MALA-within-Gibbs samplers for high-dimensional distributions with sparse conditional structure

Markov chain Monte Carlo (MCMC) samplers are numerical methods for drawing samples from a given target probability distribution. We discuss one particular MCMC sampler, the MALA-within-Gibbs sampler, from the theoretical and practical perspectives. We first show that the acceptance ratio and step size of this sampler are independent of the overall problem dimension when (i) the target distribution has sparse conditional structure, and (ii) this structure is reflected in the partial updating strategy of MALA-within-Gibbs. If, in addition, the target density is blockwise log-concave, then the sampler's convergence rate is independent of dimension. From a practical perspective, we expect that MALA-within-Gibbs is useful for solving high-dimensional Bayesian inference problems where the posterior exhibits sparse conditional structure at least approximately. In this context, a partitioning of the state that correctly reflects the sparse conditional structure must be found, and we illustrate this process in two numerical examples. We also discuss trade-offs between the block size used for partial updating and computational requirements that may increase with the number of blocks

MALA-within-Gibbs samplers for high-dimensional distributions with sparse conditional structure

Markov chain Monte Carlo (MCMC) samplers are numerical methods for drawing samples from a given target probability distribution. We discuss one particular MCMC sampler, the MALA-within-Gibbs sampler, from the theoretical and practical perspectives. We first show that the acceptance ratio and step size of this sampler are independent of the overall problem dimension when (i) the target distribution has sparse conditional structure, and (ii) this structure is reflected in the partial updating strategy of MALA-within-Gibbs. If, in addition, the target density is blockwise log-concave, then the sampler's convergence rate is independent of dimension. From a practical perspective, we expect that MALA-within-Gibbs is useful for solving high-dimensional Bayesian inference problems where the posterior exhibits sparse conditional structure at least approximately. In this context, a partitioning of the state that correctly reflects the sparse conditional structure must be found, and we illustrate this process in two numerical examples. We also discuss trade-offs between the block size used for partial updating and computational requirements that may increase with the number of blocks

Localization for MCMC: sampling high-dimensional posterior distributions with local structure

We investigate how ideas from covariance localization in numerical weather prediction can be used in Markov chain Monte Carlo (MCMC) sampling of high-dimensional posterior distributions arising in Bayesian inverse problems. To localize an inverse problem is to enforce an anticipated “local” structure by (i) neglecting small off-diagonal elements of the prior precision and covariance matrices; and (ii) restricting the influence of observations to their neighborhood. For linear problems we can specify the conditions under which posterior moments of the localized problem are close to those of the original problem. We explain physical interpretations of our assumptions about local structure and discuss the notion of high dimensionality in local problems, which is different from the usual notion of high dimensionality in function space MCMC. The Gibbs sampler is a natural choice of MCMC algorithm for localized inverse problems and we demonstrate that its convergence rate is independent of dimension for localized linear problems. Nonlinear problems can also be tackled efficiently by localization and, as a simple illustration of these ideas, we present a localized Metropolis-within-Gibbs sampler. Several linear and nonlinear numerical examples illustrate localization in the context of MCMC samplers for inverse problems

A priori testing of sparse adaptive polynomial chaos expansions using an ocean general circulation model database

This work explores the implementation of an adaptive strategy to design sparse ensembles of oceanic simulations suitable for constructing polynomial chaos surrogates. We use a recently developed pseudo-spectral algorithm that is based on a direct application of the Smolyak sparse grid formula and that allows the use of arbitrary admissible sparse grids. The adaptive algorithm is tested using an existing simulation database of the oceanic response to Hurricane Ivan in the Gulf of Mexico. The a priori tests demonstrate that sparse and adaptive pseudo-spectral constructions lead to substantial savings over isotropic sparse sampling in the present setting

Inference of experimental radial impurity transport on Alcator C-Mod: Bayesian parameter estimation and model selection

© 2020 IAEA, Vienna We present a fully Bayesian approach for the inference of radial profiles of impurity transport coefficients and compare its results to neoclassical, gyrofluid and gyrokinetic modeling. Using nested sampling, the Bayesian impurity transport inference (BITE) framework can handle complex parameter spaces with multiple possible solutions, offering great advantages in interpretative power and reliability with respect to previously demonstrated methods. BITE employs a forward model based on the pySTRAHL package, built on the success of the well-known STRAHL code (Dux 2003 Fusion Sci Technol. 44 708-15), to simulate impurity transport in magnetically-confined plasmas. In this paper, we focus on calcium (Ca, Z = 20) laser blow-off injections into Alcator C-Mod plasmas. Multiple Ca atomic lines are diagnosed via high-resolution x-ray imaging crystal spectroscopy and vacuum ultra-violet measurements. We analyze a sawtoothing I-mode discharge for which neoclassical and turbulent (quasilinear and non-linear) predictions are also obtained. We find good agreement in diffusion across the entire radial extent, while turbulent convection and density profile peaking are estimated to be larger in experiment than suggested by theory. Efforts and challenges associated with the inference of experimental pedestal impurity transport are discussed

Receptor-defined subtypes of breast cancer in indigenous populations in Africa: a systematic review and meta-analysis.

BACKGROUND: Breast cancer is the most common female cancer in Africa. Receptor-defined subtypes are a major determinant of treatment options and disease outcomes but there is considerable uncertainty regarding the frequency of poor prognosis estrogen receptor (ER) negative subtypes in Africa. We systematically reviewed publications reporting on the frequency of breast cancer receptor-defined subtypes in indigenous populations in Africa. METHODS AND FINDINGS: Medline, Embase, and Global Health were searched for studies published between 1st January 1980 and 15th April 2014. Reported proportions of ER positive (ER+), progesterone receptor positive (PR+), and human epidermal growth factor receptor-2 positive (HER2+) disease were extracted and 95% CI calculated. Random effects meta-analyses were used to pool estimates. Fifty-four studies from North Africa (n=12,284 women with breast cancer) and 26 from sub-Saharan Africa (n=4,737) were eligible. There was marked between-study heterogeneity in the ER+ estimates in both regions (I2>90%), with the majority reporting proportions between 0.40 and 0.80 in North Africa and between 0.20 and 0.70 in sub-Saharan Africa. Similarly, large between-study heterogeneity was observed for PR+ and HER2+ estimates (I2>80%, in all instances). Meta-regression analyses showed that the proportion of ER+ disease was 10% (4%-17%) lower for studies based on archived tumor blocks rather than prospectively collected specimens, and 9% (2%-17%) lower for those with ≥ 40% versus those with <40% grade 3 tumors. For prospectively collected samples, the pooled proportions for ER+ and triple negative tumors were 0.59 (0.56-0.62) and 0.21 (0.17-0.25), respectively, regardless of region. Limitations of the study include the lack of standardized procedures across the various studies; the low methodological quality of many studies in terms of the representativeness of their case series and the quality of the procedures for collection, fixation, and receptor testing; and the possibility that women with breast cancer may have contributed to more than one study. CONCLUSIONS: The published data from the more appropriate prospectively measured specimens are consistent with the majority of breast cancers in Africa being ER+. As no single subtype dominates in the continent availability of receptor testing should be a priority, especially for young women with early stage disease where appropriate receptor-specific treatment modalities offer the greatest potential for reducing years of life lost. Please see later in the article for the Editors' Summary