STATISTICIAN’S VIEW
(Note: The authors’ response is on the following page.)
Dear Editor,
The article “Math Is Music, Statistics Is Literature” by [Richard]
DeVeaux and [Paul] Velleman in the September 2008 issue of
Amstat News (pp.54–58) is interesting, intelligent, and persua-
sive; unfortunately, its main conclusion is unsound.
The authors say correctly that the change in the teaching of
introductory statistics began in the early 1980s, when there was
a shift from a mathematical basis to a form of literature. What
they forget is that, by then, there had been developed a differ-
ent basis for statistics that diminishes the need for the literary
element and retains the scientific emphasis.
Statistics is essentially the study of uncertainty, which, in
its commonest form, explores the uncertainty about a param-
eter θ in the light of data x. Based on the work of Sir Harold
Jeffreys in England, Bruno de Finetti in Italy, and Leonard
Jimmie Savage in America, there had, by the early ’80s, been
developed a systematic treatment of uncertainty, and hence of sta-
tistics. The basic result that these three pioneers, with others, had dem-
onstrated was that the only satisfactory way to handle uncertainty was
through probability. That is, uncertainty had to be measured in terms
of numbers that obeyed the three basic laws of probability: convexity,
addition, and multiplication. Any other type of measurement could
lead to nonsense, technically termed incoherence.
With this result available, virtually the whole of statistics could
introductory course. I disagree and have tried in my book,
be based on probability, where all analyses would proceed according
Understanding Uncertainty, to present the results needed for an intro-
to the rules of that calculus. By an accident of history, this approach
ductory statistics course, and more, using only arithmetic and sym-
came to be known as Bayesian, for the minor reason that it made
bols. Indeed, I would go further and say that every educated person
extensive use of Bayes formula, which is essentially the multiplication
ought to understand probability as the three laws, since all of us face
law of probability.
uncertainty every day of our lives and probability is the only way to
With the result “uncertainty is probability” available, one can see
handle it.
the outlines of an introductory statistics course. It begins with develop-
DeVeaux and Velleman list six points in their article. 1 to 3 and 5
ing a familiarity with the three laws, where it is neither necessary nor
fit naturally into the above scheme. 4, “focus on what we don’t know,”
desirable to present the pioneers’ development—only the conclusion.
is our central thesis, but shows that confidence intervals are the wrong
With this achieved, what we think of as statistics can begin. What is
way to express uncertainty, since they do not make a probability state-
the uncertain quantity of interest? Call it θ. What data x are available?
ment about θ. 6, “think about conditional probability,” is our main
The final uncertainty is the probability of θ, given x, p(θ|x). This is
conclusion, but shows that p-values are incoherent, for again they are
proportional to θ(x|θ)p(θ), where p(x|θ) is the uncertainty in the data,
not probability statements about θ.
where the quantity of interest to have the value θ, and p(θ) is the initial
Mathematics is the language of theoretical science; statistics that of
uncertainty. The former will be familiar to all statisticians as the likeli-
empirical science. Let us keep it that way, and keep literature where it
hood of θ, given x. In most practical situations, in order to model the
properly belongs.
uncertainty easily, it is necessary to write θ as a pair (ξ,η), where ξ is Dennis V. Lindley
the quantity of real interest and η a nuisance. Again, by the probability Minehead, UK
calculus, η can be eliminated by integration—the addition law.
The Bayesian approach, as just outlined, has the great merit of plac-
References
ing logical thinking at the heart of our subject. Every science needs its
logical structure; it is part of what makes a topic scientific; whereas, lit-
Jeffreys, H. (1998) Theory of Probability. Clarendon Press, Oxford.
erature does not. Our logic is probability; some would say conditional
de Finetti, B. (1974) Theory of Probability. John Wiley, London.
probability, but in the Bayesian view, all probability is conditional.
Any uncertainty depends on what you know, or assume, at the time
Savage, L.J. (1954) The Foundations of Statistics. John Wiley,
the uncertainty is expressed.
New York.
David Moore and George McCabe, in their book Introduction to
Moore, D.S. and McCabe, G.P (1998) Introduction to the
the Practice of Statistics, which DeVeaux and Velleman cited as influ-
Practice of Statistics. W. H. Freeman, New York.
encing current teaching, barely mention probability in their 825 pages,
and conditional probability does not occur, thought it haunts almost Lindley, D.V. (2006) Understanding Uncertainty. John Wiley,
every page. Perhaps this is because they regard it as too difficult for the Hoboken.
DECEMBER 2008 AMSTAT NEWS 17
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