Hierarchical models: Local proposal variances for RWM-within-Gibbs and MALA-within-Gibbs

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. This leads to locally optimal proposal variances that depend on the mixing components of the hierarchical model and that correspond to the classical asymptotically optimal acceptance rate of 0.234. Ignoring the local character of the optimal scaling is possible, leading to an optimal proposal variance that remains fixed for the duration of the algorithm; the corresponding asymptotically optimal acceptance rate is then shown to be lower than 0.234. Similar ideas are applied to MALA-within-Gibbs samplers, leading to efficient yet computationally affordable algorithms. Simplifications for location and scale hierarchies are presented, and findings are illustrated through numerical studies. The local and fixed approaches for the RWM- and MALA-within-Gibbs are compared to competitive samplers in the literature.