2009/11/03 by Matti Vihola
Mathematics · #math.PR #math.ST #stat.CO #stat.TH #msc:65C40 #msc:60J27 #msc:93E15 #msc:93E35
published as Electronic Journal of Probability, Vol 16, pp. 45-75, 2011 · 31 pages, 1 figure
arxiv created 2009/11/03 · arxiv updated 2011/02/09
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 Sn = Cov(X1,...,Xn) + εI, that is, the sample covariance matrix of the history of the chain plus a (small) constant ε>0 multiple of the identity matrix I. The lower bound on the eigenvalues of Sn induced by the factor εI is theoretically convenient, but practically cumbersome, as a good value for the parameter ε may not always be easy to choose. This article considers variants of the AM algorithm that do not explicitly bound the eigenvalues of Sn away from zero. The behaviour of Sn is studied in detail, indicating that the eigenvalues of Sn do not tend to collapse to zero in general.