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This chapter delves into the theory and application of reversible Markov Chain Monte Carlo (MCMC) algorithms, focusing on their role in Bayesian inference. It begins with the Metropolis–Hastings algorithm and explores variations such as component-wise updates, and the Metropolis-Adjusted Langevin Algorithm (MALA). The chapter also discusses Hamiltonian Monte Carlo (HMC) and the importance of scaling MCMC methods for high-dimensional models or large datasets. Key challenges in applying reversible MCMC to large-scale problems are addressed, with a focus on computational efficiency and algorithmic adjustments to improve scalability.
This chapter provides a comprehensive overview of the foundational concepts essential for scalable Bayesian learning and Monte Carlo methods. It introduces Monte Carlo integration and its relevance to Bayesian statistics, focusing on techniques such as importance sampling and control variates. The chapter outlines key applications, including logistic regression, Bayesian matrix factorization, and Bayesian neural networks, which serve as illustrative examples throughout the book. It also offers a primer on Markov chains and stochastic differential equations, which are critical for understanding the advanced methods discussed in later chapters. Additionally, the chapter introduces kernel methods in preparation for their application in scalable Markov Chain Monte Carlo (MCMC) diagnostics.
This chapter focuses on continuous-time MCMC algorithms, particularly those based on piecewise deterministic Markov processes (PDMPs). It introduces PDMPs as a scalable alternative to traditional MCMC, with a detailed explanation of their simulation, invariant distribution, and limiting processes. Various continuous-time samplers, including the bouncy particle sampler and zig-zag process, are compared in terms of efficiency and performance. The chapter also addresses practical aspects of simulating PDMPs, including techniques for exploiting model sparsity and data subsampling. Extensions to these methods, such as handling discontinuous target distributions or distributions defined on spaces of different dimensions, are discussed.
The development of more sophisticated and, especially, approximate sampling algorithms aimed at improving scalability in one or more of the senses already discussed in this book raises important considerations about how a suitable algorithm should be selected for a given task, how its tuning parameters should be determined, and how its convergence should be as- sessed. This chapter presents recent solutions to the above problems, whose starting point is to derive explicit upper bounds on an appropriate distance between the posterior and the approximation produced by MCMC. Further, we explain how these same tools can be adapted to provide powerful post-processing methods that can be used retrospectively to improve approximations produced using scalable MCMC.
This chapter explores the benefits of non-reversible MCMC algorithms in improving sampling efficiency. Revisiting Hamiltonian Monte Carlo (HMC), the chapter discusses the advantages of breaking detailed balance and introduces lifting schemes as a tool to enhance exploration of the parameter space. It reviews non-reversible HMC and alternative algorithms like Gustafson’s method. The chapter also covers techniques like delayed rejection and the discrete bouncy particle sampler, offering a comparison between reversible and non-reversible methods. Theoretical insights and practical implementations are provided to highlight the efficiency gains from non-reversibility.
This chapter introduces stochastic gradient MCMC (SG-MCMC) algorithms, designed to scale Bayesian inference to large datasets. Beginning with the unadjusted Langevin algorithm (ULA), it extends to more sophisticated methods such as stochastic gradient Langevin dynamics (SGLD). The chapter emphasises controlling the stochasticity in gradient estimators and explores the role of control variates in reducing variance. Convergence properties of SG-MCMC methods are analysed, with experiments demonstrating their performance in logistic regression and Bayesian neural networks. It concludes by outlining a general framework for SG-MCMC and offering practical guidance for efficient, scalable Bayesian learning.
A graduate-level introduction to advanced topics in Markov chain Monte Carlo (MCMC), as applied broadly in the Bayesian computational context. The topics covered have emerged as recently as the last decade and include stochastic gradient MCMC, non-reversible MCMC, continuous time MCMC, and new techniques for convergence assessment. A particular focus is on cutting-edge methods that are scalable with respect to either the amount of data, or the data dimension, motivated by the emerging high-priority application areas in machine learning and AI. Examples are woven throughout the text to demonstrate how scalable Bayesian learning methods can be implemented. This text could form the basis for a course and is sure to be an invaluable resource for researchers in the field.
This study aimed to explore the association between hyperglycemia in pregnancy (type 2 diabetes (T2D) and gestational diabetes mellitus (GDM)) and child developmental risk in Europid and Aboriginal women.
