Science and technology
A once-neglected statistical technique
may help to explain how the mind works
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
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.
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
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.
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
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.