# Can the adaptive Metropolis algorithm collapse without the covariance lower bound?

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 dc.contributor.author Vihola, Matti dc.date.accessioned 2012-10-26T06:23:16Z dc.date.available 2012-10-26T06:23:16Z dc.date.issued 2011 dc.identifier.citation Vihola, M. (2011). Can the adaptive Metropolis algorithm collapse without the covariance lower bound?. Electronic Journal of Probability, 16, 45-75. DOI:10.1214/EJP.v16-840. dc.identifier.issn 1083-6489 dc.identifier.uri http://hdl.handle.net/123456789/40092 dc.description.abstract The Adaptive Metropolis (AM) algorithm is based on the symmetric random-walk Metropolis algorithm. The proposal distribution has the following time-dependent covariance matrix at step $n+1$ $fi S_n = Cov(X_1,...,X_n) + \epsilon I,$ that is, the sample covariance matrix of the history of the chain plus a (small) constant $\epsilon>0$ multiple of the identity matrix $I$. The lower bound on the eigenvalues of $S_n$ induced by the factor $\epsilon I$ is theoretically convenient, but practically cumbersome, as a good value for the parameter $\epsilon$ may not always be easy to choose. This article considers variants of the AM algorithm that do not explicitly bound the eigenvalues of $S_n$ away from zero. The behaviour of $S_n$ is studied in detail, indicating that the eigenvalues of $S_n$ do not tend to collapse to zero in general. dc.language.iso eng dc.publisher Institute of Mathematical Statistics dc.relation.ispartofseries Electronic Journal of Probability dc.relation.uri http://ejp.ejpecp.org/ dc.rights This work is licensed under a Creative Commons Attribution 3.0 License. dc.rights openAccess fi dc.rights.uri http://creativecommons.org/licenses/by/3.0/ dc.subject.other Adaptive Markov chain Monte Carlo en dc.subject.other Metropolis algorithm en dc.subject.other stability en dc.subject.other stochastic approximation en dc.subject.other adaptiivinen Markov chain Monte Carlo fi dc.subject.other Metropolis-algoritmi fi dc.subject.other stabiilius fi dc.subject.other stokastinen approksimaatio fi dc.title Can the adaptive Metropolis algorithm collapse without the covariance lower bound? dc.type Article en dc.identifier.urn URN:NBN:fi:jyu-201210262789 dc.subject.kota 111 dc.contributor.laitos Matematiikan ja tilastotieteen laitos fi dc.contributor.laitos en dc.type.uri http://purl.org/eprint/type/JournalArticle dc.identifier.doi 10.1214/EJP.v16-840 dc.description.version Publisher's PDF eprint.status http://purl.org/eprint/type/status/PeerReviewed