The Nature of Statistical Evidence

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Springer Science & Business Media, Dec 21, 2007 - Mathematics - 152 pages
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Our motivation for writing this book was a dissatisfaction with the many books with titles hke Foundations of Statistics. These books provide a needed description of the subject with examples of various statistical methods; but they do not satisfy the discerning reader because they do not explain why certain conclusions may be drawn from certain data and they do not discuss how statistics, the subject, meshes with the scientific process. We naively set out to fill in these gaps, but the situation is not so simple. What is desired is a tool—the one true statistics—which can be applied to data with certainty to justify conclusions, but what is available, in fact, are various competing theories of statistics. The careful reader may correctly point out the relation of this manuscript to po- modem epistemology, a philosophy which emphasizes that all human knowing—in fields as diverse as religion and science—is culture dependent and that therefore "truth" is not absolute. Take, for example, the "truths" of Christianity and Islam— wars have been fought over their differences. In the field of statistics there are, for example, Bayesians and sampling theorists. Carson (2003) states, "Surely it is better, postmodems tell us, to encourage insights that flow from many different perspectives. . . " In statistics, postmodernism is manifest at two levels. First, the findings of statistical applications are not about the true state of nature but about models of experiments on nature.

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Contents

Overview
1
Mathematics and Its Applications 9
7
The Evolution of Natural Scientists and Their Theories
18
Law and Learning
33
Introduction to Probability
41
Attitudes Toward Chance
61
A Framework for Statistical Evidence
75
A Critique of Bayesian Inference
84
The LongRun Consequence of Behavior
97
The Nature of Statistical Evidence
116
The Science of Statistics
122
Deductive Logic
133
References
143
Subject Index
151
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Page 7 - That, if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles...
Page 13 - The long chains of simple and easy reasonings by means of which geometers are accustomed to reach the conclusions of their most difficult demonstrations, had led me to imagine that all things, to the knowledge of which man is competent, are mutually connected in the same way, and that there is nothing so far removed from us as to be beyond our reach, or so hidden that we cannot discover it, provided only we abstain from accepting the false for the true, and always preserve in our thoughts the order...
Page 143 - ON THE ALGEBRAICAL AND NUMERICAL THEORY OF ERRORS OF OBSERVATIONS AND THE COMBINATION OF OBSERVATIONS.
Page 26 - Conclusions are withheld until adequate evidence has accumulated. A conclusion is a statement which is to be accepted as applicable to the conditions of an experiment or observation unless and until unusually strong evidence to the...
Page 15 - Now metaphysics has always been the ape of mathematics. Geometry suggested the idea of a demonstrative system of absolutely certain philosophical principles; and the ideas of the metaphysicians have at all times been in large part drawn from mathematics. The metaphysical axioms are imitations of the geometrical axioms; and now that the latter have been thrown overboard, without doubt the former will be sent after them. It is evident, for instance, that we can have no reason to think that every phenomenon...
Page 20 - In order to lay the foundation of such investigations on strict mechanical principles, I shall demonstrate the laws of motion of an indefinite number of small, hard, and perfectly elastic spheres acting on one another only during impact. ' If the properties of such a system of bodies are found to correspond to those of gases, an important physical analogy will be established, which may lead to more accurate knowledge of the properties of matter. If experiments on gases are inconsistent with the hypothesis...
Page 24 - Let me now say only this, that truth is one species of good, and not, as is usually supposed, a category distinct from good, and coordinate with it. The true is the name of whatever proves itself to be good in the way of belief, and good, too, for definite, assignable reasons.
Page 20 - To see what is general in what is particular and what is permanent in what is transitory is the aim of scientific thought.
Page 61 - In applying statistical theory, the main consideration is not what the shape of the universe is, but whether there is any universe at all. No universe can be assumed, nor . . . statistical theory . . . applied unless the observations show statistical control. In this state the samples when cumulated over a suitable interval of time give a distribution of a particular shape, and this shape is reproduced hour after hour, day after day, so long as the process remains in statistical control — ie, exhibits...
Page 10 - If p and q are points, then there exists one and only one line containing p and q. Axiom 4. If L is a line, then there exists a point not on L. Axiom 5. If L is a line, and p is a point not on L, then there exists one and only one line containing p that is parallel to L. These axioms would not...

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