Can the Adaptive Metropolis Algorithm Collapse Without the Covariance Lower Bound?

Details Can the Adaptive Metropolis Algorithm Collapse Without the Covariance Lower Bound? Can the Adaptive Metropolis Algorithm Collapse Without the Covariance Lower Bound?

2009-11-03

arxiv.org/abs/0911.0522v1

Electronic Journal of Probability, Vol 16, pp. 45-75, 2011

Mathematics

Probability

31 pages, 1 figure

Scientific paper

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 $n+1$ \[ S_n = Cov(X_1,...,X_n) + \epsilon I, \] 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.

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