# Can the Adaptive Metropolis Algorithm Collapse Without the Covariance Lower Bound - Mathematics > Probability

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Abstract: The Adaptive Metropolis AM algorithm is based on the symmetric random-walkMetropolis algorithm. The proposal distribution has the followingtime-dependent covariance matrix at step $n+1$ \S n = CovX 1,

.,X n + \epsilon I, \ that is, the sample covariance matrixof the history of the chain plus a small constant $\epsilon>0$ multiple ofthe identity matrix $I$. The lower bound on the eigenvalues of $S n$ induced bythe factor $\epsilon I$ is theoretically convenient, but practicallycumbersome, as a good value for the parameter $\epsilon$ may not always be easyto choose. This article considers variants of the AM algorithm that do notexplicitly 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 nottend to collapse to zero in general.

Autor: Matti Vihola

Fuente: https://arxiv.org/