, the prior is updated to the posterior by
being shifted in the direction of the data). Thus, Bayesian methods
enable decision makers to go beyond point estimates and take into
account the uncertainty associated with a test in order to make
optimal decisions. As we show below, however, the practice of
estimating priors and likelihoods is fraught with difficulties.
Second, posterior probabilities do not always determine what
the optimal decision is. Imagine that the posterior probability of
cancer is best characterized as a probability distribution centered
around 10%. What is one to make of this fact? It obviously
depends on other factors (life expectancy associated with the
cancer, the risks and benefits associated with treatment, costs,
etc.), as well as the shape of probability distribution (e.g., how
tightly centered around 10% is the probability distribution? Is the
distribution skewed?). Accordingly, in Bayesian optimal decision
theory, the posterior probability is combined with a utility func-
tion, so that all the relevant variables (including the probability of
Figure 1. Separate probability distributions are shown for the prior, posterior, and likelihood function. The
posterior probability effectively revises the prior probability in the direction of the likelihood function.
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BOWERS AND DAVIS
cancer) can be considered when making a decision. The optimal
decision is the one that maximizes utility—or equivalently, min-
imizes loss. However, this decision will depend on which variables
affect the utility function and how the resulting utility function
relates to the probability of cancer. For example, surgery may be
the best option if the probability of cancer is very high, whereas
some other treatment may maximize utility if the probability is
moderate. If the best estimate of the probability of cancer is .02,
the best decision is to have no treatment. As noted below, the
estimation of utility functions in psychological models is not
straightforward, as both the variables that affect utility and the
shape of the function relating these variables to utility tend to be
unknown.
A Possible Confusion Regarding “Optimality”
Undoubtedly one of the reasons that Bayesian theories have
gathered such attention in psychology and neuroscience is that
claims of optimality are counterintuitive. Indeed, such claims
might at first sound absurd. If humans are optimal, why are we
unable to outrun a cheetah, outswim a dolphin, or fly? Indeed, why
don’t we run, swim, and fly at the limits set by physics? What
needs to be emphasized is that optimality means something quite
specific in the Bayesian context. The core claim is that the mind
makes or approximates optimal decisions given noisy data. The
noise that the brain has to deal with is the product of not only the
physical world itself but also suboptimal human design (e.g., it is
suboptimal that photoreceptors are located at the back of the retina
and that neurons fire differently to the same stimulus on different
occasions). But bad design is not problematic for Bayesian theories
claiming that human cognition approaches optimality. In principle,
there is no limit on how badly designed systems are, except in one
respect. The decision stage that interprets noisy, suboptimal inputs
is hypothesized to be near optimal.
In sum, Bayesian statistics provide a method of computing the
posterior probability of a hypothesis, which provides the best way
to update a prior belief given new evidence. Bayesian decision
theory defines how our beliefs should be combined with our
objectives to make optimal decisions given noisy data.
Three Types of Bayesian Theorizing in Psychology and
Neuroscience
Three approaches to Bayesian theorizing in psychology and
neuroscience can be distinguished; we refer to these as the ex-
treme, methodological, and theoretical approaches. The difference
concerns how Bayesian theories developed at a computational
level of description relate to the algorithmic level. On the extreme
view, psychology should concern itself only with developing the-
ories at a computational level. Questions of how the mind actually
computes are thought to be intractable and irrelevant. For instance,
Movellan and Nelson (2001) wrote:
Endless debates about undecidable structural issues (modularity vs.
interactivity, serial vs. parallel processing, iconic vs. propositional
representations, symbolic vs. connectionist models) may be put aside
in favor of a rigorous understanding of the problems solved by
organisms in their natural environments. (pp. 690 –691)
We do not have much to say about this approach, as it dismisses
the questions that are of interest to most cognitive psychologists.
This approach might make good sense in computer science or
robotics labs, but not, in our view, cognitive science. In any case,
such extreme views are rare.
More common is the methodological Bayesian approach. Meth-
odological Bayesians are not committed to any specific theory of
mind, at any level of description. Rather, they use Bayesian models
as tools: The models provide a measure of optimal behavior that
serves as a benchmark for actual performance. From this perspec-
tive, the striking result is how often human performance is near
optimal, and this is considered useful for constraining a theory
(whatever algorithm the mind uses, it must support behavior that
approximates optimal performance). But this approach is in no
way committed to the claim that the mind and brain compute in a
Bayesian-like way at the algorithmic level. For example, Geisler
and Ringach (2009) wrote:
There are a number of perceptual and motor tasks where humans
parallel the performance predicted by ideal Bayesian decision theory;
however, it is unknown whether humans accomplish this with simple
heuristics or by actually implementing the machinery of Bayesian
inference. (p. 3)
That is, the methodological approach is consistent with qualita-
tively different theories of mind. On the one hand, the mind might
be Bayesian-like and compute products of priors and likelihoods
for all the possible hypotheses (as in Equation 1). On the other
hand, the mind might be distinctly non-Bayesian and carry out
much simpler computations, considering only a small subset of the
available evidence to reach a near-optimal decision.
Theoretical Bayesians agree that Bayesian models developed at
the computational level constitute a useful method for constraining
theories at the algorithmic level (consistent with the methodolog-
ical Bayesian approach), but claim that the mind carries out or
approximates Bayesian computations at the algorithmic level, in
some unspecified way. For example, when describing the near-
optimal performance of participants in making predictions of un-
certain events, Griffiths and Tenenbaum (2006) wrote: “These
results are inconsistent with claims that cognitive judgments are
based on non-Bayesian heuristics” (p. 770). Indeed, Bayesian
models developed at a computational level are thought to give
insight into how neurons compute. For instance, Körding and
Wolpert (2006, p. 322) wrote: “Over a wide range of phenomena
people exhibit approximately Bayes-optimal behaviour. This
makes it likely that the algorithm implemented by the [central
nervous system] may actually support mechanisms for these kinds
of Bayesian computations.” When Chater, Oaksford, Hahn, and
Heit (2010) wrote that “Bayesian methods . . . may bridge across
each of Marr’s levels of explanation” (p. 820), we take it that they
were adopting a theoretical Bayesian perspective in which com-
putational, algorithmic, and implementational descriptions of the
mind may all be Bayesian.
Again, theoretical Bayesians are not committed to specific
claims regarding how the mind and brain realize Bayesian com-
putations at the algorithmic and implementational levels. But in
some way a Bayesian algorithm must (a) store priors in the forms
of probability distributions, (b) compute estimates of likelihoods
based on incoming data, (c) multiply these probability functions,
and (d) multiply priors and likelihoods for at least some alternative
hypotheses (the denominator in Equation 1). It is a commitment to
the above four claims that makes a theory Bayesian (as opposed to
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