A Pathological MCMC Algorithm and Its Use as a Benchmark for Convergence Assessment Techniques PDF Download
Are you looking for read ebook online? Search for your book and save it on your Kindle device, PC, phones or tablets. Download A Pathological MCMC Algorithm and Its Use as a Benchmark for Convergence Assessment Techniques PDF full book. Access full book title A Pathological MCMC Algorithm and Its Use as a Benchmark for Convergence Assessment Techniques by Christian P. Robert. Download full books in PDF and EPUB format.
Author: Christian P. Robert
Publisher:
ISBN:
Category :
Languages : en
Pages : 0
Get Book Here
Book Description
We present in this paper a particular Metropolis-type algorithm for the simulation of a beta (formula) variable whose convergence is extremely slow. The interest of this phenomenon is to provide a simple benchmark against which convergence control techniques can be tested. We illustrate this use for state-of-the-art common control methods, backing up our evaluation by additional illustrations for a more standard algorithm derived from the same principle.
Author: Christian P. Robert
Publisher:
ISBN:
Category :
Languages : en
Pages : 0
Get Book Here
Book Description
We present in this paper a particular Metropolis-type algorithm for the simulation of a beta (formula) variable whose convergence is extremely slow. The interest of this phenomenon is to provide a simple benchmark against which convergence control techniques can be tested. We illustrate this use for state-of-the-art common control methods, backing up our evaluation by additional illustrations for a more standard algorithm derived from the same principle.
Author: Christian P. Robert
Publisher: Springer Science & Business Media
ISBN: 1461217164
Category : Mathematics
Languages : en
Pages : 201
Get Book Here
Book Description
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.
Author: James E. Gentle
Publisher: Springer Science & Business Media
ISBN: 0387216111
Category : Computers
Languages : en
Pages : 427
Get Book Here
Book Description
Will provide a more elementary introduction to these topics than other books available; Gentle is the author of two other Springer books
Author: James E. Gentle
Publisher: Springer Science & Business Media
ISBN: 0387216103
Category : Computers
Languages : en
Pages : 387
Get Book Here
Book Description
Monte Carlo simulation has become one of the most important tools in all fields of science. Simulation methodology relies on a good source of numbers that appear to be random. These "pseudorandom" numbers must pass statistical tests just as random samples would. Methods for producing pseudorandom numbers and transforming those numbers to simulate samples from various distributions are among the most important topics in statistical computing. This book surveys techniques of random number generation and the use of random numbers in Monte Carlo simulation. The book covers basic principles, as well as newer methods such as parallel random number generation, nonlinear congruential generators, quasi Monte Carlo methods, and Markov chain Monte Carlo. The best methods for generating random variates from the standard distributions are presented, but also general techniques useful in more complicated models and in novel settings are described. The emphasis throughout the book is on practical methods that work well in current computing environments. The book includes exercises and can be used as a test or supplementary text for various courses in modern statistics. It could serve as the primary test for a specialized course in statistical computing, or as a supplementary text for a course in computational statistics and other areas of modern statistics that rely on simulation. The book, which covers recent developments in the field, could also serve as a useful reference for practitioners. Although some familiarity with probability and statistics is assumed, the book is accessible to a broad audience. The second edition is approximately 50% longer than the first edition. It includes advances in methods for parallel random number generation, universal methods for generation of nonuniform variates, perfect sampling, and software for random number generation.
Author: James E. Gentle
Publisher: Springer Science & Business Media
ISBN: 0387981446
Category : Mathematics
Languages : en
Pages : 732
Get Book Here
Book Description
Computational inference is based on an approach to statistical methods that uses modern computational power to simulate distributional properties of estimators and test statistics. This book describes computationally intensive statistical methods in a unified presentation, emphasizing techniques, such as the PDF decomposition, that arise in a wide range of methods.
Author: Didier Chauveau
Publisher:
ISBN:
Category :
Languages : en
Pages : 0
Get Book Here
Book Description
In this paper, we propose a methodology essentially based on the Central Limit Theorem for Markov chains to monitor convergence of MCMC algorithms using actual outputs. Our methods are grounded on the fact that normality is a testable implication of sufficient mixing. The first control tool tests the normality hypothesis for normalized averages of functions of the Markov chain over independent parallel chains started from a dispersed distribution. A second connected tool is based on graphical monitoring of the stabilization of the variance after n iterations near the limiting variance. Both methods work without knowledge on the sampler driving the chain, and the normality diagnostic leads to automated stopping rules. These stopping rules are implemented in a software toolbox whose performances are illustrated through simulations for finite and continuous state chains reflecting some typical situations and a full scale application. Comparisons are made with the binary control method of Raftery and Lewis.
Author:
Publisher:
ISBN:
Category : Mathematical statistics
Languages : en
Pages : 798
Get Book Here
Book Description
The Current Index to Statistics (CIS) is a bibliographic index of publications in statistics, probability, and related fields.
Author: Roshan Shariff
Publisher:
ISBN:
Category : Algorithms
Languages : en
Pages : 7
Get Book Here
Book Description
Sampling from a given probability distribution is a key problem in many different disciplines. Markov chain Monte Carlo (MCMC) algorithms approach this problem by constructing a random walk governed by a specially constructed transition probability distribution. As the random walk progresses, the distribution of its states converges to the required target distribution. The Metropolis-Hastings (MH) algorithm is a generally applicable MCMC method which, given a proposal distribution, modifies it by adding an accept/reject step: it proposes a new state based on the proposal distribution and the existing state of the random walk, then either accepts or rejects it with a certain probability; if it is rejected, the old state is retained. The MH algorithm is most effective when the proposal distribution closely matches the target distribution: otherwise most proposals will be rejected and convergence to the target distribution will be slow. The proposal distribution should therefore be designed to take advantage of any known structure in the target distribution. A particular kind of structure that arises in some probabilistic inference problems is that of symmetry: the problem (or its sub-problems) remains unchanged under certain transformations. A simple kind of symmetry is the choice of a coordinate system in a geometric problem; translating and rotating the origin of a plane does not affect the relative positions of any points on it. The field of group theory has a rich and fertile history of being used to characterize such symmetries; in particular, topological group theory has been applied to the study of both continuous and discrete symmetries. Symmetries are described by a group that acts on the state space of a problem, transforming it in such a way that the problem remains unchanged. We consider problems in which the target distribution has factors, each of which has a symmetry group; each factor's value does not change when the space is transformed by an element of its corresponding symmetry group. This thesis proposes a variation of the MH algorithm where each step first chooses a random transformation of the state space and then applies it to the current state; these transformations are elements of suitable symmetry groups. The main result of this thesis extends the acceptance probability formula of the textbook MH algorithm to this case under mild conditions, adding much-needed flexibility to the MH algorithm. The new algorithm is also demonstrated in the Simultaneous Localization and Mapping (SLAM) problem in robotics, in which a robot traverses an unknown environment, and its trajectory and a map of the environment must be recovered from sensor observations and known control signals. Here the group moves are chosen to exploit the SLAM problem's natural geometric symmetries, obtaining the first fully rigorous justification of a previous MCMC-based SLAM method. New experimental results comparing this method to existing state-of-the-art specialized methods on a standard range-only SLAM benchmark problem validate the strength of the approach.
Author:
Publisher:
ISBN:
Category : Economics
Languages : en
Pages : 940
Get Book Here
Book Description
Author:
Publisher:
ISBN:
Category : Classification
Languages : en
Pages : 940
Get Book Here
Book Description
