6533b82ffe1ef96bd129469f
RESEARCH PRODUCT
Can the Adaptive Metropolis Algorithm Collapse Without the Covariance Lower Bound?
Matti Viholasubject
Statistics and ProbabilityFOS: Computer and information sciencesIdentity matrixMathematics - Statistics TheoryStatistics Theory (math.ST)Upper and lower boundsStatistics - Computation93E3593E15Combinatorics60J27Mathematics::ProbabilityLaw of large numbers65C40 60J27 93E15 93E35stochastic approximationFOS: MathematicsEigenvalues and eigenvectorsComputation (stat.CO)Metropolis algorithmMathematicsProbability (math.PR)Zero (complex analysis)CovariancestabilityUniform continuityBounded function65C40Statistics Probability and Uncertaintyadaptive Markov chain Monte CarloMathematics - Probabilitydescription
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.
| year | journal | country | edition | language |
|---|---|---|---|---|
| 2011-01-01 |