By D. R. Cox
Publisher: Cambridge University Press
Print Publication Year: 2006
Online Publication Date:March 2011
Online ISBN:9780511813559
Hardback ISBN:9780521866736
Paperback ISBN:9780521685672
Chapter DOI: http://dx.doi.org/10.1017/CBO9780511813559.006
Subjects: Statistical theory and methods, Quantitative biology, biostatistics and mathematical modeling
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Summary. This chapter discusses the nature of probability as it is used to represent both variability and uncertainty in the various approaches to statistical inference. After some preliminary remarks, the way in which a frequency notion of probability can be used to assess uncertainty is reviewed. Then two contrasting notions of probability as representing degree of belief in an uncertain event or hypothesis are examined.
General remarks
We can now consider some issues involved in formulating and comparing the different approaches.
In some respects the Bayesian formulation is the simpler and in other respects the more difficult. Once a likelihood and a prior are specified to a reasonable approximation all problems are, in principle at least, straightforward. The resulting posterior distribution can be manipulated in accordance with the ordinary laws of probability. The difficulties centre on the concepts underlying the definition of the probabilities involved and then on the numerical specification of the prior to sufficient accuracy.
Sometimes, as in certain genetical problems, it is reasonable to think of θ as generated by a stochastic mechanism. There is no dispute that the Bayesian approach is at least part of a reasonable formulation and solution in such situations. In other cases to use the formulation in a literal way we have to regard probability as measuring uncertainty in a sense not necessarily directly linked to frequencies. We return to this issue later.
pp. i-iv
pp. v-viii
pp. ix-xii
pp. xiii-xvi
pp. 1-16
2 - Some concepts and simple applications : Read PDF
pp. 17-29
3 - Significance tests : Read PDF
pp. 30-44
4 - More complicated situations : Read PDF
pp. 45-63
5 - Interpretations of uncertainty : Read PDF
pp. 64-95
6 - Asymptotic theory : Read PDF
pp. 96-132
7 - Further aspects of maximum likelihood : Read PDF
pp. 133-160
8 - Additional objectives : Read PDF
pp. 161-177
9 - Randomization-based analysis : Read PDF
pp. 178-193
Appendix A - A brief history : Read PDF
pp. 194-196
Appendix B - A personal view : Read PDF
pp. 197-200
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