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Bayesian statistical decision theory --- Statistique bayésienne --- Problems, exercices, etc. --- Problèmes et exercices --- Statistique bayésienne --- Problèmes et exercices --- Statistique

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The exponential increase in the use of MCMC methods and the corre­ sponding applications in domains of even higher complexity have caused a growing concern about the available convergence assessment methods and the realization that some of these methods were not reliable enough for all-purpose analyses. Some researchers have mainly focussed on the con­ vergence to stationarity and the estimation of rates of convergence, in rela­ tion with the eigenvalues of the transition kernel. This monograph adopts a different perspective by developing (supposedly) practical devices to assess the mixing behaviour of the chain under study and, more particularly, it proposes methods based on finite (state space) Markov chains which are obtained either through a discretization of the original Markov chain or through a duality principle relating a continuous state space Markov chain to another finite Markov chain, as in missing data or latent variable models. The motivation for the choice of finite state spaces is that, although the resulting control is cruder, in the sense that it can often monitor con­ vergence for the discretized version alone, it is also much stricter than alternative methods, since the tools available for finite Markov chains are universal and the resulting transition matrix can be estimated more accu­ rately. Moreover, while some setups impose a fixed finite state space, other allow for possible refinements in the discretization level and for consecutive improvements in the convergence monitoring.

Convergence --- Markov processes --- Monte Carlo method --- Convergence. --- Markov processes. --- Monte Carlo method. --- Discretization (Mathematics) --- Engineering & Applied Sciences --- Applied Mathematics --- Applied mathematics. --- Engineering mathematics. --- Applications of Mathematics. --- Engineering --- Engineering analysis --- Mathematical analysis --- Mathematics --- Mathematical models. --- Analysis, Markov --- Chains, Markov --- Markoff processes --- Markov analysis --- Markov chains --- Markov models --- Models, Markov --- Processes, Markov --- Stochastic processes --- Functions

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Bayesian statistical decision theory --- Acqui 2006 --- Statistique bayesienne

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Statistical science --- Mathematical statistics --- Business economics --- Computer. Automation --- informatica --- statistiek --- econometrie --- statistisch onderzoek

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Monte Carlo statistical methods, particularly those based on Markov chains, have now matured to be part of the standard set of techniques used by statisticians. This book is intended to bring these techniques into the class­ room, being (we hope) a self-contained logical development of the subject, with all concepts being explained in detail, and all theorems, etc. having detailed proofs. There is also an abundance of examples and problems, re­ lating the concepts with statistical practice and enhancing primarily the application of simulation techniques to statistical problems of various dif­ ficulties. This is a textbook intended for a second-year graduate course. We do not assume that the reader has any familiarity with Monte Carlo techniques (such as random variable generation) or with any Markov chain theory. We do assume that the reader has had a first course in statistical theory at the level of Statistical Inference by Casella and Berger (1990). Unfortu­ nately, a few times throughout the book a somewhat more advanced no­ tion is needed. We have kept these incidents to a minimum and have posted warnings when they occur. While this is a book on simulation, whose actual implementation must be processed through a computer, no requirement is made on programming skills or computing abilities: algorithms are pre­ sented in a program-like format but in plain text rather than in a specific programming language. (Most of the examples in the book were actually implemented in C, with the S-Plus graphical interface.

Mathematical statistics --- Mathematical statistics. --- Monte Carlo method. --- Statistique mathématique --- Monte-Carlo, Méthode de --- Monte Carlo method --- 519.245 --- 681.3*G3 --- 519.2 --- #ABIB:astp --- 519.5 --- Artificial sampling --- Model sampling --- Monte Carlo simulation --- Monte Carlo simulation method --- Stochastic sampling --- Games of chance (Mathematics) --- Mathematical models --- Numerical analysis --- Numerical calculations --- Stochastic processes --- Mathematics --- Statistical inference --- Statistics, Mathematical --- Statistics --- Probabilities --- Sampling (Statistics) --- Stochastic approximation. Monte Carlo methods --- Probability and statistics: probabilistic algorithms (including Monte Carlo);random number generation; statistical computing; statistical software (Mathematics of computing) --- Probability. Mathematical statistics --- Statistical methods --- 519.2 Probability. Mathematical statistics --- 681.3*G3 Probability and statistics: probabilistic algorithms (including Monte Carlo);random number generation; statistical computing; statistical software (Mathematics of computing) --- 519.245 Stochastic approximation. Monte Carlo methods --- Statistique mathématique --- Monte-Carlo, Méthode de --- Statistics . --- Statistical Theory and Methods. --- Statistical analysis --- Statistical data --- Statistical science --- Econometrics --- Statistiques