Multiplicative random walk Metropolis-Hastings on the real line

In this article we propose multiplication based random walk Metropolis Hastings (MH) algorithm on the real line. We call it the random dive MH (RDMH) algorithm. This algorithm, even if simple to apply, was not studied earlier in Markov chain Monte Carlo literature. The associated kernel is shown to have standard properties like irreducibility, aperiodicity and Harris recurrence under some mild assumptions. These ensure basic convergence (ergodicity) of the kernel. Further the kernel is shown to be geometric ergodic for a large class of target densities on $\mathbb{R}$. This class even contains realistic target densities for which random walk or Langevin MH are not geometrically ergodic. Three simulation studies are given to demonstrate the mixing property and superiority of RDMH to standard MH algorithms on real line. A share-price return data is also analyzed and the results are compared with those available in the literature.

💡 Research Summary

The paper introduces a novel Metropolis‑Hastings (MH) algorithm for sampling from probability densities defined on the real line, called the Random Dive Metropolis‑Hastings (RDMH) algorithm. Unlike the traditional random‑walk Metropolis‑Hastings (RWMH) that proposes additive moves, RDMH generates proposals by multiplicatively “diving” from the current state. Formally, given the current state (x\in\mathbb{R}), a proposal (y) is drawn as (y = x\cdot U), where (U) is a strictly positive random variable defined on ((0,\infty)). The distribution of (U) is chosen to be symmetric on the log‑scale, i.e. the density satisfies (f_U(u)=f_U(1/u)/u^{2}). This symmetry guarantees that the proposal kernel is reversible after the log‑transformation and that the acceptance probability has a simple closed form:

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