On the robustness of optimal scaling for random walk Metropolis algorithms.
Published: 2006
Language: eng
Number of pages: 198
ISBN-13: 9780494220412
Description: In this thesis, we study the optimal scaling problem for sampling from a target distribution of interest using a random walk Metropolis (RWM) algorithm. In order to implement this method, the selection of a proposal distribution is required, which is assumed to be a multivariate normal distribution with independent components. We investigate how the proposal scaling (i.e. the variance of the normal distribution) should be selected for best performance of the algorithm.Specifically, we prove that the sequence of stochastic processes formed by say the i*-th component of each Markov chain usually converges (as d &rarr; infinity) to a Langevin diffusion process whose speed measure varies according to target scaling vector. The demonstrations involve an appropriate resealing of space and time, as well as   L1  convergence of generators.We illustrate with a number of examples how practitioners can take advantage of these results.We first propose a method to determine the optimal form for the proposal scaling as a function of d. This assists us in producing a necessary and sufficient condition for the algorithm to adopt the same limiting behavior as with iid targets, resulting in an asymptotically optimal acceptance rate (AOAR) of 0.234. We show that when this condition is violated the limiting process of the algorithm is altered, yielding AOARs which might drastically differ from the usual 0.234. We also demonstrate that inhomogeneous proposal distributions are sometimes essential to obtain of a nontrivial limit. Finally, we illustrate how our results can be applied to the special case of correlated targets whose distribution is jointly normal.The d-dimensional target distribution we consider is formed of independent components, each of which has its own scaling term   q-2j  (d) (j = 1,..., d). This constitutes an extension of the d-dimensional  iid target considered by Roberts, Gelman & Gilks (1997) who showed that for large d, the acceptance rate should be tuned to 0.234 for optimal performance of the algorithm. In a similar fashion, we show that for the aforementioned framework, the relative efficiency of the algorithm can be characterized by its overall acceptance rate.
Notes: Source: Dissertation Abstracts International, Volume: 68-01, Section: B, page: 0369.

Thesis (Ph.D.)--University of Toronto, 2006.

Electronic version licensed for access by U. of T. users.