PANDORA is a longitudinal birth cohort recruited from a hyperglycemia in pregnancy register, and from normoglycemic women in antenatal clinics. The Wave 1 substudy included 308 children who completed developmental and behavioral screening between age 18 and 60 months. Developmental risk was assessed using the Ages and Stages Questionnaire (ASQ) or equivalent modified ASQ for use with Aboriginal children. Emotional and behavioral risk was assessed using the Strengths and Difficulties Questionnaire. Multivariable logistic regression was used to assess the association between developmental scores and explanatory variables, including maternal T2D in pregnancy or GDM.
After adjustment for ethnicity, maternal and child variables, and socioeconomic measures, maternal hyperglycemia was associated with increased developmental “concern” (defined as score ≥1 SD below mean) in the fine motor (T2D odds ratio (OR) 5.30, 95% CI 1.77–15.80; GDM OR 3.96, 95% CI 1.55–10.11) and problem-solving (T2D OR 2.71, 95% CI 1.05–6.98; GDM OR 2.54, 95% CI 1.17–5.54) domains, as well as increased “risk” (score ≥2 SD below mean) in at least one domain (T2D OR 5.33, 95% CI 1.85–15.39; GDM OR 4.86, 95% CI 1.95–12.10). Higher maternal education was associated with reduced concern in the problem-solving domain (OR 0.27, 95% CI 0.11–0.69) after adjustment for maternal hyperglycemia.
Maternal hyperglycemia is associated with increased developmental concern and may be a potential target for intervention so as to optimize developmental trajectories.
Immediate facial nerve reconstruction is the standard of care following radical parotidectomy; however, quality of life comparisons with those undergoing limited superficial parotidectomy without facial nerve sacrifice is lacking.
Method
Patients who underwent parotidectomy were contacted to determine quality of life using the University of Washington Quality of Life and Parotidectomy Specific Quality of Life questionnaires. A total of 29 patients (15 in the radical parotidectomy and 14 in the limited superficial parotidectomy groups) completed and returned questionnaires.
Results
Using the University of Washington Quality of Life Questionnaire, similar quality of life was noted in both groups, with the radical parotidectomy group having significantly worse speech and taste scores. Using the Parotidectomy Specific Quality of Life Questionnaire, the radical parotidectomy group reported significantly worse speech, eye symptoms and eating issues.
Conclusion
Those undergoing radical parotidectomy with reconstruction had comparable overall quality of life with the limited superficial parotidectomy group. The Parotidectomy Specific Quality of Life Questionnaire better identified subtle quality of life complaints. Eye and oral symptoms remain problematic, necessitating better rehabilitation and more focused reconstructive efforts.
Identifying predictors of patient outcomes evaluated over time may require modeling interactions among variables while addressing within-subject correlation. Generalized linear mixed models (GLMMs) and generalized estimating equations (GEEs) address within-subject correlation, but identifying interactions can be difficult if not hypothesized a priori. We evaluate the performance of several variable selection approaches for clustered binary outcomes to provide guidance for choosing between the methods.
Methods:
We conducted simulations comparing stepwise selection, penalized GLMM, boosted GLMM, and boosted GEE for variable selection considering main effects and two-way interactions in data with repeatedly measured binary outcomes and evaluate a two-stage approach to reduce bias and error in parameter estimates. We compared these approaches in real data applications: hypothermia during surgery and treatment response in lupus nephritis.
Results:
Penalized and boosted approaches recovered correct predictors and interactions more frequently than stepwise selection. Penalized GLMM recovered correct predictors more often than boosting, but included many spurious predictors. Boosted GLMM yielded parsimonious models and identified correct predictors well at large sample and effect sizes, but required excessive computation time. Boosted GEE was computationally efficient and selected relatively parsimonious models, offering a compromise between computation and parsimony. The two-stage approach reduced the bias and error in regression parameters in all approaches.
Conclusion:
Penalized and boosted approaches are effective for variable selection in data with clustered binary outcomes. The two-stage approach reduces bias and error and should be applied regardless of method. We provide guidance for choosing the most appropriate method in real applications.
Metals have been central to the development of human civilization from the Bronze Age to the present although historically, mining and smelting have been the cause of local environmental pollution with the potential to harm human health. Despite problems from artisanal mining in some developing countries, modern mining for Western standards now uses the best available mining technology combined with environmental monitoring, mitigation and remediation measures to control emissions to the environment. The relocation and removal of large quantities of mineral and waste could also release chemicals into the environment including surface water, ground water and soil during the mining lifecycle. There are only few published methods available for prioritizing hazardous chemicals. These fail to recognize differences between organic and inorganic chemicals make it necessary to develop separate screening and prioritization procedures for those two different classes of chemicals. In this study, we focus on the development of screening and prioritization procedure in risk assessment for inorganic chemicals with particular reference to those used, generated and released in mining operations.
