| Título: | Can the Adaptive Metropolis Algorithm Collapse Without the Covariance Lower Bound? |
| Autores: | Vihola, Matti; University of Jyväskylä |
| Fecha: | 2011-01-01 |
| Publicador: | Electronic journal of probability |
| Fuente: |
 |
| Tipo: |
Peer-reviewed Article |
| Tema: |
adaptive Markov chain Monte Carlo; Metropolis algorithm; stability; stochastic approximation 65C40; 60J27; 93E15; 93E35 |
| Descripción: | 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 $n+1$, $S_n=\mathrm{Cov}(X_1,\ldots,X_n)+\varepsilon I$, that is, the sample covariance matrix of the history of the chain plus a (small) constant $\varepsilon>0$ multiple of the identity matrix $I$ . The lower bound on the eigenvalues of $S_n$ induced by the factor $\varepsilon I$ is theoretically convenient, but practically cumbersome, as a good value for the parameter $\varepsilon$ 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. In dimension one, it is shown that $S_n$ is bounded away from zero if the logarithmic target density is uniformly continuous. For a modification of the AM algorithm including an additional fixed component in the proposal distribution, the eigenvalues of $S_n$ are shown to stay away from zero with a practically non-restrictive condition. This result implies a strong law of large numbers for super-exponentially decaying target distributions with regular contours. |
| Idioma: | No aplica |
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