Abstract. This chapter provides a survey of the main methods in non-uniform random variate generation, and highlights recent research on the subject. Classical paradigms such as inversion, rejection, guide tables, and transformations are reviewed. We provide information on the expected time complexity of various algorithms, before addressing modern topics such as indirectly specified distributions, random processes, and Markov chain methods.

We give a new method for generating perfectly random samples from the stationary distribution of a Markov chain. The method is related to coupling from the past (CFTP), but only runs the Markov chain forwards in time, and never restarts it at previous times in the past. The method is also related to an idea known as PASTA (Poisson arrivals see time averages) in the operations research literature. Because the new algorithm can be run using a read-once stream of randomness, we call it read-once CFTP. The memory and time requirements of read-once CFTP are on par with the requirements of the usual form of CFTP, and for a variety of applications the requirements may be noticeably less. Some perfect sampling algorithms for point processes are based on an extension of CFTP known as coupling into and from the past; for completeness, we give a read-once version of coupling into and from the past, but it remains unpractical. For these point process applications, we give an alternative...

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We introduce a new sampling algorithm, the equi-energy sampler, for efficient statistical sampling and estimation. Complementary to the widely used temperature-domain methods, the equi-energy sampler, utilizing the temperature–energy duality, targets the energy directly. The focus on the energy function not only facilitates efficient sampling, but also provides a powerful means for statistical estimation, for example, the calculation of the density of states and microcanonical averages in statistical mechanics. The equi-energy sampler is applied to a variety of problems, including exponential regression in statistics, motif sampling in computational biology and protein folding in biophysics.

We study the slice sampler, a method of constructing a reversible Markov chain with a specified invariant distribution. Given an independence Metropolis-Hastings algorithm it is always possible to construct a slice sampler that dominates it in the Peskun sense. This means that the resulting Markov chain produces estimates with a smaller asymptotic variance. Furthermore the slice sampler has a smaller second-largest eigenvalue than the corresponding independence MetropolisHastings algorithm. This ensures faster convergence to the distribution of interest. A sufficient condition for uniform ergodicity of the slice sampler is given and an upper bound for the rate of convergence to stationarity is provided. Keywords: Auxiliary variables, Slice sampler, Peskun ordering, Metropolis-Hastings algorithm, Uniform ergodicity. 1 Introduction The slice sampler is a method of constructing a reversible Markov transition kernel with a given invariant distribution. Auxiliary variables ar...

: Perfect simulation algorithms based on Propp and Wilson (1996) have so far been of limited use for sampling problems of interest in statistics. We specify a new family of perfect sampling algorithms obtained by combining MCMC tempering algorithms with dominated coupling from the past, and demonstrate that our algorithms will be useful for sample based inference. Perfect tempering algorithms are less efficient than the MCMC algorithms on which they typically depend. However, samples returned by perfect tempering are distributed according to the intended distribution, so that these new sampling algorithms do not suffer from the convergence problems of MCMC. Perfect tempering is related to rejection sampling. When rejection sampling has been tried, but has proved impractical, it may be possible to convert the rejection algorithm into a perfect tempering algorithm, with a significant gain in algorithm efficiency. Keywords: Bayesian inference; Dominated coupling from the past; Exact samp...

In 1996, Propp and Wilson introduced Coupling from the Past (CFTP), an algorithm for generating a sample from the exact stationary distribution of a Markov chain. In 1998, Fill proposed another so{ called perfect sampling algorithm. These algorithms have enormous potential in Markov Chain Monte Carlo (MCMC) problems because they eliminate the need to monitor convergence and mixing of the chain. This article provides a brief introduction to the algorithms, with an emphasis on understanding rather than technical detail. 1 Setting A Markov chain is a sequence of random variables fX t g that can be thought of as evolving over time, and where the distribution of X t+1 depends on X t , but not on X t 1 ; X t 2 ; : : : . When used in Markov chain Monte Carlo (MCMC) algorithms, Markov chains are usually constructed from a Markov transition kernel K, a conditional probability density on X such that X t+1 jX t K(X t ; X t+1 ). Interest is usually in the stationary distribution of the chain, ...

Abstract. The emergence in the past years of Bayesian analysis in many methodological and applied fields as the solution to the modeling of complex problems cannot be dissociated from major changes in its computational implementation. We show in this review how the advances in Bayesian analysis and statistical computation are intermingled. Key words and phrases: Monte Carlo methods, importance sampling, Markov chain Monte Carlo (MCMC) algorithms.

This paper investigates a particular type of slice sampler algorithm, the polar slice sampler. This algorithm is shown to have convergence properties which are essentially independent of the dimension of the problem, at least for log-concave densities. For such densities, the algorithm provably converges (from appropriate starting point) to within 0:01 of stationarity in total variation distance in a number of iterations given as a computable function of the spherical asymmetry of the density. In particular, for spherically symmetric log-concave densities, in arbitrary dimension, with appropriate starting point, we prove that the algorithm converges in at most 525 iterations. Simulations are done which confirm the polar slice sampler's excellent performance. 1. Introduction.

Markov Chain Monte Carlo has long become a very useful, established tool in statistical physics and spatial statistics. Recent years have seen the development of a new and exciting generation of Markov Chain Monte Carlo methods: perfect simulation algorithms. In contrast to conventional Markov Chain Monte Carlo, perfect simulation produces samples which are guaranteed to have the exact equilibrium distribution. In the following we provide an example-based introduction into perfect simulation focussed on the method called Coupling From The Past.