Earthquakes associated with gas production have been recorded in the northern part of the Netherlands since 1986. The Huizinge earthquake of 16 August 2012, the strongest so far with a magnitude of ML = 3.6, prompted reassessment of the seismicity induced by production from the Groningen gas field. An international research programme was initiated, with the participation of many Dutch and international universities, knowledge institutes and recognised experts.
The prime aim of the programme was to assess the hazard and risk resulting from the induced seismicity. Classic probabilistic seismic hazard and risk assessment (PSHA) was implemented using a Monte Carlo method. The scope of the research programme extended from the cause (production of gas from the underground reservoir) to the effects (risk to people and damage to buildings). Data acquisition through field measurements and laboratory experiments was a substantial element of the research programme. The existing geophone and accelerometer monitoring network was extended, a new network of accelerometers in building foundations was installed, geophones were placed at reservoir level in deep wells, GPS stations were installed and a gravity survey was conducted.
Results of the probabilistic seismic hazard and risk assessment have been published in production plans submitted to the Minister of Economic Affairs, Winningsplan Groningen 2013 and 2016 and several intermediate updates. The studies and data acquisition further constrained the uncertainties and resulted in a reduction of the initially assessed hazard and risk.
This paper reviews the evolution of a sequence of seismological models developed and implemented as part of a workflow for Probabilistic Seismic Hazard and Risk Assessment of the seismicity induced by gas production from the Groningen gas field. These are semi-empirical statistical geomechanical models derived from observations of production-induced seismicity, reservoir compaction and structure of the field itself. Initial versions of the seismological model were based on a characterisation of the seismicity in terms of its moment budget. Subsequent versions of the model were formulated in terms of seismic event rates, this change being driven in part by the reduction in variability of the model forecasts in this domain. Our approach makes use of the Epidemic Type After Shock model (ETAS) to characterise spatial and temporal clustering of earthquakes and has been extended to also incorporate the concentration of moment release on pre-existing faults and other reservoir topographic structures.
This study examined the response of forage crops to composted dairy waste (compost) applied at low rates and investigated effects on soil health. The evenness of spreading compost by commercial machinery was also assessed. An experiment was established on a commercial dairy farm with target rates of compost up to 5 t ha−1 applied to a field containing millet [Echinochloa esculenta (A. Braun) H. Scholz] and Pasja leafy turnip (Brassica hybrid). A pot experiment was also conducted to monitor the response of a legume forage crop (vetch; Vicia sativa L.) on three soils with equivalent rates of compost up to 20 t ha−1 with and without ‘additive blends’ comprising gypsum, lime or other soil treatments. Few significant increases in forage biomass were observed with the application of low rates of compost in either the field or pot experiment. In the field experiment, compost had little impact on crop herbage mineral composition, soil chemical attributes or soil fungal and bacterial biomass. However, small but significant increases were observed in gravimetric water content resulting in up to 22.4 mm of additional plant available water calculated in the surface 0.45 m of soil, 2 years after compost was applied in the field at 6 t ha−1 dried (7.2 t ha−1 undried), compared with the nil control. In the pot experiment, where the soil was homogenized and compost incorporated into the soil prior to sowing, there were significant differences in mineral composition in herbage and in soil. A response in biomass yield to compost was only observed on the sandier and lower fertility soil type, and yields only exceeded that of the conventional fertilizer treatment where rates equivalent to 20 t ha−1 were applied. With few yield responses observed, the justification for applying low rates of compost to forage crops and pastures seems uncertain. Our collective experience from the field and the glasshouse suggests that farmers might increase the response to compost by: (i) increasing compost application rates; (ii) applying it prior to sowing a crop; (iii) incorporating the compost into the soil; (iv) applying only to responsive soil types; (v) growing only responsive crops; and (vi) reducing weed burdens in crops following application. Commercial machinery incorporating a centrifugal twin disc mechanism was shown to deliver double the quantity of compost in the area immediately behind the spreader compared with the edges of the spreading swathe. Spatial variability in the delivery of compost could be reduced but not eliminated by increased overlapping, but this might represent a potential 20% increase in spreading costs.