A comparison of centring parameterisations of Gaussian process-based models for Bayesian computation using MCMC

Abstract

Markov chain Monte Carlo (MCMC) algorithms for Bayesian computation for Gaussian process-based models under default parameterisations are slow to converge due to the presence of spatial-and other-induced dependence structures. The main focus of this paper is to study the effect of the assumed spatial correlation structure on the convergence properties of the Gibbs sampler under the default non-centred parameterisation and a rival centred parameterisation (CP), for the mean structure of a general multi-process Gaussian spatial model. Our investigation finds answers to many pertinent, but as yet unanswered, questions on the choice between the two. Assuming the covariance parameters to be known, we compare the exact rates of convergence of the two by varying the strength of the spatial correlation, the level of covariance tapering, the scale of the spatially varying covariates, the number of data points, the number and the structure of block updating of the spatial effects and the amount of smoothness assumed in a Matérn covariance function. We also study the effects of introducing differing levels of geometric anisotropy in the spatial model. The case of unknown variance parameters is investigated using wellknown MCMC convergence diagnostics. A simulation study and a real-data example on modelling air pollution levels in London are used for illustrations. A generic pattern emerges that the CP is preferable in the presence of more spatial correlation or more information obtained through, for example, additional data points or by increased covariate variability.

Multilevel Linear Models, Gibbs Samplers and Multigrid Decompositions (with Discussion)

“…In particular, we derive analytic expressions for the convergence rates of the Gibbs Sampler for various multilevel linear models and explore the dependence of these rates on the model structure, the choice of parametrization and the introduction of identifiability constraints. The theoretical results given in this paper extend and improve substantially on existing literature (Roberts and Sahu, 1997;Yu and Meng, 2011;Bass and Sahu, 2016a;Gao and Owen, 2017) both in terms of generality of hierarchical structure and the availability of explicit rates. We also show by simulations that the understanding gained from the Gaussian case can be extrapolated to more general settings.…”

“…Another important class of models that would be worth approaching with methodologies analogous to the ones developed here are models based on Gaussian processes commonly used, for example, in spatial statistics (see e.g. Bass and Sahu, 2016b).…”

Auxiliary Gradient-Based Sampling Algorithms

“…This problem is challenging because of the high dependence between these hyperparameters and the state vector x . To deal with this, several approaches have been presented in the literature (Knorr‐Held and Rue, ; Papaspiliopoulos et al ., ; Murray and Adams, ; Filippone et al ., ; Hensman et al ., ; Bass and Sahu, ). Next, we discuss a new sampling move that is tailored to the auxiliary sampler based on the latent variable z .…”

“…We also need to infer 20 kernel hyperparameters, i.e. a lengthscale 2 k and a variance σ 2 xk for each class. All sampling algorithms were applied so that the full latent field is sampled in a single step, i.e.…”

Efficient inference of generalized spatial fusion models with flexible specification

“…This implies faster convergence of the chains; hence, fewer numbers of iteration are required for model inference. In nonfusion Gaussian process models, the strength of spatial correlation among other parameters has an effect on the efficiency of different parameterizations (Bass & Sahu, ; Papaspiliopoulos, Roberts, & Sköld, ). However, it is beyond the scope of this study to investigate those effects in the spatial fusion model setting.…”