CHAPTER 1Introduction 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.” Curvesper 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 seventeenthcentury mathematician and philosopher René Descartes (Bell, 1937). Descartes realized that the concept diameter, when applied to a circle, resembled the parameter of a parabola, 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 wordperimeter, 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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