Adaptive Sequential Monte Carlo (SMC) for Posterior Inference and Model Selection among Complex Geological Priors Encoded through Deep Generative Neural Networks

Abstract:

In geoscientific modelling and inversion studies, conceptual model uncertainty is commonly ignored even when it is the main source of uncertainty. Bayesian model selection enables comparison and ranking of alternative conceptual subsurface models, encapsulated by spatial geologically-based prior models, according to the support provided by direct and indirect hydrogeological and geophysical data. Deep generative neural networks have been proven efficient in encoding such complex spatial priors and allow for a very strong dimensionality reduction that comes at the price of enhanced non-linearity. For instance, even state-of-the-art Markov chain Monte Carlo (MCMC) sampling methods may struggle to locate and explore the posterior latent space of such encodings. Furthermore, for Bayesian model selection the key target is not the posterior probability density function (PDF), but the normalizing constant in Bayes’ theorem. Calculating this so-called evidence, a multi-dimensional integral over the model parameter space, is typically significantly more expensive than estimating posterior PDFs. Here, we evaluate a recent adaptive sequential Monte Carlo (SMC) approach that is building on Annealed Importance Sampling (AIS); a method that directly targets the evidence through a particle approximation. Both techniques rely on importance sampling over a sequence of transitional distributions between the prior and the posterior PDF. The main differences are that the adaptive SMC method tunes the step size between neighboring distributions and performs resampling among particles when the variance of the particle weights becomes too large. Compared to classical MCMC, an advantage of both methods is that the underlying proposal distribution of the Markov steps can easily be tuned on-the-go without violating detailed balance conditions. We consider two categorical training images and associated synthetic cross-hole ground penetrating radar (GPR) tomography data. We find that the adaptive SMC method is faster and more reliable in locating the posterior PDF than state-of-the-art adaptive MCMC and is better adapted for parallelization, while showing superior evidence and posterior PDF estimates than AIS.