Markov Chain Monte Carlo in small worlds - University of Idaho.pdfVIP

Stat Comput (2006) 16: 193–202 DOI 10.1007/s11222-006-6966-6 Markov Chain Monte Carlo in small worlds Yongtao Guan · Roland Fleißner · Paul Joyce · Stephen M. Krone Received: November 2004 / Accepted: December 2005 C Springer Science + Business Media, LLC 2006 Abstract As the number of applications for Markov Chain improve the performance of MCMC it preserves the simplic- Monte Carlo (MCMC) grows, the power of these methods ity of the underlying algorithm. as well as their shortcomings become more apparent. While MCMC yields an almost automatic way to sample a space Keywords Markov Chain Monte Carlo · according to some distribution, its implementations often fall Metropolis-Hastings algorithm · Proposal distributions · short of this task as they may lead to chains which converge Small-world networks · Importance sampling too slowly or get trapped within one mode of a multi-modal space. Moreover, it may be difficult to determine if a chain is 1. Introduction only sampling a certain area of the space or if it has indeed reached stationarity. Markov Chain Monte Carlo (Gamerman, 1997) is a sampling In this paper, we show how a simple modification of scheme for surveying a space S with a prescribed probability the proposal mechanism results in faster convergence of the measure π . It has particular importance in Bayesian analy- chain and helps to circumvent the problems described above. sis, where x ∈ S represents a vector of parameters and π (x ) This mechanism, which is based on an idea from the field is the posterior distribution of the parameters conditional on of “small-world” networks, amounts to adding occasional the data. MCMC can as well be used to solve the so-called “wild” proposals to any local proposal s