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{\LARGE\textbf{Bayes rules}}
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\textbf{A once-neglected statistical technique may help to explain how
the mind works}
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SCIENCE, being a human activity, is not immune to fashion. For example,
one of the first mathematicians to study the subject of probability
theory was an English clergyman called Thomas Bayes, who was born in
1702 and died in 1761. His ideas about the prediction of future events
from one or two examples were popular for a while, and have never been
fundamentally challenged. But they were eventually overwhelmed by those
of the ``frequentist'' school, which developed the methods based on
sampling from a large population that now dominate the field and are
used to predict things as diverse as the outcomes of elections and
preferences for chocolate bars.
Recently, however, Bayes's ideas have made a comeback among computer
scientists trying to design software with human-like intelligence.
Bayesian reasoning now lies at the heart of leading internet search
engines and automated ``help wizards''. That has prompted some
psychologists to ask if the human brain itself might be a
Bayesian-reasoning machine. They suggest that the Bayesian capacity to
draw strong inferences from sparse data could be crucial to the way the
mind perceives the world, plans actions, comprehends and learns
language, reasons from correlation to causation, and even understands
the goals and beliefs of other minds.
These researchers have conducted laboratory experiments that convince
them they are on the right track, but only recently have they begun to
look at whether the brain copes with everyday judgments in the real
world in a Bayesian manner. In research to be published later this year
in Psychological Science, Thomas Griffiths of Brown University in Rhode
Island and Joshua Tenenbaum of the Massachusetts Institute of Technology
put the idea of a Bayesian brain to a quotidian test. They found that it
passes with flying colours.
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\textbf{Prior assumptions}
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The key to successful Bayesian reasoning is not in having an extensive,
unbiased sample, which is the eternal worry of frequentists, but rather
in having an appropriate ``prior'', as it is known to the cognoscenti.
This prior is an assumption about the way the world works---in essence, a
hypothesis about reality---that can be expressed as a mathematical
probability distribution of the frequency with which events of a
particular magnitude happen.
The best known of these probability distributions is the ``normal'', or
Gaussian distribution. This has a curve similar to the cross-section of
a bell, with events of middling magnitude being common, and those of
small and large magnitude rare, so it is sometimes known by a third
name, the bell-curve distribution. But there are also the
Poisson\footnote{In this summary (but not in the original paper) the
term `Poisson distribution' is used when the gamma distribution is
meant.} distribution, the Erlang distribution, the power-law
distribution and many even weirder ones that are not the consequence of
simple mathematical equations (or, at least, of equations that
mathematicians regard as simple).
With the correct prior, even a single piece of data can be used to make
meaningful Bayesian predictions. By contrast frequentists, though they
deal with the same probability distributions as Bayesians, make fewer
prior assumptions about the distribution that applies in any particular
situation. Frequentism is thus a more robust approach, but one that is
not well suited to making decisions on the basis of limited
information---which is something that people have to do all the time.
Dr Griffiths and Dr Tenenbaum conducted their experiment by giving
individual nuggets of information to each of the participants in their
study (of which they had, in an ironically frequentist way of doing
things, a total of 350), and asking them to draw a general conclusion.
For example, many of the participants were told the amount of money that
a film had supposedly earned since its release, and asked to estimate
what its total ``gross'' would be, even though they were not told for how
long it had been on release so far.
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Besides the returns on films, the participants were asked about things
as diverse as the number of lines in a poem (given how far into the poem
a single line is), the time it takes to bake a cake (given how long it
has already been in the oven), and the total length of the term that
would be served by an American congressman (given how long he has
already been in the House of Representatives). All of these things have
well-established probability distributions, and all of them, together
with three other items on the list---an individual's lifespan given his
current age, the run-time of a film, and the amount of time spent on
hold in a telephone queuing system---were predicted accurately by the
participants from lone pieces of data.
There were only two exceptions, and both proved the general rule, though
in different ways. Some 52\% of people predicted that a marriage would
last forever when told how long it had already lasted. As the authors
report, ``this accurately reflects the proportion of marriages that end
in divorce'', so the participants had clearly got the right idea. But
they had got the detail wrong. Even the best marriages do not last
forever. Somebody dies. And ``forever'' is not a mathematically tractable
quantity, so Dr Griffiths and Dr Tenenbaum abandoned their analysis of
this set of data.
The other exception was a topic unlikely to be familiar to 21st-century
Americans---the length of the reign of an Egyptian Pharaoh in the fourth
millennium BC. People consistently overestimated this, but in an
interesting way. The analysis showed that the prior they were applying
was an Erlang distribution, which was the correct type. They just got
the parameters wrong, presumably through ignorance of political and
medical conditions in fourth-millennium BC Egypt. On congressmen's
term-lengths, which also follow an Erlang distribution, they were spot
on.
Indeed, one of the most impressive things Dr Griffiths and Dr Tenenbaum
have shown is the range of distributions the mind can cope with. Besides
Erlang, they tested people with examples of normal distributions,
power-law distributions and, in the case of baking cakes, a complex and
irregular distribution. They found that people could cope equally well
with all of them, cakes included. Indeed, they are so confident of their
method that they think it could be reversed in those cases where the
shape of a distribution in the real world is still a matter of debate.
To prove the point, they actually did such a reversal in the case of
telephone-queue waiting times. Traditionally, these have been assumed to
follow a Poisson\footnote{See previous footnote} distribution, but some recent research suggests they
actually follow a power law. Analysing the participants' responses
suggests that a power law, indeed, it is.
How the priors are themselves constructed in the mind has yet to be
investigated in detail. Obviously they are learned by experience, but
the exact process is not properly understood. Indeed, some people
suspect that the parsimony of Bayesian reasoning leads occasionally to
it going spectacularly awry, with whatever process it is that forms the
priors getting further and further off-track rather than converging on
the correct distribution.
That might explain the emergence of superstitious behaviour, with an
accidental correlation or two being misinterpreted by the brain as
causal. A frequentist way of doing things would reduce the risk of that
happening. But by the time the frequentist had enough data to draw a
conclusion, he might already be dead.
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\textit{The Economist} January 7th 2006
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The paper referred to is `Optimal predictions in everyday cognition'.
T L Griffiths and J R Tenenbaum, \textit{Psychological Science}
\textbf{17} (9), 767--773.
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