PrefaceMost statistical work is concerned directly with the provision and implementationof methods for study design and for the analysis and interpretation of data.The theory of statistics deals in principle with the general concepts underlyingall aspects of suchwork and from this perspective the formal theory of statisticalinference is but a part of that full theory. Indeed, from the viewpoint of individualapplications, it may seem rather a small part. Concern is likely to be moreconcentrated on whether models have been reasonably formulated to addressthe most fruitful questions, on whether the data are subject to unappreciatederrors or contamination and, especially, on the subject-matter interpretation ofthe analysis and its relation with other knowledge of the field.Yet the formal theory is important for a number of reasons. Without somesystematic structure statistical methods for the analysis of data become a collectionof tricks that are hard to assimilate and interrelate to one another, orfor that matter to teach. The development of new methods appropriate for newproblems would become entirely a matter of ad hoc ingenuity. Of course suchingenuity is not to be undervalued and indeed one role of theory is to assimilate,generalize and perhaps modify and improve the fruits of such ingenuity.Much of the theory is concerned with indicating the uncertainty involved inthe conclusions of statistical analyses, and with assessing the relative merits ofdifferent methods of analysis, and it is important even at a very applied level tohave some understanding of the strengths and limitations of such discussions.This is connected with somewhat more philosophical issues connected withthe nature of probability. A final reason, and a very good one, for study of thetheory is that it is interesting.The object of the present book is to set out as compactly as possible thekey ideas of the subject, in particular aiming to describe and compare the mainideas and controversies over more foundational issues that have rumbled on atvarying levels of intensity for more than 200 years. I have tried to describe thevarious approaches in a dispassionate way but have added an appendix with amore personal assessment of the merits of different ideas.Some previous knowledge of statistics is assumed and preferably someunderstanding of the role of statistical methods in applications; the latterunderstanding is important because many of the considerations involved areessentially conceptual rather than mathematical and relevant experience isnecessary to appreciate what is involved.The mathematical level has been kept as elementary as is feasible and ismostly that, for example, of a university undergraduate education in mathematicsor, for example, physics or engineering or one of the more quantitativebiological sciences. Further, as I think is appropriate for an introductory discussionof an essentially applied field, the mathematical style used here eschewsspecification of regularity conditions and theorem–proof style developments.Readers primarily interested in the qualitative concepts rather than their developmentshould not spend too long on the more mathematical parts of thebook.The discussion is implicitly strongly motivated by the demands of applications,and indeed it can be claimed that virtually everything in the book hasfruitful application somewhere across the many fields of study to which statisticalideas are applied. Nevertheless I have not included specific illustrations.This is partly to keep the book reasonably short, but, more importantly, to focusthe discussion on general concepts without the distracting detail of specificapplications, details which, however, are likely to be crucial for any kind ofrealism.The subject has an enormous literature and to avoid overburdening the readerI have given, by notes at the end of each chapter, only a limited number of keyreferences based on an admittedly selective judgement. Some of the referencesare intended to give an introduction to recentwork whereas others point towardsthe history of a theme; sometimes early papers remain a useful introduction toa topic, especially to those that have become suffocated with detail. A briefhistorical perspective is given as an appendix.The book is a much expanded version of lectures given to doctoral students ofthe Institute of Mathematics, Chalmers/Gothenburg University, and I am verygrateful to Peter Jagers and NannyWermuth for their invitation and encouragement.It is a pleasure to thank Ruth Keogh, Nancy Reid and Rolf Sundberg fortheir very thoughtful detailed and constructive comments and advice on a preliminaryversion. It is a pleasure to thank also Anthony Edwards and DeborahMayo for advice on more specific points. I am solely responsible for errors offact and judgement that remain.introductory,setting out the formulation of problems, outlining in a simple casethe nature of frequentist and Bayesian analyses, and describing some specialmodels of theoretical and practical importance. The discussion continues withthe key ideas of likelihood, sufficiency and exponential families.Chapter 4 develops some slightly more complicated applications. The longChapter 5 is more conceptual, dealing, in particular, with the various meaningsof probability as it is used in discussions of statistical inference. Most of the keyconcepts are in these chapters; the remaining chapters, especially Chapters 7and 8, are more specialized.Especially in the frequentist approach, many problems of realistic complexityrequire approximate methods based on asymptotic theory for their resolutionand Chapter 6 sets out the main ideas. Chapters 7 and 8 discuss various complicationsand developments that are needed from time to time in applications.Chapter 9 deals with something almost completely different, the possibilityof inference based not on a probability model for the data but rather onrandomization used in the design of the experiment or sampling procedure.I have written and talked about these issues for more years than it is comfortableto recall and am grateful to all with whom I have discussed the topics,especially, perhaps, to those with whom I disagree. I am grateful particularlyto David Hinkley with whom I wrote an account of the subject 30 years ago.The emphasis in the present book is less on detail and more on concepts but theeclectic position of the earlier book has been kept.