Complexity Analysis of Accelerated MCMC Methods for Bayesian Inversion

We study Bayesian inversion for a model elliptic PDE with unknown diffusion coefficient. We provide complexity analyses of several Markov Chain-Monte Carlo (MCMC) methods for the efficient numerical evaluation of expectations under the Bayesian posterior distribution, given data $\delta$. Particular attention is given to bounds on the overall work required to achieve a prescribed error level $\varepsilon$. Specifically, we first bound the computational complexity of "plain" MCMC, based on combining MCMC sampling with linear complexity multilevel solvers for elliptic PDE. Our (new) work versus accuracy bounds show that the complexity of this approach can be quite prohibitive. Two strategies for reducing the computational complexity are then proposed and analyzed: first, a sparse, parametric and deterministic generalized polynomial chaos (gpc) "surrogate" representation of the forward response map of the PDE over the entire parameter space, and, second, a novel Multi-Level Markov Chain Monte Carlo (MLMCMC) strategy which utilizes sampling from a multilevel discretization of the posterior and of the forward PDE. For both of these strategies we derive asymptotic bounds on work versus accuracy, and hence asymptotic bounds on the computational complexity of the algorithms. In particular we provide sufficient conditions on the regularity of the unknown coefficients of the PDE, and on the approximation methods used, in order for the accelerations of MCMC resulting from these strategies to lead to complexity reductions over "plain" MCMC algorithms for Bayesian inversion of PDEs.

Sparse Deterministic Approximation of Bayesian Inverse Problems

We present a parametric deterministic formulation of Bayesian inverse problems with input parameter from infinite dimensional, separable Banach spaces. In this formulation, the forward problems are parametric, deterministic elliptic partial differential equations, and the inverse problem is to determine the unknown, parametric deterministic coefficients from noisy observations comprising linear functionals of the solution. We prove a generalized polynomial chaos representation of the posterior density with respect to the prior measure, given noisy observational data. We analyze the sparsity of the posterior density in terms of the summability of the input data's coefficient sequence. To this end, we estimate the fluctuations in the prior. We exhibit sufficient conditions on the prior model in order for approximations of the posterior density to converge at a given algebraic rate, in terms of the number $N$ of unknowns appearing in the parameteric representation of the prior measure. Similar sparsity and approximation results are also exhibited for the solution and covariance of the elliptic partial differential equation under the posterior. These results then form the basis for efficient uncertainty quantification, in the presence of data with noise

A double-blind placebo-controlled trial of paroxetine in the management of social phobia (social anxiety disorder) in South Africa

CITATION: Stein, D. J. et al. 1999. A double-blind placebo-controlled trial of paroxetine in the management of social phobia (social anxiety disorder) in South Africa. South African Medical Journal, 89(4):402-403.The original publication is available at http://www.samj.org.zaBackground. Social phobia, also known as social anxiety disorder, is a highly prevalent disorder with significant morbidity. Patients with social phobia frequently develop co-morbid psychiatric disorders such as depression and substance abuse, and the disorder impacts significantly on social and occupational functioning. It has been suggested that the selective serotonin reuptake inhibitors (SSRIs) are useful in the management of this disorder, but few controlled trials have been undertaken in this regard. There are also few data on the pharmacotherapy of social phobia in South Africa. Methods. A double-blind randomised placebo-controlled multi-site flexible-dose trial of paroxetine was undertaken over 12 weeks among patients with a primary diagnosis of social phobia. Primary response measures were the Global Improvement item on the Clinical Global Impression scale (CGI) and mean change from baseline in the patient-rated Liebowitz Social Anxiety Scale (LSAS) total score. Ninety-three patients participated at 9 South African sites; their data are reported here. Results. There was a significant drug effect on both the CGI Global Improvement score and the LSAS at 12 weeks. In addition, there was no significant difference in overall rate of adverse experiences between those on paroxetine and those on placebo. Conclusions. Paroxetine is both effective and safe in the acute treatment of social phobia. The findings here are consistent with those of previous controlled studies of the SSRIs as well as with previous work done in the USA on the use of paroxetine in the treatment of this disorder. Early diagnosis and treatment of social phobia should be encouraged. However, further research on long-term pharmacotherapy of social phobia is needed.Publisher’s versio