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.
Adaptive MCMC methods, in which parameters governing the evolution of the Markov chain can be learned during sampling @cite_24 @cite_45 , are an active current field of research addressing multimodality @cite_20 @cite_2 @cite_28 . Although adaptive algorithms can be more problematic to analyse, nevertheless adaptation could be added to the Skipping Sampler, for example in the underlying RWM proposal and the number of draws, and we reserve this for future work. Conversely, as an MH sampler, the Skipping Sampler can be plugged into existing sampling methods as appropriate, for instance the Gibbs Sampler @cite_18 @cite_9 and the Sequential Monte Carlo or Slice Samplers. An application to Slice Sampling is explored in Section .
@cite_18 @cite_28 @cite_9 @cite_24 @cite_45 @cite_2 @cite_20
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Abstract This article describes a new Metropolis-like transition rule, the multiple-try Metropolis, for Markov chain Monte Carlo (MCMC) simulations. By using this transition rule together with adaptive direction sampling, we propose a novel method for incorporating local optimization steps into a MCMC sampler in continuous state-space. Numerical studies show that the new method performs significantly better than the traditional Metropolis-Hastings (M-H) sampler. With minor tailoring in using the rule, the multiple-try method can also be exploited to achieve the effect of a griddy Gibbs sampler without having to bear with griddy approximations, and the effect of a hit-and-run algorithm without having to figure out the required conditional distribution in a random direction. One of the main shortcomings of Markov chain Monte Carlo samplers is their inability to mix between modes of the target distribution. In this paper we show that advance knowledge of the location of these modes can be incorporated into the MCMC sampler by introducing mode-hopping moves that satisfy detailed balance. The proposed sampling algorithm explores local mode structure through local MCMC moves (e.g. diffusion or Hybrid Monte Carlo) but in addition also represents the relative strengths of the different modes correctly using a set of global moves. This “mode-hopping” MCMC sampler can be viewed as a generalization of the darting method [1]. Recently, Stochastic Gradient Markov Chain Monte Carlo (SG-MCMC) methods have been proposed for scaling up Monte Carlo computations to large data problems. Whilst these approaches have proven useful in many applications, vanilla SG-MCMC might suffer from poor mixing rates when random variables exhibit strong couplings under the target densities or big scale differences. In this study, we propose a novel SG-MCMC method that takes the local geometry into account by using ideas from Quasi-Newton optimization methods. These second order methods directly approximate the inverse Hessian by using a limited history of samples and their gradients. Our method uses dense approximations of the inverse Hessian while keeping the time and memory complexities linear with the dimension of the problem. We provide a formal theoretical analysis where we show that the proposed method is asymptotically unbiased and consistent with the posterior expectations. We illustrate the effectiveness of the approach on both synthetic and real datasets. Our experiments on two challenging applications show that our method achieves fast convergence rates similar to Riemannian approaches while at the same time having low computational requirements similar to diagonal preconditioning approaches. We propose Adaptive Incremental Mixture Markov chain Monte Carlo (AIMM), a novel approach to sample from challenging probability distributions defined on a general state-space. While adaptive MCMC methods usually update a parametric proposal kernel with a global rule, AIMM locally adapts a semiparametric kernel. AIMM is based on an independent Metropolis-Hastings proposal distribution which takes the form of a finite mixture of Gaussian distributions. Central to this approach is the idea that the proposal distribution adapts to the target by locally adding a mixture component when the discrepancy between the proposal mixture and the target is deemed to be too large. As a result, the number of components in the mixture proposal is not fixed in advance. Theoretically, we prove that there exists a process that can be made arbitrarily close to AIMM and that converges to the correct target distribution. We also illustrate that it performs well in practice in a variety of challenging situations, including high-dimensional and multimodal target distributions.