The pOint of view behind the present work is that the connection between a statistical model and a statistical analysis-is a dua lity (in a vague sense). In usual textbooks on mathematical statistics it is often so that the statistical model is given in advance and then various in ference principles are applied to deduce the statistical ana lysis to be performed. It is however possible to reverse the above procedure: given that one wants to perform a certain statistical analysis, how can this be expressed in terms of a statistical model? In that sense we think of the statistical analysis and the stati stical model as two ways of expressing the same phenomenon, rather than thinking of the model as representing an idealisation of "truth" and the statistical analysis as a method of revealing that truth to the scientist. It is not the aim of the present work to solve the problem of giving the correct-anq final mathematical description of the quite complicated relation between model and analysis. We have rather restricted ourselves to describe a particular aspect of this, formulate it in mathematical terms, and then tried to make a rigorous and consequent investigation of that mathematical struc ture.

This book surveys results in the area sometimes denoted as "partial exchangeability" of "de Finetti type theorems". It is to be seen as an attempt to give sense to the general idea that there is a strong coupling between a statistical model and the statistical analysis. So strong that there is a canonical mathematical construction leading from the analysis to the model. Special sections are devoted to the study of sufficiency, of triviality of tails of Markov chains, studied e.g. by coupling methods, Martin boundaries and projective limits of Markov kernels and Polish spaces. In addition, many examples of extreme point models are treated in detail. This book is intended for researchers and graduate students in mathematical statistics and probability.

In the summer of 1988 in Providence, the AMS celebrated its centennial with a wide range of mathematical activities. Among those was a symposium, Mathematics into the Twenty-first Century, which brought together a number of the top research mathematicians who will likely have a significant impact on the mathematics of this century. This book contains the lectures presented by 16 of the 18 individuals who spoke during the symposium. Written by some of the major international figures in mathematical research, this group of articles covers a panorama of the vital areas of mathematics at the turn of the 21st century and gives the general mathematical reader a broad perspective on some of the major trends in research.

Algorithmic Learning in a Random World describes recent theoretical and experimental developments in building computable approximations to Kolmogorov's algorithmic notion of randomness. Based on these approximations, a new set of machine learning algorithms have been developed that can be used to make predictions and to estimate their confidence and credibility in high-dimensional spaces under the usual assumption that the data are independent and identically distributed (assumption of randomness). Another aim of this unique monograph is to outline some limits of predictions: The approach based on algorithmic theory of randomness allows for the proof of impossibility of prediction in certain situations. The book describes how several important machine learning problems, such as density estimation in high-dimensional spaces, cannot be solved if the only assumption is randomness.

Bayesian statistics is a dynamic and fast-growing area of statistical research and the Valencia International Meetings provide the main forum for discussion. These resulting proceedings form an up-to-date collection of research.

An examination of the difficulties that statistical theory and, in particular, estimation theory can encounter within the area of dependent data. This is achieved through the study of the theory of branching processes starting with the demographic question: what is the probability that a family name becomes extinct? Contains observations on the generation sizes of the Bienaymé-Galton-Watson (BGW) process. Various parameters are estimated and branching process theory is contrasted to a Bayesian approach. Illustrations of branching process theory applications are shown for particular problems.

Keeping pace with the latest developments in all branches of statistical science. Encyclopedia of Statistical Sciences is the number one source of information on statistical theory, methods, and applications for researchers and clinicians. This new volume is the last of three updates designed to bring the Encyclopedia in line with new and emerging topics and important advances in statistical science made over the past decade. Each self-contained entry is written by a leader in the field and easily understood by readers with a modest statistical background. In addition to the main selections, which feature fascinating discussions of developments in various branches of the statistical sciences, readers will find a series of shorter entries ranging in subject matter from the lives of pioneers in statistics to updates of earlier articles and reviews of statistical agencies and journals. Up-to-date bibliographies, thorough cross-referencing, and extensive indexing facilitate quick access to specific information and provide an indispensable platform for further study and research. A cumulative index and listing of all the entries in the 13 volumes of the Encyclopedia, together with the corresponding authors, are included. With the publication of this update installment, the Encyclopedia of Statistical Sciences retains its position as the only cutting-edge reference of choice for those working in statistics, probability theory, biostatistics, quality control, and economics and in applications of statistical methods in sociology, engineering, computer and communication science, biomedicine, psychology, and many other areas.

This third volume of case studies presents detailed applications of Bayesian statistical analysis, emphasising the scientific context. The papers were presented and discussed at a workshop held at Carnegie-Mellon University, and this volume - dedicated to the memory of Morrie Groot-reproduces six invited papers, each with accompanying invited discussion, and nine contributed papers with the focus on econometric applications.