Matt Jones and Bradley Love
Bayesian Fundamentalism or Enlightenment? On the explanatory status and
theoretical contributions of Bayesian models of cognition
Word Counts
Abstract 60
Text 998
References 460
Total 1518
Probabilistic models as theories of children’s minds
Alison Gopnik
Dept. of Psychology
University of California at Berkeley
Berkeley, CA 94720
510-967-3587
Gopnik@berkeley.edu
Alisongopnik.com
Abstract
Our research program proposes that children have representations and learning
mechanisms that can be characterized as causal models of the world – coherent,
structured, hypotheses with consistent relationships to probabilistic patterns of
evidence. We also propose that Bayesian inference is one mechanism by which
children learn these models from data. These proposals are straightforward
psychological hypotheses and far from “Bayesian fundamentalism” .
Text
So who exactly are these Bayesian fundamentalists? Since I don’t know or know
of any researchers who fit Jones and Love’s description, I can’t defend them.
Instead, let me describe what my colleagues and I have actually done over the
last ten years. Our research program has proposed that children have
representations and learning mechanisms that can be characterized as causal
models of the world – coherent, structured, hypotheses with consistent and
predictable relationships to probabilistic patterns of evidence, and that Bayesian
inference is one mechanism by which children may learn these models from
data.
This proposal has exactly the same status as any other claim in psychology.
Since the cognitive revolution, it has been a commonplace that we can explain
the mind in terms of representations and rules. It is also a commonplace of the
philosophy of cognitive science that these representations and rules can involve
many levels of abstraction, from high-level representations like “intuitive theories”
to neural representations of three-d structure. In vision science, for example, the
poster child of successful cognitive science, explanations at many levels from
ideal observer theories to neural implementations have been mutually
illuminating.
Bayesian learning is just one part of a broader approach, better called the
“probabilistic model” approach, (See Gopnik & Schulz, 2007 and Griffiths et al,
2010; for applications to cognitive development see Gopnik et al. 2004; and
special sections of Developmental Science (2007) and Cognition (in press).) The
central advance has not been Bayes’ law itself, but the ability to formulate
structured representations, such as causal graphical models, or “Bayes nets”,
(Spirtes, Glymour & Scheines, 1993; Pearl, 2000) or hierarchical causal models,
category hierarchies or grammars, that can be easily combined with probabilistic
learning, such as Bayesian inference.
I agree with Jones and Love that formal models are only useful if they can help
address empirical questions. For developmental psychologists the great question
is how children can learn as much as they do about the world around them. In
the past “theory theorists” proposed that children learn by constructing
hypotheses and testing them against evidence. But if this is a deterministic
process then the “poverty of the stimulus” problem becomes acute. In contrast, if
the child is a probabilistic learner, weighing the evidence to strengthen support
for one hypothesis over another, we can explain how children are gradually able
to revise their initial theories in favor of better ones.
Empirically, we’ve discovered, as a result of extensive and often challenging
experiments, that young children do indeed behave like probabilistic learners –
rather than simply using associationist mechanisms to match the patterns in the
data or fiddling with details of innate core knowledge. The framework lets us
make precise predictions about how children will behave, for example, that they
will infer a common cause structure with one set of evidence, but infer a causal
chain in an almost identical task with slightly different evidence (Schulz et al.
2007a). From a more specifically Bayesian perspective it also allows us to make
predictions about the probability of particular responses. The empirical likelihood
that a child will make one prediction or another is closely matched to the
posterior distribution of hypotheses that support those predictions (eg Schulz et
al. 2007b).
The ultimate test of any perspective is whether it generates new and interesting
empirical research. Researchers inspired by this approach have already begun to
make important developmental discoveries, discoveries that don’t fit either a
connectionist/dynamic or nativist picture. 9-month-olds, for example, can make
causal inferences that go beyond association (Sobel and Kirkham, 2006) 20month-olds can infer a person’s desire from a non-random sampling pattern
(Kushnir, Xu and Wellman, 2010) , and 4-year-olds discover new abstract
variables and rules from only a few data points (Schulz et al. 2008; Lucas et al,
2010), integrate new evidence and prior knowledge (Schulz et al. 2007b; Kushnir
& Gopnik 2007, Sobel et al, 2004) and rationally experiment to uncover new
causal structure (Schulz & Bonawitz, 2007).
The framework has also allowed us to identify cases where children behave in
ways that are interestingly different from ideal inference engines. For example,
Kushnir & Gopnik (2005) showed that children weight their own actions more
heavily than they normatively should for purposes of causal inference. Bonawitz
et al. (2010) have shown that two-year-olds have difficulty integrating purely
correlational information and information about the outcomes of actions.
The precision of these models also allows us to test competing hypotheses. For
example, we recently described formal models of learning by imitation which
gave different weights to statistical evidence and a demonstrator’s intentions
(Buchsbaum et al. in press). Comparing the models with the data allowed us to
show that young children make prior assumptions about why people act, and that
these assumptions can explain why they sometimes “overimitate”, reproducing
everything that the teacher does, and also predict the degree to which
overimitation will occur.
It would certainly be helpful to relate these more abstract accounts to algorithmic
level instantiations, this has already been done to some extent (Danks, 2003)
and we are working on precisely this problem in my lab and others at the moment
(Bonawitz et al., 2010). Ultimately, we would also like to relate these
representations to neural implementations. It might also be interesting to relate
these findings to memory or attention. But this is simply the usual scientific work
of connecting different research programs, and we can’t tell which connections
will be illuminating beforehand. There is nothing privileged about information
processing or neuroscience that means that such research is about
“mechanisms” while research into representations is not. This would be like
rejecting the explanation that my computer saved the file when I pushed ctrl-s
because ctrl-s is the save command in Windows, since this explanation doesn’t
refer to the computer’s working memory or wiring diagrams.
I would be happy if Jones and Love saw this research program as a part of the
Bayesian enlightenment. If so, however, it is an enlightenment that has been in
place for at least ten years.
References
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et al. (2010). Just do it? Investigating the gap between prediction and action in
toddlersʼ causal inferences. Cognition, 115(1), 104-17.
Buschbaum, D., Griffiths, T. L., & Gopnik, A. (In Press). Children's Imitation of
Causal Action Sequences is Influenced by Statistical and Pedagogical Evidence.
Cognition.
Danks, D. (2003). Equilibria of the Rescorla–Wagner model. Journal of
Mathematical Psychology, 47(2), 109-121.
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