We introduce the Skipping Sampler, a novel algorithm to efficiently sample from the restriction of an arbitrary probability density to an arbitrary measurable set. Such conditional densities can arise in the study of risk and reliability and are often of complex nature, for example having multiple isolated modes and non-convex or disconnected support. The sampler can be seen as an instance of the Metropolis-Hastings algorithm with a particular proposal structure, and we establish sufficient conditions under which the Strong Law of Large Numbers and the Central Limit Theorem hold. We give theoretical and numerical evidence of improved performance relative to the Random Walk Metropolis algorithm.
Many methods for sampling and integrating over a measurable subset @math use prior knowledge of the subset. In Splitting (see for example @cite_32 and references therein) and related approaches such as Sequential Monte Carlo @cite_34 , it is necessary to define a sequence of nested subsets which decrease to @math . If satisfaction of the constraint is determined implicitly (for example by a black box' function) then it may be unclear how to construct such a sequence. In methods using Importance Sampling (see for example @cite_29 @cite_32 ) a biased sampling distribution should be constructed based on knowledge of the set @math , to sample from @math more frequently, and integrals are corrected using certain weights.
@cite_29 @cite_34 @cite_32
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Importance sampling methods can be iterated like MCMC algorithms, while being more robust against dependence and starting values. The population Monte Carlo principle consists of iterated generations of importance samples, with importance functions depending on the previously generated importance samples. The advantage over MCMC algorithms is that the scheme is unbiased at any iteration and can thus be stopped at any time, while iterations improve the performances of the importance function, thus leading to an adaptive importance sampling. We illustrate this method on a mixture example with multiscale importance functions. A second example reanalyzes the ion channel model using an importance sampling scheme based on a hidden Markov representation, and compares population Monte Carlo with a corresponding MCMC algorithm. We propose a methodology to sample sequentially from a sequence of probability distributions that are defined on a common space, each distribution being known up to a normalizing constant. These probability distributions are approximated by a cloud of weighted random samples which are propagated over time by using sequential Monte Carlo methods. This methodology allows us to derive simple algorithms to make parallel Markov chain Monte Carlo algorithms interact to perform global optimization and sequential Bayesian estimation and to compute ratios of normalizing constants. We illustrate these algorithms for various integration tasks arising in the context of Bayesian inference. Copyright 2006 Royal Statistical Society.