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{\LARGE\textbf{In praise of Bayes}}

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\textbf{Bayesianism is a controversial  but increasingly popular
approach to statistics that offers many  benefits---although not
everyone is persuaded of its validity}

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IT IS not often that a  man born 300 years ago suddenly springs back to
life. But that is what has  happened to the Reverend Thomas Bayes, an
18th-century Presbyterian minister and  mathematician---in spirit, at
least, if not in body. Over the past decade the  value of a statistical
method outlined by Bayes in a paper first published in  1763 has become
increasingly apparent and has resulted in a blossoming of  ``Bayesian''
methods in scientific fields ranging from archaeology to computing. 
Bayes' fans have restored his tomb and posted pictures of it on the
Internet,  and a celebratory bash is planned for next year to mark the
300th anniversary of  his birth. There is even a Bayes
songbook---though, since Bayesians are an  academic bunch, it is
available only in the obscure file formats that are used  for scientific
papers.

Proponents of the  Bayesian approach argue that it has many advantages
over traditional,  ``frequentist'' statistical methods. Expressing
scientific results in Bayesian  terms, they suggest, makes them easier
to understand and makes borderline or  inconclusive results less prone
to misinterpretation. Bayesians claim that their  methods could make
clinical trials of drugs faster and fairer, and computers  easier to
use. There are even suggestions that Bayes's ideas could prompt a 
re-evaluation of fundamental scientific concepts of evidence and
causality. Not  bad for an old dead white male.

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\textbf{Previous convictions}

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The essence of the  Bayesian approach is to provide a mathematical rule
explaining how you should  change your existing beliefs in the light of
new evidence. In other words, it  allows scientists to combine new data
with their existing knowledge or  expertise.

The canonical example is  to imagine that a precocious newborn observes
his first sunset, and wonders  whether the sun will rise again or not.
He assigns equal prior probabilities to  both possible outcomes, and
represents this by placing one white and one black  marble into a bag.
The following day, when the sun rises, the child places  another white
marble in the bag. The probability that a marble plucked randomly  from
the bag will be white (ie, the child's degree of belief in future
sunrises)  has thus gone from a half to two-thirds. After sunrise the
next day, the child  adds another white marble, and the probability (and
thus the degree of belief)  goes from two-thirds to three-quarters. And
so on. Gradually, the initial belief  that the sun is just as likely as
not to rise each morning is modified to become  a near-certainty that
the sun will always rise.

In a Bayesian analysis,  in other words, a set of observations should be
seen as something that changes  opinion, rather than as a means of
determining ultimate truth. In the case of a  drug trial, for example,
it is possible to evaluate and compare the degree to  which a sceptic
and an enthusiast would be convinced by a particular set of  results.
Only if the sceptic can be convinced should a drug be licensed for use. 

This is far more subtle  than the traditional way of presenting results,
in which an outcome is deemed  statistically significant only if there
is a better than 95\% chance that it  could not have occurred by chance.
The problem, according to Robert Matthews, a  mathematician at Aston
University in Birmingham, is that medical researchers  have failed to
understand that subtlety. In a paper to be published shortly in  the
\textit{Journal of Statistical Planning and Inference}, he sets out to 
demystify the Bayesian approach, and explains how to apply it after the
event to  existing data.

Patients in clinical  trials will soon benefit. Bayesian methods offer
the possibility of modifying a  trial while it is being conducted,
something that is impossible with traditional  statistics. Andy Grieve
and his colleagues at Pfizer, a drug firm, are intending  to do just
that.

Traditionally,  dose-allocation trials---in which the aim is to
establish the most effective dose  of a new drug---involve giving
different groups of patients different doses and  evaluating the results
once the trial has finished. This is fine from a  statistical point of
view, but unfair on those patients who turn out to have  been given
non-optimal doses. Rather than analysing the results at the end of a 
trial, Dr Grieve's method will evaluate patients' responses during it,
and  adjust the doses accordingly. The advantage of this over the
traditional  approach that it maximises the medical benefit to all
participants. It also  means that fewer people are needed to conduct a
trial, because participants on  non-optimal doses can have those doses
changed to increase the amount of data  collected near the optimal dose.

Pfizer is intending to  conduct a trial using this new method, and the
plan is to re-analyse the data  once it is completed in ways that will
satisfy both Bayesians and non-Bayesians.  This kind of parallel
approach is likely to become increasingly common, at least  until
Bayesianism has been more widely accepted.

Bayesian methods can  also be used to decide between several competing
hypotheses, by seeing which is  most consistent with the available data.
This idea was recently used to  determine the date of construction of
``Seahenge'', an ancient timber circle found  off the coast of Norfolk,
in eastern England. Results from tree-ring dating were  inconclusive,
suggesting such divergent dates as 2019\textsc{bc},  2050\textsc{bc} and
2454\textsc{bc}. So six samples from  the monument's central stump were
radiocarbon-dated, and the results were used  to evaluate the three
tree-ring possibilities via Bayesian analysis. The  evidence was
overwhelmingly in favour of 2050\textsc{bc}, and was  inconsistent with
either of the other two tree-ring dates.

Decision-making using  Bayesian methods has many applications in
software, as well. Perhaps the  best-known example is Microsoft's Office
Assistant, which appears as a somewhat  irritating anthropomorphic
paper-clip that tries to help the user. When a user  calls up the
assistant, Bayesian methods are used to analyse recent actions in  order
to try to work out what the user is attempting to do, with this 
calculation constantly being modified in the light of new actions. 
(Unfortunately, a non-Bayesian approach is used to decide when to make
the  paper-clip pop up on its own, adding to the annoyance of many
users.) According  to Eric Horvitz, a statistician in Microsoft's
research division, future  products will try to determine users'
intentions more broadly, so as to speed  things up. Having worked out
which link on a web article a user is most likely  to click on, for
example, the computer could fetch the corresponding article in  advance,
so that it appears more quickly.

Bayes is still, however,  the focus of much controversy. Larry
Wasserman, a statistician at Carnegie  Mellon University, in Pittsburgh,
says that although Bayesianism is becoming  more acceptable, it is no
panacea, and when used indiscriminately it becomes  ``more a religion
than a science''. Perhaps the grandest claims made for Bayesian  methods
are those of Judea Pearl, a computer scientist at the University of 
California, Los Angeles. Dr Pearl has suggested that by analysing
scientific  data using a Bayesian approach it may be possible to
distinguish between  correlation (in which two phenomena, such as
smoking and lung cancer, occur  together) and causation (in which one
actually causes the other). This kind of  claim makes many scientists,
including many Bayesians, throw up their hands in  horror. Evidently
there is life in the old reverend yet.

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  \textit{The Economist} September 30th 2000
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