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

Plane wave discontinuous Galerkin methods: exponential convergence of the hp-version

We consider the two-dimensional Helmholtz equation with constant coefficients on a domain with piecewise analytic boundary, modelling the scattering of acoustic waves at a sound-soft obstacle. Our discretisation relies on the Trefftz-discontinuous Galerkin approach with plane wave basis functions on meshes with very general element shapes, geometrically graded towards domain corners. We prove exponential convergence of the discrete solution in terms of number of unknowns

Closing the Implementation Gap: Bringing Clean Air to the Region

This report identifies 25 clean air measures that can positively impact human health, crop yields, climate change and socio-economic development, as well as contribute to achieving the Sustainable Development Goals. Implementing these measures could help 1 billion people breathe cleaner air by 2030 and reduce global warming by a third of a degree Celsius by 2050

Breaking the barriers between intelligence, investigation and evaluation: A continuous approach to define the contribution and scope of forensic science.

Forensic science has been evolving towards a separation of more and more specialised tasks, with forensic practitioners increasingly identifying themselves with only one sub-discipline or task of forensic science. Such divisions are viewed as a threat to the advancement of science because they tend to polarise researchers and tear apart scientific communities. The objective of this article is to highlight that a piece of information is not either intelligence or evidence, and that a forensic scientist is not either an investigator or an evaluator, but that these notions must all be applied in conjunction to successfully understand a criminal problem or solve a case. To capture the scope, strength and contribution of forensic science, this paper proposes a progressive but non-linear continuous model that could serve as a guide for forensic reasoning and processes. In this approach, hypothetico-deductive reasoning, iterative thinking and the notion of entropy are used to frame the continuum, situate forensic scientists' operating contexts and decision points. Situations and examples drawn from experience and practice are used to illustrate the approach. The authors argue that forensic science, as a discipline, should not be defined according to the context it serves (i.e. an investigation, a court decision or an intelligence process), but as a general, scientific and holistic trace-focused practice that contributes to a broad range of goals in various contexts. Since forensic science does not work in isolation, the approach also provides a useful basis as to how forensic scientists should contribute to collective and collaborative problem-solving to improve justice and security

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

A review of likelihood ratios in forensic science based on a critique of Stiffelman "No longer the Gold standard: Probabilistic genotyping is changing the nature of DNA evidence in criminal trials".

Stiffelman [1] gives a broad critique of the application of likelihood ratios (LRs) in forensic science, in particular their use in probabilistic genotyping (PG) software. These are discussed in this review. LRs do not infringe on the ultimate issue. The Bayesian paradigm clearly separates the role of the scientist from that of the decision makers and distances the scientist from comment on the ultimate and subsidiary issues. LRs do not affect the reasonable doubt standard. Fact finders must still make decisions based on all the evidence and they must do this considering all evidence, not just that given probabilistically. LRs do not infringe on the presumption of innocence. The presumption of innocence does not equate with a prior probability of zero but simply that the person of interest (POI) is no more likely than anyone else to be the donor. Propositions need to be exhaustive within the context of the case. That is, propositions deemed relevant by either defense or prosecution which are not fanciful must not be omitted from consideration