CHAPTER 1
Introduction
1.1 Background
The first sentence of I. R.
Savage’s book,
Statistics: Uncertainty and Behavior
(1968), is: “Probabilities are quantitative expressions
of uncertainty.”
Thus, for Savage, the term
uncertainty
denotes a more general concept than the
principle of quantum mechanics that earned W. K.
Heisenberg a 1932 Nobel Prize.
But what exactly is
statistical uncertainty?
One of the many possible
answers to this question concerns what Savage calls
“quantitative expressions.”
Curves
per se
and quantitative expressions for
curves can be matched well or matched poorly.
For this reason we place much emphasis on
the distinctions and connections between model parameters and curve properties.
Parameters
are symbols within some mathematical expression that may vary between problems, but not
within a given context or problem.
In this sense they are representational devices that are
intermediates between constants and variables.
DESCARTES
An early recorded application of a parameter was published by the seventeenth
century
mathematician and philosopher René Descartes (Bell, 1937).
Descartes realized that the
concept
diameter
, when applied to a circle, resembled the
para
meter of a
para
bola, the curve
property "latus rectum."
Then, as now, the notational linkage of two different entities, the
first a symbol and the second a curve property like the diameter or latus rectum of a
particular curve, was a major advantage of the parametric approach.
The word
parameter
sounds like the word
perimeter
, yet the role played by any
parameter is contrary to that played by any perimeter.
Rather than defining boundaries,
parameters extend flexibility.
When a mathematical expression does not contain any
parameters, it is completely specified to the degree that it can be tabled.
On the other hand,

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