|
Abstract:
|
The performance of RWM- and MALA-within-Gibbs algorithms for sampling from hierarchical models is studied. For the RWM-within-Gibbs, asymptotically optimal tunings for Gaussian proposal distributions featuring a diagonal covariance matrix are developed using existing scaling analyses. The principal difference with traditional optimal scaling results lies in the local character of the optimal proposal variances obtained, meaning that they vary from one iteration to the next. The concept of local proposal variances has been discussed in Girolami & Calderhead (2011) and Bédard (2015); in the latter, scaling analyses of the RWM algorithm for hierarchical target densities are performed. Although theoretically appealing, local proposal variances had to be obtained numerically in that context, which turned out to be rather impractical. With the RWM-within-Gibbs sampler, these variances may now be found analytically in a large number of cases, leading to a personalized version of the proposal variance in a given iteration. Similar ideas are applied to MALA-within-Gibbs, leading to efficient yet computationally affordable algorithms. In an attempt to quantify the benefit, in terms of efficiency, of using local proposal variances in the RWM- and MALA-within-Gibbs, we present numerical illustrations. In several cases, local versions of MALA-within-Gibbs can outperform fancy variants included in the MCMC toolbox.
|
